Wolf Prize in Mathematics, vol.2

1. Raoul Bott, 2. Raoul Bott their holomorphic 3. S. S. Chern,

REFERENCES Vector fields and characteristic numbers, Michigan Math. J. 14 (1967), 231-244. and S. S. Chern, Hermitian vector bundles and the equidistribulion of the zeroes of sections, Acta Math. 114 (1965), 71-112. The geometry ofG-structures, Bull. Amer. Math. Soc. 72 (1966), 167-219.

HARVARD UNIVERSITY

BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 78, Number 6, November 1972

ON CHARACTERISTIC CLASSES OF T-FOLIATIONS BY R. BOTT 1 AND A. HAEFLIGER Communicated by Michael Altiyah, April 13, 1972

1. Introduction. In this note we describe and apply a general procedure for constructing characteristic classes of T-foliations. Our work was inspired by that of Godbillon-Vey [4] and Gelfand-Fuks [5] on the one hand, and that of Chern-Simons [3] on the other. We know of several essentially equivalent procedures—some more in the line of [3] (see for example [2])—but the one we present here fits naturally and well nigh unavoidably into the Gelfand-Fuks theory and we have been told that Malgrange and Gelfand have also, quite independently, come upon it. We will take the following point of view toward characteristic classes and foliations. Consider a pseudogroup T whose elements are diffeomorphisms of open sets in R". A T-foliation on a smooth (that is C°°) manifold M is by definition a maximal family F of submersions of open sets U in Af, such that for every x e U n V there exists an element yyv e T with fv — yvv °fv in some vicinity of x. Given two T-foliations, F on M and F on M\ a morphism from F to F' is by definition a smooth map f:M

->M'

such that for every "local projection" f^e F' the composition

f6°f:f~lV^R" is in F. With this concept of morphism the T-foliations form a category C(T) and we define a characteristic class of r-foliations with coefficients in a group R, as a natural transformation a : C ( r ) - / / * ( ;/?). Thus a(F) e H*(M; R) and iff: F' -* F is a morphism, then AMS 1970 subject classifications. Primary 53A10, 53C35; Secondary 53A05, 28A75. 1 Supported in part by National Science Foundation grant GP-7952X3, while at the Institute for Advanced Study, Princeton, New Jersey. Copyright £) American Mathematical Society 1972

1040

R. BOTT AND A. HAEKLIGliR

(Nowmbei

a(/-'F)=/*x(F). Alternatively, one could define this notion in terms of the classifying space Br—which we assume to be realized as a CW-complex [6]. PROPOSITION There is a natural bijection of the R-valued characteristic classes on V-foliations and the cohomology group H*{BT\ R).

2. Relation with the cohomology of formal vector-fields. Suppose now that T is a transitive Lie-pseudogroup acting on R" and let a (I~) denote "the Lie algebra of formal T vector-fields" associated to T. Here a vectorfield defined on U c R" is called a T vector-field if the local one parameter group which it engenders is in T, and a {T) is defined as the inverse limit a ( 0 = lima'(J") of the k jets at 0 of T vector-fields. In the pseudogroup T let T0 be the set of elements of T keeping 0 fixed and set I~J equal to the k jets of elements in T0. Then the T% form an inverse system of Lie groups and we can find a subgroup K c lim r£, whose projection on every I~S is a maximal compact subgroup for k > 0. This follows from the fact that the kernel of the projection r £ + ' -+ r j is a vector space for k > 0. The subgroup K is unique up to conjugation, and its Lie algebra k can be identified with a subalgebra ofa(r). For our purposes we now need the cohomology of basic elements rel K in a(r), H(a{r); K), which is defined in the following manner: Let y4{a'(r)} denote the algebra of multilinear alternate forms on a*(r), and let /J{a(D} be the direct limit of the A{ak (r)}. The bracket in a(r) there induces a differential on ,4{a(r)} in the usual way, and we write H{a(r)} for the resulting cohomology group. The relative group H*(a(f); K) is now defined as the cohomology of the subcomplex of /4{a(r)} consisting of elements which are invariant under the natural action of K, and annihilated by all inner products with elements of k. THEOREM

1. Let F be a Y-foliation on M. There is an algebra homo-

morphism (p:H{a(D;K}-+

H(M;R)

which is a natural transformation on the category C(T). Note that H(a(T); K) can be thought of as the continuous cohomology of the topological category T. To construct

1972)

ON CHARACTERISTIC CLASSES OF T-FOLIATIONS

1041

of r . It is a principal T^-bundle. On the other hand, T acts transitively on the left on Pk(r). Denote by 4(P°°(r)) the direct limit of the algebras A(Pk(T)) of differential forms on P*(r). The invariant forms with respect to the action of T constitute a differential subalgebra denoted by Ar. PROPOSITION.

Ar is naturally isomorphic to A{a(T)).

Now let F be a foliation on M. Let Pk(F) be the differentiate bundle over M whose fiber over x e M is the space of k jets at x of local projections fv e F such that fv{x) — 0. This is a To-principal bundle. Its restriction to U is isomorphic to the inverse image by fv of the bundle Pk(r): If g* is the k jet at 0 of an element g of T and g(0) e fv(U), the isomorphism maps (x, gk) on the k jet at x of g" lfv. Hence it is readily seen that the differential algebra of T-invariant forms on Pk(r) is mapped in the algebra A(Pk(F)) of differential forms on Pk(F). Hence if we denote by /t(P°°(F)) the direct limit of the A{Pk(F)), we get an injective homomorphism q> of A(a(r)) in A(P™(F)) commuting with the differential. This homomorphism is compatible with the action of K, hence induces a homomorphism on the subalgebra of /C-basic elements. But the algebra /l(Plc(F);K) of K-basic elements in A(Pk(F)) is isomorphic to the algebra of differential forms on P\F)/K which is a bundle over M with contractible fiber TD/K. Hence H(A(Pk(F); K)) is isomorphic, via the de Rham theorem, to H(M;R). The homomorphism q> is therefore obtained as the composition //(o(T); K) - H(A(P~{F); K)) = H(M; R). REMARK. In the same spirit, one can for instance also consider the category of differentiable fiber bundles E, with fiber a compact comogeneous space B = G/K, with structural group G, and a foliation transverse to the fibers. Also let vB be the topological Lie algebra of tangent vector fields on B (cf. [5]). Then, there is a natural homomorphism

H(vB; K) - H{E) which is naturally related to the preceding one. 3. The basic examples. By extending the computation of Gelfand-Fuks [5] the group H(a; K) can be computed in certain cases, and we will describe the results in several instances of topological interest; namely when T consists of the holomorphic, orientation preserving, and arbitrary diffeomorphisms of open sets in R". We denote these T's by T„C, T„+ and T„ respectively.

1042

R. BOTT AND A. HAEKLIGER

[November

In the following E(ul,u2,...) denotes an exterior algebra in the indicated us and R[cu... ,cj denotes the polynomial ring over R in the variables c , , . . . , cq, with degree c, = 2i. We also write # [ c , ,...,<:„] for the ring /?[<:,,..., c„]/(elements of degree > 2q\ and denote by WO„ the differential complex: WOn =

E(u1,u3,...,u2k+1)®R[cl,...,cJ,

dcj = 0; dut = c, for odd i; 2k + 1 largest odd integer
while

H*(a(T„);K) = H(WOn), H*(a(r„+); K) = H(WO„),

n odd, 2

= HiWOJlxl/iX

- c„),

neven.

REMARKS. (1) Every n-manifold of course carries a natural T„ structure, the one whose projections contain all the coordinate charts of M. Under

Wn = £ ( M , , M 2 , . . . , U B ) ® y ? [ c , , . . . , c j ,

du-, = c,.

Then H{a(r„)} = H(Wn). This group occurs naturally in our context when one seeks characteristic classes for T-foliations with trivialized normal bundles. We turn next to the complex case. Here one has THEOREM

3. For V„C the unitary group plays the role of K, and H*(r„C;K)®C^

where WU„ is the complex E[ul,..., ® C with duk = ck — Z\.

H(WUJ u„] ® R[c\,...,

c„] ® R[cx

r„]

Here, of course,

. It seems to be very difficult, especially in

19721

ON CHARACTERISTIC CLASSES OF T-FOL1ATIONS

1043

the real case, to find foliations which distinguish between the different classes defined by

0„).

Now suppose G is semisimple. Then according tq the deep theory of BorelHarish-Chandra, G contains a group D acting discretely on G/K and such that D\G/K is compact. From this fact and Poincare duality, it follows that H*(a; K) injects into H*(D\G/K) and this in tum easily leads to the conclusion: PROPOSITION.

/ / G is semisimple, then (p:H(g;K)-+H(BrGIH)

is injective. Using this principle, one obtains a weak independence result for the classes ulc", c* = c\' • • • C aj + 2a 2 + • • • + na„ = n in H2n+1(WOn\ For this purpose define a minimal variety V to be the quotient of a simple semisimple complex Lie group by a maximal parabolic subgroup. (Examples are the complex Grassmannians, the quadrics and, in general, each simple group G has no more than / minimal varieties associated to it where / = rank G.) These varieties are compact so that if V has complex dimension n, then c"(V) is well-defined. The q> image of the classes /Xjc" in H2n+ '(WOJ are as independent as the Chem numbers c"(V) of the minimal varieties V o/dim c V = n. INDEPENDENCE THEOREM.

COROLLARY.

In degree (2n + 1) the image of

These two classes correspond to the complex projective space and the

1044

R. BOTT AND A. HAEFLIGER

quadric, which exist in every dimension. Unfortunately, they are also the only ones when n is a prime. However, these classes in turn imply that many higher dimensional ones are also nontrivial. In the complex case the corresponding classes are much easier to detect, and there they, in fact, take on a continuum of values. Furthermore, they play a universal role for the infinitesimal Lefschetz theorem (see [2]). Precisely, one has the following: THEOREM

4. Let Fx be the restructure

defined on Cn + 1 — 0, by the

vector-field dz0

dz2

dzn

Then one has the formula f

JS2""

(/< , 0 [ F J = 3c*W/c"(X)

where 3 denotes imaginary part and the c,(A) are taken to be the ith elementary symmetric functions of the system k = {A 0 ,..., A„}.

BIBLIOGRAPHY 1. R. Bott, On a topological obstruction to integrability, Proc. Internat. Congress Math. (Nice, 1970), vol. 1, Gauthier-Villars, Paris, 1971, pp. 27-36. 2. , The Lefschetz formula and exotic characteristic classes, Proc. Differential Geometry Conference, Rome, May 1971. 3. S. Chern and J. Simons, Some cohomology classes in principal fiberbundles and their applications to Riemannian geometry, Proc. Nat. Acad. Sci. U.S.A. 68 (1971), 791-794. 4. Godbillon-Vey, Un invariant des feuilletages de codimension 1, C. R. Acad. Sci. Paris 273 (1971), 92. 5. I. M. GelTand and D. B. Fuks, Cohomologies of the Lie algebra of tangent vector fields of a smooth manifold. 1, II, Funkcional. Anal, i Prilozen. 3 (1969), no. 3, 32-52; ibid. 4 (1970), 23-32. (Russian) MR 41 # 1067. 6. A. Haefliger, Feuilletages sur les varietes ouvertes, Topology 9 (1970), 183-194. MR 41 # 7709. DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY, CAMBRIDGE, MASSACHUSETTS

02138 Current address (A. Haefliger): Universite de Geneve, Geneve, Switzerland

AN EQUIVARIANT SETTING OF THE MORSE THEORY

l

by Raoul BOTT 2

This being a "Feiertag" in the true sense of the word, and mine being the last lecture—as well as being so far out of line of the topic of this symposium—you will permit me a few words of a general nature; an explanation so to speak, why I have been invited to these festivities. I expect we all find periods in our life when it seems that Providence is at the helm and all that is asked of us is to give it free reign. To me the year 1949-50 is such a period and it is as clearly etched in my mind now as it was ten, -twenty, -thirty! years ago. For then, quite providentially at the last possible moment I was asked to the Institute for Advanced Study at Princeton a brash brand new Ph.D from Carnegie Tech, and I had the whole miraculous world of pure mathematics burst upon me. At that time Princeton was inhabited by giants. One saw Einstein and Godell stroll arm in arm—I recall a lecture by Dirac with Einstein, Pauli, von Neumann, Hermann Weyl and Oppenheimer in the audience—and it was my great good fortune that on this exciting trip providence had also provided me with a wonderful guide, a very dear friend, and at the same time a quite exacting tutor. That young man is our birthday-child of today. I of course have a store of anecdotes from that time, and maybe I can divulge some of them later tonight. Here let me mention only one. 1 bombarded Ernst with so many stupid questions that in desperation he finally imposed a fine of 25 cents (!) on any conjecture he could disprove in less than five minutes. This should give you some idea of the inflation of the past thirty years and also help to explain Ernst's vast fortune at this time. Of course I had many other teachers there: Reidemeister, Steenrod— Ernst and I attended his course on Fiber-bundles—and also friends, many of whom I am delighted to see here today: de Rham, Erdos, Beno Eckmann, 1 Presented at the Symposium iiber Logik und Algorithmic in honour of Ernst SPECKER, Zurich, February 1980. 2 This work was supported in part through funds provided by the National Science Foundation under grant 33-966-7566-2.

— 272 —

Chandrasekharan. How wonderful to see you all—and how nice it would have been if already in those days we had known that this occasion awaited us just thirty years hence. But it is time to show you that Ernst did teach me some topology there in Princeton. I apologize to have nothing to tell you which would be more apropos to this conference. (There is some irony in that, for in 1949 networks, graphs and such like were my interests. It nearly seems that in our interaction in Princeton, Ernst and I interchanged our momentum.) But before I launch into a subject which I fell in love with, also in that year of 1949, let me close with one serious word. Thank you Ernst for having been a friend and teacher to me, and I expect to most of us here, in the truest sense: for you aways teach as a friend, and what you teach is always more than the subject. Now to work. I would like to show you a formula, which seems to me the appropriate generalization of the Morse theory to the equivariant situation. Recall then first of all that the "Morse theory" attaches to every "generic" smooth function / , on a compact manifold, M, a polynomial •^t ( / ) = X ?Ap» where the summation is over the critical points o f / a n d p

if/? is such a critical point, i.e.

then lp denotes the number of negative eigenvalues of the matrix Hpf

~ WdiP \p '

The germ "generic" is here meant in the sense that all these second order matrices be nonsingular. This immediately implies that the critical points of/are finite in number and so Jtt ( / ) is indeed a well defined polynomial. Now the fundamental first result of the Morse theory is that this polynomial has as a lower bound the Poincare Polynomial of M: P,(M)

= Itk

dim

Hk(M)

where Hk (M) denotes the fc-th cohomology group of M (relative to some field). I will not try to define P, (M) in greater detail here—suffice it to say that it is our oldest and most trustworthy topological invariant of a space, and that its coefficients are called the Betti-numbers of M.

— 273 — Precisely, then the Morse theorem can be stated as follows: In the situation envisaged there exists a polynomial Q (t) with nonnegative coefficients such that (*)

Jt,U)-Pt{M)

=(1+0Q(0-

Note that this certainly implies that Mt (f) is > P, (Af) coefficient by coefficient. In fact it is a quite considerable refinement of that inequality. For instance, it immediately implies the following Lacunary principle of Morse: If the product of any two consecutive coefficients of Jt', ( / ) vanish, then Jtt{f)

=P,{M),

and M is "free of torsion". For example the 2-coordinate is a function on the 2-sphere: x2 + y2 + z2 = 1, with a minimum at z = 0 and a maximum at z — 1. Hence Jl, (z) = 1 + t2 and so P, (S2) = 1 + t2 also. In short, as a zeroeth approximation to Pt (M), one may use J(t ( / ) for an economical/, i.e. one having as few critical points as M will allow. Put differently, we may think of the Morse theory as quantitatively predicting the minimal critical behavior forced by the topology of the situation. The problem I want to address here now, is, how other constraints on a function force critical points, and in particular I would like to measure the effect which an assumed symmetry o f / h a s on its critical behavior. Two examples will serve to set the stage: Example 1. L e t / b e a smooth real valued function on the line R, which is invariant under the group of integers, Z, acting on R by translation: / ( x + 1) = / ( x ) . What then is the minimal critical behavior o f / ? Answer. Clearly such an / descends to a function / z : R/Z -> R on the circle obtained by identifying the points of R under the group-action, and as such it must satisfy the Morse inequalities of a function o n S 1 .

— 274 —

Example 2. Let the circle S1 act on 5 2 e R 3 by rotation about the 2-axis in 3-space. What is the appropriate critical behavior forced on a function/invariant under this action? If we attempt to treat this case as before by descending to the function / s , : S2/Sx -* R induced by/, something goes wrong, because S 2/S1 is clearly just an interval - 1 < z < 1, and hence its Poincare polynomial is just 1. Thus the Morse inequalities would only force one critical point which is patently absurd. Now the main difference between these two examples is that in the first one the action of the group was—what we call "free" and in the second it was not. By the way, an action of a group G on a space X is "free" if (roughly) the G-orbit of each point x is homeomorphic to G, and these orbits behave uniformly under the variation of x. This is clearly the case in the first example, and fails near the North and South poles of 5 2 in the second. In our context it is now easily seen that indeed, if G acts freely on M, a n d / i s G invariant then the Morse inequalities of the induced function fG on M/G express the critical behavior that is forced o n / b y the symmetry G. Note that in this case M/G will also be a manifold in its own right! However when the action is not free, M/G will in general not be a manifold at all and its Poincare series seems to have little to do with the additional critical behavior forced o n / b y the G-symmetry. Now in homotopy theory this phenomenon is well-known and the homotopy theorists long ago devised a method of dealing with it. The point is that they defined a homotopy quotient for any action of a group G on a space X—notated XG—which, when the action is free reduces to X/G in the homotopy category, and in other cases produces a new space out of the action. This space XG reflects the singularities of the action in a—at times—surprising manner. The plan of the homotopy theorists is as follows: First note that if G acts freely on a space W, then the diagonal action of G on X x w will be free whatever the action of G on X was. Secondly, recall that contractible spaces do not affect homotopy in any way. In line with these two principles the homotopy quotient XG is defined as follows: We first select a fixed space W on which G acts satisfying two conditions: a) G acts freely on W b) W is contractible.

— 275 — Then define Xc by: Xc = X x WIG . Note by the way that according to this procedure the homotopy quotient of the "worst possible of actions" i.e. the action of G on a point/? is given by pG =

p

x w/G = W/G .

This space—which we call the classifying space of G, and denote by BG is absolutely fundamental in modern homotopy theory. The reason is, that it turns out that a space W satisfying a, b si essentially uniquely determined by G (in homotopy theory) so that finally BG is a space which depends on G alone and somehow is an amalgam of its own topology and its abstract group structure. The following is a table of BG's for some garden variety of groups.

TABLE OF CLASSIFYING SPACES

G

W

Z

W/G

P,(W/G) 1

R

R/Z = S

R"

l

R"/Z" = S x ... x S'

(1+/)"

S(H)

RP^

1

s(H)

CP,

(i-?2r'

(7(2)

2-framesin//

G2 (//)

(1 - f 2 ) " 1 (1 - f 4 ) " 1

U(n)

n frames in //

G„(//)

Z" Z2

(7(1) = s

1

1+ t

(1 - r 2 ) - ' ... (1 -t2")'1

.

Here Z denotes the integers, Z" the direct product of Z with itself n times, Z 2 the group { + 1 } , and U (n) of course the unitary group. The first two Ws of course come to mind immediately, on the other hand the rest should strike you as way out. In all of these think of H as a complex infinite-dimensional Hilbert space, and of S (//) as its unit sphere. The space of /i-frames on H is then the space of n-tuples { xv ..., xn} of elements in S (H) which are mutually orthogonal. U (n) clearly acts on these

and the quotient gives precisely the Grassmannian G„ (H) of all «-dimensional subspaces of H. For n = 1, this is simply the projective space.

— 276 — AH these examples then rely on the beautiful fact that the unit sphere in an infinite dimensional Hilbert Space is contractible! Note also that most of these spaces BG have Poincare Series in accordance with the fact that they are infinite-dimensional manifolds. We are now finally ready for the equivariant version of the Morse inequalities. Assume then that M and G are compact, that G acts smoothly on M and t h a t / i s a G-invariant smooth function on M. The critical points of/then naturally fall into orbits {0} of the group action and we now also assume t h a t / i s nondegenerate in its category—that is—that the Hessian o f / i s nondegenerate in the directions normal to the critical orbits. It then follows t h a t / h a s only a finite number of critical orbits {0 ; }" = 1 . Finally let {/>,} " = i be a set of points one in each orbit Ot, and let Hi be the stability group of />;. Thus O^G/Hi. i = l,...,n. With all this understood, define the equivariant Morse Series o f / b y : J??(f) = Itx^P,(BHp) where p ranges over the {/>,}. Also define the equivariant Poincare series of M by: P?(M)

= Itk dim

Hk(MG).

(This is then simply the Poincare series of the homotopy quotient.) The theorem—due to Atiyah and myself—is now: For simplicity assume that all the stability groups are connected and both G and M are compact. Then the equivariant Morse Series of f and Poincare Series of M satisfy the Morse inequalities: (**)

J??(f)-P?(M)

=(1+06(0

where Q (t) = q0 + qx t + ..., with q{ > 0. The proof is really not very difficult, indeed / descends to a function fG on MG in the obvious manner and then one simply has to apply the theory of nondegenerate "critical manifolds" to see that ^,{fo) =^?(fh

Q-E.D.

By the way when one allows for disconnected stability groups, the formula persists with appropriate local coefficient systems. As concrete instance of (**), consider the function / ( x , y,z) = z

— 277 —

on our two sphere S2 of Example 2. It is clearly invariant under Sl; Further its critical points are just the North and South poles.—Both are fixed under the action of 5 ' , thus H = Si and hence 1

t2

•*?(/) = , -I2 + v 1 - r 1- r The right-hand side of (**) is also quite computable and one finds that c

i

Pt (S )

=

l

+ '2

i

so

^ a t the height function z, is again a perfect Morse

function in the equivariant sense. Although I am pleased with (**) per se—as I am with all nice formulas— let me just say one word on how M. Atiyah and I chanced upon it, and how we put it to use. Of late the physicists have been much concerned with the Yang-Mills functional, and its classical extrema. These are then solutions of certain nonlinear differential equations, and it occurred to us that the Morse theory should enable us to get at some information about the topology of the space of these solutions. This of course leads to an infinite dimensional analogue of the situation described—but it turns out that in the end it is the formula (**) which leads to the correct result. For instance applied to the solution space of the Yang-Mills problem— let's call it Min,—for a U (3) bundle over a Riemann Surface of genus g, (**) gives us the following formula for the Poincare polynomial of Min: *) jP,(Min) = (r5 + l)2g(t3 + l)2g - (t2 + \)2t*g~2(l + t)2g(l + P)2* + (1 + f 2 + t 4 ) r 6 f l - 2 ( l + 0 4 9

I display it proudly to show you that although we are often accused (especially by Ernst) of dealing in chimerial concepts, they do at times solidify into very concrete information.—More concrete than any I have seen here—for that matter. This formula is also noteworthy in that it was not new\ It was derived by a completely different method—using counting procedures in finite fields going back to C. L. Siegel, and the "WeilConjectures". The original impetus for that derivation was done by Harder, [2] who did the U (2) case. The formula (***) occurs in a paper by Ramanan and Usha.

— 278 — In closing let me bring this formula into the context of our seminar by offering it as a provisional proof of the consistancy of mathematics; to tide us over until you, dear Ernst, or some of your friends devise a more convincing one. Harvard, March 1980 I append the following bibliography in case someone is interested in following some aspect of the subject. A paper, scheduled for the London Math. Journal, is under preparation by Atiyah and myself.

BIBLIOGRAPHY 1 BOTT R. and H. SAMELSON. Applications of the theory of Morse to symmetric spaces. Amer. J. of Math. 80 (1968) 964-1029. 2 HARDER G. Eine Bemerkung Zu einer Arbeit von P. E. Newstead. Jour, fur Math. 242(1910) 16-25. 3 HARDER G. and M. S. NARASIMHAN. On the cohomology groups of moduli spaces of vector bundles over curves. Math. Ann. 212 (1975) 215-248. 4 MUMFORD D. and P. E. NEWSTEAD. Periods of a moduli space of bundles on curves. Amer. J. Math. 90 (1968) 1201-1208. 5 NARASIMHAN M. S. and S. RAMANAN. Moduli of vector bundles on a compact Riemann surface. Ann. of Math. 89 (1969) 19-51. 6 NARASIMHAN M. S. and S. RAMANAN. Vector bundles on curves. Proceedings of the Bombay Colloquium of Algebraic Geometry 335-346 Oxford University Press, 1969. 7 NARASIMHAN M. S. and C. S. SESHADRI. Stable and Unitary vector bundles on a compact Riemann surface. Ann. of Math. 82 (1965) 540-567. 8 NEWSTEAD P. E. Topological properties of some spaces of stable bundles. Topology 6 (1967) 241-262. 9 Stable bundles of rank 2 and odd degree over a curve of genus 2. Topology 7 (1968) 205-215. 10 Characteristic classes of stable bundles of rank 2 over an algebraic curve. Trans. Amer. Math. Soc. 169 (1972) 337-345. 11 Rationality of moduli spaces of stable bundles. (To appear). 12 SESHADRI C. S. Space of unitary vector bundles on a compact Riemann surface. Ann. of Math. 85 (1967) 303-336. 13 Moduli of 7T-vector bundles over an algebraic curve. Questions on Algebraic Varieties Roma 1970. (Recule Raoul H. Bott Department of Mathematics Harvard University Cambridge Mass. 02138 USA

15 juillet 1980)

Wolf Prize in Mathematics, Vol. 2 (pp. 61-117) eds. S. S. Chem and F. Hirzebruch © 2001 World Scientific Publishing Co.

Curriculum Vitae

Born in Mendoza. Argentina, on September 14, 1920 Civil Engineering Degree, University of Buenos Aires, Argentina, 1947 Doctor of Philosophy in Mathematics, University of Chicago, USA, 1950 Died in Chicago on April 16, 1998 Honors Academies — Member of the American Academy of Arts and Sciences, Boston, Massachusetts, 1958 — Correspondent Member (1959), and Member (1984), of the National Academy of Exact, Physical and Natural Sciences, Buenos Aires, Argentina — Member of the National Academy of Sciences of the United States, Washington D.C., 1968 — Correspondent Member of the Royal Academy of Sciences, Madrid, Spain, 1970 — Member of the Latin American Academy of Sciences, 1983 — Foreign Associate of the Institut de France, Paris, France, 1984 — Member of the Third World Academy of Sciences, Trieste, Italy, 1984 Prizes — Latin American Prize in Mathematics, awarded by IPCLAR (Instituto para la Promocion de las Ciencias, Letras, Artes y Realizaciones), Santa Fe, Argentina, 1969 — Bocher Memorial Prize, awarded by American Mathematical Society, 1979 — Consagracion Nacional Prize, Argentina, 1989 — Wolf Prize, awarded by the Wolf Foundation, Jerusalem, Israel, 1989 — Steele Prize, American Mathematical Society, 1989 — National Medal of Science, United States of America, 1991 Degrees, Positions, Scholarships — Doctor Honoris Causa University of Buenos Aires, Argentina, 1969 — Doctor of Science, Honoris Causa Technion, Haifa, Israel, 1989 — Doctor of Science, Honoris Causa Ohio State University, 1995 — Doctor Honoris Causa Universidad Autonoma de Madrid, Spain, 1997 — Louis Block Professor of Mathematics University of Chicago, 1968-1972 — University Professor of Mathematics University of Chicago, 1975-1985 — Honorary Professor University of Buenos Aires, 1975 — Rockefeller Foundation Fellow, University of Chicago, 1949-1950 Teaching, Visiting and Research Positions — Visiting Associate Professor, Ohio State University, Columbus, 1950-1953 — Temporary Member, Institute for Advanced Study, Princeton, New Jersey, 1953-1955

62

— Associate Professor, Massachusetts Institute of Technology, 1955-1959 — Professor, University of Chicago, 1959-1968 — For other teaching positions see Degrees, Positions, Scholarships — Visiting Professor at various times at the following universities: University of Buenos Aires, Cornell University, Stanford University, National University of Bogota, Colombia, College de France, Paris, University of Paris (Sorbonne), Autonoma and Complutense Universities, Madrid, University of Rome, Gottingen University Other Professional Activities — ' Former Associate Editor of the following journals: Transactions of the American Mathematical Society, Illinois Journal of Mathematics, Journal of Functional Analysis, Duke Mathematical Journal, Journal of Differential Equations, Advances in Mathematics — Former member of the Council of the American Mathematical Society — Consultant of the Organizing Committe of the International Congress of Mathematicians, Nice, France, 1970 — Former member of the Editorial Committee of the American Mathematical Society — Member of the Research Career of the National Council of Scientific and Technical. Research of Argentina

63

List of Publications

1949 1. (with Gonzalez Domfnguez, A. and Zygmund, A.), "Nota sobre los valores limites de funciones analiticas", Rev. Union Mat. Argentina 14, pp. 16-19. 1950 2. "On theorems of M. Riesz and A. Zygmund", Proc. Am. Math. Soc. 1, pp. 533-535. 3. "On the behaviour of harmonic functions at the boundary", Trans. Amer. Math. Soc. 68, pp. 47-54. 4. (with Zygmund, A.), "On the theorem of Hausdorff-Young and its extensions", Contributions to Fourier Analysis, Annals of Math. Studies 25. Princeton University Press, pp. 166-188. 5. "On a theorem of Marcinkiewicz and Zygmund", Trans. Amer. Math. Soc. 68, pp. 55-61. 6. (with Zygmund, A.), "Note on the boundary values of functions of several complex variables", Contributions to Fourier Analysis, Annals of Math. Studies, 25. Princeton University Press, pp. 145-165. 7. (with Zygmund, A.), On singular integrals in the theory of the potential, Proc. Int. Congress of Mathematicians, Vol. 1. 1951 8. "On the differentiability of absolutely continuous functions", Rev. Mat. Univ. Parma 2, pp. 203-213. 9. (with Zygmund, A.), "Note on the interpolation of linear operations", Studia Math. 12, pp. 194-204. 10. (with Zygmund, A.), "On the existence of certain singular integrals", Acta Math. 88, pp. 85-139. 11. (with Klein, G.), "On an extremun problem concerning trigonometrical polynomials", Studia Math. 12, pp. 166-169. 1952 12. (with Pepinsky, R.), On the phases of Fourier coefficients of positive real periodic functions, computing methods and the phase problem in X-ray crystal analysis. Dept. of Physics, Perm. State College, pp. 339-349. 13. "A general ergodic theorem", Ann. Math. (2) 58, pp. 183-191. 1953 14. (with Mann, H. B.), "On the moments of stochastic integrals", Sankhya 12, pp. 347-350.

64

15. "Solution of the scalar radiation problem for surfaces of revolution", Ohio State Univ. Research Foundation, Antenna Lab. A. F. 18 (600) 88, 478-17, 4 pp. 1954 16. (with Arens, R.), Analytic functions of Fourier transforms, Segundo Simposio sobre algunos problemas matematicos que se estan estudiando en Latinoamerica. Centro de la UNESCO de Cooperation Cientifica para America Latina, Montevideo, Uruguay, pp. 39-52. 17. Singular integrals, Segundo Symposium sobre algunos problemas matematicos que se estan estudiando en America Latina, Julio 1954, Centro de la UNESCO de Cooperation Cientifica para America Latina, Montevideo, Uruguay, pp. 319-328. 18. "The multipole expansion of radiation fields", J. Rational Mech. Anal. 3, pp. 523-537. 19. (with Zygmund, A.), "Singular integrals and periodic functions", Studia Math. 14, pp. 249-271. 1955 20. (with Zygmund, A.), "On a problem of Mihlin", Trans. Amer. Math. Soc. 78, pp. 209-224. 21. (with Arens, R.), "Analytic functions of several Banach algebra elements", Ann. Math. 2, pp. 204-216. 22. "Sur les mesures invariantes", G. R. Acad. Sci. Paris 240, pp. 1960-1962. 23. (with Devinatz, A.), "On the Fourier-Stieltjes transforms", Canad. J. Math. 7, pp. 453-461. 24. (with Devinatz, A.), "Sur certaines courbes dans I'espace de Hilbert", C. R. Acad. Sci. Paris 241, pp. 539-541. 25. (with Devinatz, A.), "Sur certaines courbes a courbure constante dans I'espace de Hilbert", C. R. Acad. Sci. Paris 241, pp. 586-587. 1956 26. (with Zygmund, A.), "A note on the interpolation of sublinear operations", Amer. J. Math. 78, pp. 282-288. 27. (with Zygmund, A.), "On singular integrals", Amer. J. Math. 78, pp. 289-309. 28. Ideals in Abelian group algebras. Symposium on Harmonic Analysis and Related Integral Transforms, Technical Report, Dept. of Math., Cornell Univ., 12 pp. 29. (with Zygmund, A.), "Algebras of certain singular integral operators", Amer. J. Math. 78, pp. 310-320. 30. (with Zygmund, A.), "Singular integral operators and differential equations", Amer. J. Math. 79, pp. 901-921. 1957 31. (with Zygmund, A.), "On a problem of Mihlin", Trans. Amer. Math. Soc. 84, pp. 559-560. Addenda to the paper.

65

1958 32. "Uniqueness in the Cauchy problem for partial differential equations", Amer. J. Math. 80, pp. 16-36. 1959 33. (with Spitzer, F. and Widom, H.), "Inversion of Toeplitz matrices", Illinois J. Math. 3, pp. 490-498. 1960 34. "Integrales singulares y sus aplicaciones a ecuaciones diferenciales hiperbolicas", Cursos y Seminarios de Matemdtica, Fasc. 3, Univ. Buenos Aires, 121 pp. 35. (with Zygmund, A.), "A note on local properties of solutions of elliptic differential equations", Proc. Nat. Acad. Sci, U.S.A. 46, pp. 1385-1389. 36. (with Zygmund, A.), "On the differentiability of functions which are of bounded variation in Tonelli's sense", Rev. Un. Mat. Arg. 20, pp. 102-121. 1961 37. "Lebesgue spaces of differentiable functions and distributions", Proc. Symp. Pure Math. IV, pp. 33-49, Amer. Math. Soc. 38. (with Zygmund, A.), "Local properties of solutions of elliptic partial differential equations", Studia Math. 20, pp. 171-225. 39. Existence and uniqueness theorems for systems of partial differential equations, Proc. Symp. Fluid Dynamics and Applied Math., University of Maryland, pp. 147-195, Gordon and Breach. 1962 40. (with Benedek, A. and Panzone, R.), "Convolution operators on Banach space valued functions", Proc. Nat. Acad. Sci. U.S.A. 48, No. 3, pp. 356-365. 1963 41. Intermediate spaces and interpolation, Studia Math., Seria Specjalna, Z. I., pp. 31-34. 42. Boundary value problems for elliptic equations, Outlines of the Joint SovietAmerican Symposium on Partial Differential Equations, pp. 303-304 August. 1964 43. (with Zygmund, A.), "Higher gradients of harmonic functions", Studia Math. 24, pp. 211-226. 44. "Intermediate spaces and interpolation, the complex method", Studia Math. 24, pp. 113-190. 1965 45. "Commutators of singular integral operators", Proc. Nat. Acad. Sci. 53, pp. 1092-1099.

U.S.A.

1966 1

46. "Spaces between L and L°° and the theorem of Marcinkiewicz", Studia Math. 26, pp. 273-299. 47. "Singular integrals", Bull, of the A.M.S. 72, pp. 427-466. 48. "Estimates for integral operators", A. F. Office of Scientific Research, 65-1866, pp. 3-12. 49. "Algebras of singular integral operators", A.M.S. Proc. Symp. Pure Math. 10, pp. 18-55. 50. (with Weiss, M. and Zygmund, A.), "On the existence of singular integrals", A.M.S. Proc. Symp. Pure Math. 10, pp. 56-73. 1967 51. "The analytic calculation of the index of elliptic equations", Proc. Nat. Acad. Sci. U.S.A. 57, pp. 1193-1194. 1968 52. "Ergodic theory and translation invariant operators", Proc. Nat. Acad. Sci. U.S.A. 59, pp. 349-353. 1969 53. A priori estimates for singular integral operators, Centro Internazionale Matematico Estivo, pp. 87-141. 1970 54. "Uniqueness of distributions", Rev. Un. Mat. Arg. 25, pp. 37-65. 1971 55. (with Vaillancourt, R.), "On the boundedness of pseudo-differential operators", J. Math. Soc. Japan 23, pp. 274-278. 1972 56. (with Vaillancourt, R.), "A class of bounded pseudo-differential operators", Proc. Nat. Acad. Sci. U.S.A. 69, pp. 1185-1187. 57. "Estimates for singular integral operators in terms of maximal functions", Studia Math. 44, pp. 563-581. 1973 58. (with Zygmund, A.), "On singular integrals", Studia Math. 46, pp. 297-299. Addendum to the paper. 59. "A note on biquadratic forms", J. Linear Alg. Appl. 7, pp. 175-177. 1975 60. (with Torchinskyt, A.), "Parabolic maximal functions associated with a distribution", Adv. Math. 16, pp. 1-63.

67 1976 61. Lecture Notes on Pseudo-Differential Operators and Elliptic Boundary Value Problems. I Cursos de Matematica, Instituto Argentino de Matematica, 83 pp. 62. "An inequality for integrals", Studia Math. 57, pp. 275-277. 63. "On an integral of Marcinkiewicz", Studia Math. 57, pp. 279-284. 64. "Inequalities for the maximal function relative to a metric", Studia Math. 57, pp. 297-306. 1977 65. "On the Cauchy integral on Lipschitz curves and related operators", Proc. Nat. Acad. Sci. U.S.A. 74, pp. 1324-1327. 66. (with Torchinsky, A.), "Parabolic maximal functions associated with a distribution II", Adv. Math. 25, pp. 101-171. 67. "An atomic decomposition of distributions in parabolic Hp spaces", Adv. Math. 25, pp. 216-225. 1978 68. (with Scott, R.), "Sobolev inequalities for p > 0", Studia Math. 62, pp. 75-92. 69. (with Calderon, C. P., Fabes, E., Jodeit, M. and Riviere, N. M.), "Applications of the Cauchy integral on Lipschitz curves", Bull, of the A.M.S. 84, pp. 287-290. 70. (with Zygmund, A.) "Singular integrals with variable kernels", Appl. Anal. 7, pp. 221-238. 1979 71. (with Zygmund, A.), "A note on singular integrals", Studia Math. 65, pp. 77-87. 72. (with Alvarez Alonso, J. D.), Functional calculi for pseudo-differential operators, Fourier Analysis. Proceedings of Seminar at El Escorial. Asociacion Matematica Espahola, pp. 3-61. 1980 73. Commutators, Singular Integrals on Lipschitz Curves and Applications. Proc. Int. Congress of Mathematicians, Helsinki, pp. 67-73. 74. "On an inverse boundary value problem", Soc. Brasileira Mat. Atas 12, pp. 67-73. 1981 75. On the Radon transform and some of its generalizations, Conference on Harmonic Analysis in Honor of Antoni Zygmund II, pp. 673-689, Wadsworth Mathematics Series. 1984 76. (with Capri, O. N.), "On the convergence in L1 of singular integrals", Studia Math. 78, pp. 321-327.

68

77. (with Alvarez Alonso, J. D.), "Functional calculi for pseudo-differential operators II", Studies in Appl. Math., Adv. Math. Supp. Studies 8, pp. 27-72. 1985 78. "Boundary value problems for the Laplace equation in Lipschitzian domains", Recent Progress in Fourier Analysis, North-Holland, Notas Mat. 101, pp. 38-48. 1986 79. Integrates singulares y operadores pseudodiferenciales. Historia y perspectiva, Anales Acad. Nac. Ciencias Exac. Fis. y Nat., Buenos Aires, Vol. 38, pp. 33-45. 1987 80. "Reflexiones sobre el aprendizaje y ensenanza de la Matematica", Rev. Education Mat, Un. Mat. Arg. 3, No. 1, pp. 3-13. 1988 81. Presentation of Dr. Eduardo Zarantonello as correspondent member of the Academia National de Ciencias Exactas, Fisicas y Naturales. Buenos Aires, Argentina. Annals of the Academia, Vol. 40. 1989 82. Presentation of Dr. Misha Cotlar as correspondent member of the Academia National de Ciencias Exactas, Fisicas y Naturales. Buenos Aires, Argentina. Annals of the Academia, Vol. 41. 1990 83. Presentation of Dr. Carlos Segovia Fernandez as member of the Academia National de Ciencias Exactas, Fisicas y Naturales. Buenos Aires, Argentina. Annals of the Academia, Vol. 42. 1991 84. (with Sagher, Y.), The Hilbert Transform of the Gaussian. Proceedings of the 1989 International Conference on a.e. Convergence in Probability and Ergodic Theory, Evanston, Illinois (eds. Alexandra Bellow and Roger L. Jones), Academic Press. 1995 85. (with Bellow, A. and Krengel, U.), Hopf's ergodic theorem for particles with different velocities and the "strong sweeping out property", Canad. Math. Bull. 38(1), pp. 11-15. 1999 86. (with Bellow, A.), A weak type inequality for convolution products, Harmonic Analysis and Partial Differential Equations, Essays in Honor of Alberto

69

P. Calderon (eds. Michael Christ, Carlos E. Kenig and Cora Sadosky), The Univ. of Chicago Press, pp. 41-48. 2000

87. (with Calderon, C. P.), "A representation formula and its applications to singular integrals", Indiana J. Math. 49(1), pp. 1-6.

Singular Integrals: The Roles of Calderon and Zygmund EliasM. Stein

Editor's Note: Alberto Pedro Calderon died on April 16, 1998. A memorial article appears elsewhere In this Issue. raries and predecessors, is contained in his famous treatise Trigonometrical Series, published in 1935. The time in which this took place may be viewed as the concluding decade of the brilliant century of classical harmonic analysis: the approximately one hundred-year span which began with Dirichlet and Riemann, continued with Cantor and Lebesgue among others, and culminated with the achievements of Kolmgorov, M. Rlesz, and Hardy and Littlewood. It was during that last decade that Zygmund began to turn his attention from the one-dimensional situation to problems in higher dimensions. At first this represented merely an Incidental interest, but then later he followed it with increasZygmund's Vision: 1 9 2 7 - 1 9 4 9 ing dedication, and eventually it was to become the In the first period of his scientific work, from 1923 main focus of his scientific work. I want now to deto the middle 1930s, Zygmund devoted himself to scribe how this point of view developed with what is now called "classical" harmonic analysis: Zygmund. that Is, Fourier and trigonometric series of the circle, related power series of the unit disc, conjugate In outline, the subject of one-dimensional harfunctions, Riemannian theory connected to uniquemonic analysis as it existed in that period can be ness, lacunary series, etc. An account of much of understood in terms of what were then three what he did, as well as the work of his contempoclosely interrelated areas of study and which in many ways represented the central achievements EliasM. Stein is professor of mathematics at Princeton Uniof the theory: real-variable theory, complex analyversity. His e-mail address is s t e i n@math. sis, and the behavior of Fourier series. Zygmund's princeton.edu. This essay, published in the Notices by arrangement with first excursion into questions of higher dimenthe University of Chicago Press, is a modified version of sions dealt with the key issue of real-variable thethe article entitled "Calderon and Zygmund's theory of sin-ory, the averaging of functions. The question was gular Integrals", to appear In the proceedings of the 1996as follows. The classical theorem of Lebesgue guarconference In honor of Calderon's seventy-fifth birthday anteed that for almost every x held at the University of Chicago. That article also contains an extended list of references and further bibliolira ±\f(y)dy = f(x), graphical notes which have been omitted from the pre- (1.1) sent version.

he subject m a t t e r of this essay is Alberto Calderon's pivotal role in the creation of the modern theory of singular integrals. In that great enterprise Calderon had the good fortune of working with Antonl Zygmund, who was at first his teacher and mentor and later his collaborator. For that reason any account of that theory has to be In part the story of the efforts of both Zygmund and Calderon. With this in mind, I shall explain the various goals that motivated them, describe some of their shared accomplishments and later work of Calderon, and discuss briefly the wide influence of their achievements.

T

NOTICES OF THE AMS

VOLUME 4S, NUMBER 9

71 where / ranges over Intervals and when f is an in- fact that M. Riesz proved the V boundedness tegrable function on the line R1. properties of the Hilbert transform In higher dimensions it is natural to ask whether f~H(f) = B£ Jj(x-y)f a similar result holds when the Intervals I are replaced by appropriate generalizations in R". The p fact that this is the case when the I's are replaced by applying a contour integral to (F) , where F is the analytic function whose boundary limit has f by balls (or more general sets with bounded "ecas its real part. It should be noted that the Hilbert centricity") was well known at that time. What transform has a simple expression as a Fourier mulmust have piqued Zygmund's interest in the subtiplier, that is, ject was his realization (in 1927) that a paradoxical set constructed by Nikodym showed that the WHS) = S-^j^fd), answer is irretrievably false when the J's are taken to be rectangles (each containing the point in queswhere " denotes the Fourier transform; from this tion) but with arbitrary orientation. To this must the L2 boundedness is an immediate consequence be added the counterexample found by Saks sevvia Plancherel's theorem. eral years later, which showed that the desired analogue of (1.1) still failed even if we now re(ii) The theory of the Hardy spaces W stricted the rectangles to have a fixed orientation These arose in part as substitutes for V, when (e.g., with sides parallel to the axes) as long as one p < 1, and were by their very nature complexallowed f to be a general function in I 1 . function-theory constructs. (It should be noted, It was at this stage that Zygmund effectively however, that for 1 < p < oo they were essentially transformed the subject at hand by an important equivalent with V by Riesz's theorem.) The clasadvance: he proved that the wished-for conclusical space H p consists of analytic functions F in sion (when the sides are parallel to the axes) held the unit disc for which if fwas assumed to belong to V, with p > 1. He accomplished this by proving an inequality for 27T what is now known as the "strong" maximal funcsup f \F(rew)fdd<<x>. r<\ J tion. The original Hardy-Littlewood maximal func0 tion Involved a supremum over the averages in (1.1); in the definition of the strong maximal func- The main tool used in their study was the Blaschke tion the intervals containing x are replaced by rec- product of their zeroes in the unit disc. Using it, tangles with sides parallel to the axes. Shortly afone could reduce matters to elements F e.Hp with terwards in Jessen, Marcinkiewicz, and Zygmund no zeroes, and from these one could pass to (1935) this was refined to the requirement that f G = Fpl2\ the latter was in H2 and hence could be belong to I(logl)™"1 locally. treated by more standard (L2) methods. This study of the extension of (1.1) to R" was (iii) The Littlewood-Paley theory the first step taken by Zygmund. It is reasonable This proceeded by studying the dyadic decomto guess that it reinforced his fascination with position in frequency space and had many appliwhat was then developing as a long-term goal of cations; among them was the Marcinkiewicz mulhis scientific efforts: the extension of the central results of harmonic analysis to higher dimensions. tiplier theorem. This gave conditions on a Fourier But a great obstacle stood in the way: it was the multiplier, in terms of certain differential incrucial role played by complex function theory in equalities, that were sufficient to guarantee that the whole of one-dimensional Fourier analysis, it defined a bounded operator on V. The theory and for this there was no ready substitute. initiated and exploited certain basic "square funcIn describing this special role of complex meth- tions", and these we originally studied by complexods we shall content ourselves with highhghting variable techniques closely related to what were some of the main points. The theory can be forused in Hp spaces. mulated equally in the unit disc or the upper half(iv) The boundary behavior of harmonic functions plane, and we shall freely pass back and forth beThe main result obtained here (Privalov (1923), tween these settings. Marcinkiewicz and Zygmund (1938), and Spencer (i) The conjugate function and its basic properties (1943)) stated that for any harmonic function As is well known, the Hilbert transform comes ul.re'9) in the unit disc the following three propdirectly from the Cauchy integral formula. Closely erties are equivalent for almost all boundary points connected with this is the fact that the Hilbert ew. transform of a function f is obtained by passing to the Poisson integral of fin the upper half-plane, (1.2) u has a nontangential limit at ew, taking the conjugate harmonic function and passing back to boundary values. We also recall the (1.3) u is nontangentially bounded at e'e, OCTOBER 1998

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72 the "area integral" (S(u)W))2 = (1.4)

2

IT |Vu(z)| dxdy is finite, He")

where T(ew) is a nontangential approach region with vertex ew. The crucial first step in the proof was the application of the conformal map (to the unit disc) of the famous saw-tooth domain in Figure l. 1 This mapping allowed one to reduce the implication (1.3) => (1.2) to the special case of bounded harmonic functions in the unit disc (Fatou's t h e o r e m ) and also played a corresponding role in the other p a r t s of the proof. It is ironic t h a t complex m e t h o d s with their great power and success in the onedimensional theory actually stood in the way of progress to Figure 1. Saw-tooth domain. higher dimensions and appeared to block further progress. The only way past, as Zygmund foresaw, required a further development of "real" methods. Achievement of this objective was to take more than one generation and in some ways is not yet complete. The mathematician with whom he was to Initiate the effort to realize much of this goal was Alberto Calderon. Calderon and Zygmund: 1950-1957 Zygmund spent the academic year 1948-49 in Argentina, and there he met Calderon. Zygmund brought him back to the University of Chicago, and soon thereafter (in 1950), under his direction, Calderon obtained his doctoral thesis. The dissertation contained three parts, the first about ergodic theory, which will not concern us here. It is the second and third parts that interest us, and these represented breakthroughs in the problem of freeing oneself from complex methods and, in particular, in extending to higher dimensions some of the results described in (iv) above. In a general way we can say that his efforts here already typified the style of much of his later work: he begins

by conceiving some simple but fundamental ideas that go to the heart of the matter and then develops and exploits these insights with great power. In proving (1.3) » (1.2) we may assume that u is bounded inside the saw-tooth domain O. that arose in (iv) above: this region is the union of approach regions r(e' 8 ), ("cones") with vertex e'e, for points ew e E, and E a closed set. Calderon introduced the auxiliary harmonic function U, with If the Poisson integral of Xcf, and observed that all the desired facts flowed from the dominating properties of U: namely, u could be split as u = m + U2, where u\ is the Poisson integral of a bounded function (and hence has nontangential limits a.e.), while by the maximum principle |H2l ^ ctf, and therefore U2 has (nontangential) limits => 0 at a.e. point of E. The second idea (used to prove the Implication (1.2) => (1.4)) has as its starting point the simple identity Aw2 = 2|Vu| 2

(2.1)

valid for any harmonic function. This can be combined with Green's theorem

J'

(BAA-AAB)dx dy-\{

where A = u 2 and B is another Ingeniously chosen auxiliary function depending on the domain O only. This allowed him to show t h a t J J y | V u | 2 d x d y < °o, which is an i n t e g r a t e d n revision of (1.4). It may be noted that the above methods and the conclusions they imply make no use of complex analysis and are very general in nature. It is also a fact that these ideas played a significant role in the later real-variable extension of the Hp theory. Starting in the year 1950, a close collaboration developed between Calderon and Zygmund which lasted almost thirty years. While their joint research dealt with a number of different subjects, their preoccupying interest and most fundamental contributions were in the area of singular integrals. In this connection the first issue they addressed was—to put the matter simply—the extension to higher dimensions of the theory of the Hilbert transform. A real-variable analysis of the Hilbert transform had been carried out by Besicovitch, Titchmarsh, and Marcinkiewicz, and this is what needed to be extended in the R" setting. A reasonable candidate for consideration presented itself. It was the operator f - Tf, with

^Figure 1 is based on a diagram in Zygmund 11959], Vol(2.2) n, p. 200. 1132

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T(f)(x) = p.v. J K(y)f(x - y) dy,

VOLUME 45, NUMBER 9

when K was homogeneous of degree -n, satisfied some regularity, and in addition satisfied the cancellation condition j K(x) dcr(x) = 0. Ixl-l Besides the Hilbert transform (which is the only real example when n = 1), higher-dimensional examples include the operators that arise as second derivatives of the fundamental solution operator for t h e Laplacian (which can be written as Ixfix (A)"1), as well as the related Riesz transforms 5^-(-A)" 1/2 . For the Hilbert transform, n is equal to 1 and K(x) is equal to ^ ; the Riesz transforms are given (up to a constant multiple) by K(x) = Xjl\x\"*\j =l n.

he p u b l i s h e d (in 19S6) an a c c o u n t of Marcinkiewicz's theorem and various generalizations and extensions he had since found. In it he conceded that the paper of Marcinkiewicz "... seems to have escaped attention and does not find allusion to it in the existing literature." The second point, like the first, also involves important work of Marcinkiewicz. He had been Zygmund's brilliant student and collaborator until his death at the beginning of World War II. It is a mystery why no reference was made to the paper by Marcinkiewicz [1939b] and the multiplier theorem in it. This t h e o r e m had b e e n p r o v e d by Marcinkiewicz in an n-dimensional form (as a product "consequence" of the one-dimensional form). As an application the V inequalities for the operators s ^ r f A - 1 ) were obtained 3 ; these he had proved at the behest of Schauder.

All of this is the subject matter of their historic memoir "On the existence of singular integrals", which appeared in Acta Mathematica in 1952. There is probably no paper in the last fifty years which had such widespread influence in analysis. The ideas in this work are now so well known that I will only outline its contents. It can be viewed as having three parts. First, there is the Calderon-Zygmund lemma and t h e c o r r e s p o n d i n g Calderon-Zygmund decomposition. The main thrust of the former is as a substitute for F. Riesz's "rising sun" lemma, which F. Riesz used in re-proving Lebesgue's theorem about the almost everywhere differentiability of monotone functions and which had implicitly Figure 2. Calderon-Zygmund decomposition of a nonnegative function /'at played a key role in the earlier treat1 altitude a. The cubes Q = Qt have the property that a < IQI" Iqfdx s 2"a. ment of the Hilbert transform. A In the complement of the balls Bk, a favorable L1 estimate holds. schematic representation of their decomposition is given in Figure 2. 2 Second, using their decomposition, they then proved the weak-type L1 and V, 1 < p < °o, estiAs has already been indicated, the n-dimenmates for the operator Tin (2.2). As a preliminary sional singular integrals had their main motivation step they disposed of the I 2 theory of T using in the theory of partial differential equations. In Plancherel's theorem. their further work, Calderon and Zygmund pursued Third, they applied these results to the examthis connection, following the trail that had been ples mentioned above, and in addition they proved explored earlier by Giraud, Tricomi, and Mihlin. a.e. convergence for the singular integrals in quesStarting from those ideas (in particular the notion tion. of "symbol"), they developed their version of the It should not detract from one's great admirasymbolic calculus of "variable-coefficient" anation for this work to note two historical anomalies logues of the singular integral operators. To decontained in it. The first is the fact that there is scribe these results, one considers an extension of no mention of Marcinkiewicz's interpolation thethe class of operators arising in (2.2), namely, of orem or of the p a p e r in which it a p p e a r e d the form (Marcinkiewicz [1939a]), even though its ideas play (2.3) T(f)(x) = ao(x)f(x)+p.v. J K(x, y)f(x - y) dy, a significant role. In the Calderon-Zygmund paper the special case that is needed is in effect reproved. The explanation for this omission is that where K(x, y) is for each x a singular integral kernel of the type (2.2) in y, which depends smoothly Zygmund had simply forgotten about the exisand boundedly on x; also ao(x) is a smooth and tence of Marcinkiewicz's note. To make amends, bounded function. 2 7 still have a graphic recollection of a similar picture shown me by Harold Widom in 1952-53, when we were 3In truth, he had done this for their periodic analogues, both graduate students at the University of Chicago. but this is a technical distinction. OCTOBER

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1133

74 To each operator T of this kind there corresponds its symbol a(x, 5), defined by

At MIT we would meet quite often, and over time an easy conversational relationship developed between us. I do recall that we, in the small group who were interested in singular integrals then, felt a certain separateness from the larger community of analysts—not that this isolation was self-imposed, but more because our subject matter was seen by our colleagues as somewhat arcane, rarefied, and possibly not very relevant. However, this did change, and a fuller acceptance eventually came. I want to relate now how this occurred.

(2.4) a(x,?) = a0(x) + £(x,5), where K(x, J) denotes the Fourier transform of K(x, y) in the y variable. Thus a(x, J) is homogeneous of degree 0 in the g variable (reflecting the homogeneity of K(x, y) of degree -n In y), and it depends smoothly and boundedly on x. Conversely, to each function a(x, g) of this kind there exists a (unique) operator (2.3) for which (2.4) holds. One says that a is the symbol of Tand also writes T = Ta. The basic properties that were proved were, first, the regularity properties (2.5) Ta:I%~ tf., where L\ are the usual Sobolev spaces, involving W norms of the function and its partial derivatives through order k, with 1 < p < oo. Also proved were the basic facts of symbolic manipulations (2.6)

Starting from the calculus of singular integral operators that he had worked out with Zygmund, Calderon obtained a number of important applications to hyperbolic and elliptic equations. His most dramatic achievement was in the uniqueness of the Cauchy problem (Calderon [1958)). There he succeeded in a broad and decisive extension of the results of Holmgren (for the case of analytic coefficients) and Carleman (In the case of two dimensions). Calderon's theorem can be formulated as follows. Suppose u is a function which in the neighborhood1 of the origin in R" satisfies the equation of rrfi order: ,, ,, 3mu ^ . ,d"u

Tai • Ta2 = Ta, a2 + Error

(2.7) (Ta)* = Ta+ Error where the Error operators are smoothing of order 1, in the sense that Error: L% - l£ +1 . A consequence of the symbolic calculus is the factorizability of any linear partial differential operator L of order m,

1= I aa(x)(ff, where the coefficients aa are assumed to be smooth and bounded. On can write (2.8) I = r a (-A) m/2 + (Error)' for an appropriate symbol a, where the operator (Error)' refers to an operator that maps L\ — ijjLm+l, for k s m - 1. It seemed clear that this symbolic calculus should have wide applications to the theory of partial differential operators and to other parts of analysis. This was soon to be borne out. Acceptance: 1957-1965

At this stage of my narrative I would like to share some personal reminiscences. I had been a student of Zygmund at the University of Chicago, and In 1956 at his suggestion I took my first teaching position at MIT, where Calderon was at that time. I had met Calderon several years earlier when he came to Chicago to speak about the "method of rotations" in Zygmund's seminar. I still remember my feelings when I saw him there; these first impressions have not changed much over the years: I was struck by the sense of his understated elegance, reserve, and quiet charisma.

(3

-"

sxjrl^s^'

where the summation is taken over all indices a = (a1,...,a„), with |a| £ n and a„ < m. We also assume that u satisfies the null initial Cauchy conditions ,3.r) • *»W| .o, j.o m-1. Besides (3.1) and (3.1') it suffices that the coefficients aa belong to C1+e, that the characteristics are simple, and that n 4 3 or m < 3. Under these hypotheses u vanishes identically In a neighborhood of the origin. Calderon's approach was to reduce matters to a key "pseudo-differential inequality" (in a terminology that was used later). This inequality is complicated but somewhat reminiscent of a differential inequality that Carleman had used in two dimensions. The essence of it is that a

j ' 0 k | | | p + ( i ' + iQ)(--A)1'2H||2dr
where u(0) = 0 Implies u s 0 if (3.2) holds for k — oo. Here P and Q are singular integral operators of the type (2.3), with real symbols and P invertible; we have written f = x„, and the norms are L2

NOTICES OF THE AMS

VOLUME 45, NUMBER 9

75

ference of the dimennorms taken with respect to the variables sion of the null space xi,...»Xn-i. The function <^k is meant to behave k and the codhnension of like t~ , which, when k -* « , emphasizes the effect taking place near fk = 0. In fact, in (3.2) we can the range oftiheoperator and is an invariant take#fe(f) = (r + l/fer . The proof of assertions like (3.2) is easier in the under deformations. special case when all the operators commute; their The problem of detergeneral form is established by using the basic facts mining it was connected with a number (2.6) and (2.7) of the calculus. The paper by Calderon was at first not well re- of interesting issues in ceived, hi fact, 1 learned from himtibatit was re- geometry and topology. jected when submitted to'what was then the lead- The result of the "Seeley calculus" proved ing journal in partial differential equations, Communications of Pure and Applied Mathemat- quite useful in this conics, text: the proofs proceeded by appropriate deformations, and matAt about that time, because of the applicability ters were facilitated if these could be carried'' of singular Integrals to partial differential equations, Calderon became interested in formulating out in the more flexible the facts about singular integrals in the setting of context of "general" manifolds. This required the analysis of the effect symbols instead of recoordinate changes had on such operators. A hint stricting attention to that the problem was tractable camefromthe ob- the polynomial symbols servation that the class of kernels, K(y), of the type coming from differenarising in (2.2) was invariant under linear (inverttial operators. A conible) changes of variables y ~ L(y). (The fact that temporaneous account f (I(y)) satisfied the same regularity and homo- of this development geneity that Kiy) did was immediate; that the can- (during the period cellation property also holds for K(L(y)) is a little 1961-64) may be found less obvious.) hi the notes of the semR. Seeley was Calderon's student at that time, inar on the Atiyahand he dealt with this problem in his thesis (1959). Singer index theorem Suppose x -* ip(x) is a local diffeomorpMsm. Then (see Palais [1965J); for the result is that modulo error terms (which are an historical survey of "smoothing" of one degree) the operator (2.3) is some of the backttansformed into another operator of the same ground, see also Seeley kind, [1967]. Tfmm • afa) = f(x) + p.v. J V (x, y)f(x - y) dy%

Alberto Pedro Calderon, circa 1979

Antoni Zygmursd, circa 1979.

tzmmmmmmmmmm mmmmmmmmmmmmm

but now

With the activity surrounding the index theorem, it suddenly seemed as if everyone was interested in the algebra of singular integral operators. However, one further step was needed to make this a K'(xty) - K'(lHri,Lx(y)), household tool for analysts: it required a change where I x is thetypeartransformation given by the of point of view. Even though this change of perJacobian matrix 23j£a. On the level of symbols this spective was not .major, it was significant psychomeant that the new symbol a' was determined by logically and methodologically, since it allowed the old symbol according to the formula one to think more simply about certain aspects of the subject and because it suggested various extensions. with L'x the transpose inverse of JLX. Hence the symThe idea was merely to change the role of the bol is actually a function on the cotangent space definitions of the operators, from (2.3) for singuof the manifold. lar integrals to pseudo-differential operators The result of Seeley was not only highly satisTa(f)M = J a(x, g)?©e27rfx*f d£t factory as to its conclusions, it was also very timely (3.3) in terms of events that were about to take place. following an intervention by Gelfand (1960), in- with symbol a. (Here f is the Fourier transform terest grew hi catenating the "index" of an ellip?(©- S e~2fTtx^f(x)dx.) tic operator on a manifold. This index is die difand

OcTOiii 1998

a'0(x)« m(4?(x))

Nonas OF THE AMS

1135

76 Although the two operators are identical (when a(x, g) = ao(x) + K(x, g)), the advantage lies in the emphasis In (3.3) on the i 2 theory and Fourier transform and the wider class of operators that can be considered, in particular, differential operators. The formulation (3.3) allows one to deal more systematically with the composition of such operators and incorporate the lower-order terms In the calculus. One way of doing this was to adopt a wider class of symbols of "homogeneous type": roughly speaking, a(x, g) belongs to this class (and is of order m) if a(x, g) is for large g asymptotically the sum of terms homogeneous in g of degrees m-j, with j = 0,1,2 The change in point of view described above came Into Its full flowering with the papers of Kohn and Nirenberg (1965) and Hormander (1965), (after some work by Unterberger and Bokobza (1964) and Seeley (1965)). It is in this way that singular Integrals were subsumed by pseudo-differential operators. Despite this, singular integrals, with their formulation in terms of kernels, still retained their primacy when treating real-variable issues, issues such as IP or i 1 estimates (and2even for some of the more intricate parts of the L theory). The central role of the kernel representation of these operators became, if anything, more pronounced In the next twenty years.

The first major Insight arose in answer to the following: Question: Suppose MA is the operator of multiplication (by the function A), MA:f~A-f. What are the least regularity assumptions on A needed to guarantee that the commutator [T, MA\ is bounded on I 2 whenever X is of order 1? In R1 if Thappens to be ^ , then [T, MA] = MA', and so the condition is exactly

Calderon's New Theory of Singular Integrals: 1965-

(4.4) dWX) = i p.v. J

In the years 1957-58 there appeared the fundamental work of DeGlorgl and Nash dealing with smoothness of solutions of partial differential equations, with minimal assumptions of regularity of the coefficients. One of the most striking results, for elliptic equations, was that any solution u of the equation

Calderon proved that f - Ci(ft is bounded on I2(K) If (4.3) holds. There are two crucial points that I want to emphasize about the proof of this theorem. The first Is the reduction of the boundedness of the bilinear term if,g)-~ {Ci(f),g) to a corresponding property of a particular bilinear mapping, (F,G) - B(F,G), defined for (appropriate) holomorphlc functions in the upper half-plane {z = x + iy, y > 0} by

(4.1,

i(u)sgA(a(jU)g)=o

In an open ball satisfies an a priori Interior regularity as long as the coefficients are uniformly elliptic, l.e., (4.2)

Qlgl^XayMJjgjSQlgl2. t,j

In fact, no regularity Is assumed about the atj except for the boundedness Implicit in (4.2), and the result is that u Is Holder continuous with an exponent depending only on the constants c\ and C2.

erties of the operators (2.3) and their calculus when the dependence on x was e.g. of class C<1+e)—this hope was not to be realized. Further understanding about these things could be achieved only If one were ready to look In a somewhat different direction. 1 want to relate now how this came about.

(4.3)

A'er(Bl).

In a remarkable paper Calderon [1965] showed that this is also the case more generally. The key case, containing the essence of the result he proved, arose when T = H^, with H the Hilbert transform. Then Tis actually | ^ |, its symbol is 2rr 1g1, and IT, MA] is the "commutator" Q,

(4.5)

A

™:$,y)fWdy.

B(F, G)(x) = i f F' (x + iy)G(x + iy) dy. Jo

This B Is a primitive version of a "para-product" (in this context, the justification for this terminology Is the observation that F(x)-G(x) = B(F,G)(x) + B(G,F)(x)). It Is, in fact, not too difficult to see that f - Ci(f) is bounded on i 2 ® 1 ) if B satisfies the Hardy-space estimate (4.6) l|B(F,G)||„, sc||F||H2||G||„2. The second major point in the proof is the assertion needed to establish (4.6). It Is the converse part of the equivalence

Calderon was intrigued by this result. He initially expected, as he told me, that one could obtain such conclusions and others by refining the calculus of singular integral operators (3.2), making minimal assumptions of smoothness on a<>(x) and (4.7) \\S(F)\\v ~ IIFIIH' K(x, y). While this was plausible—and indeed in his work with Zygmund they had already derived prop- for the area integral S (which appeared in (1.4)). NOTICES OF THE AMS

VOLUME 45, NUMBER 9

77 The theorem of Calderon, and in particular the methods he used, inspired a number of significant developments in analysis. The first came because of the enigmatic nature of the proof: a deep I 2 theorem had been established by methods (using complex function theory) that did not seem susceptible to a general framework. In addition, the non-translation-lnvariance character of the operator Ci made Plancherel's theorem of no use here. It seemed likely that a method of "almost-orthogonal" decomposition—pioneered by Coflar for the classical Hilbert transform—might well succeed in this case also. This led to a reexamination of Cotlar's lemma (which had originally applied to the case of commuting self-adjoint operators). A general formulation was obtained as follows: Suppose that on a Hilbert space T = Z Tj. Then (4.8)

liril2 < £sup{\\TjTj*+k\\ + \\Tj*TJ+k\\}. k

J

Despite the success in proving (4.8) this alone was not enough to re-prove Calderon's theorem. As understood later, the missing element was a certain cancellation property. Nevertheless, the general form of Cotlar's lemma, (4.8), quickly led to a number of highly useful applications, such as singular integrals on nilpotent groups (intertwining operators), pseudo-differential operators, etc. Calderon's theorem also gave added impetus to the further evolution of the real-variable Hp theory. This came about because the equivalence (4.7) and its generalizations allowed one to show that the usual singular integrals (2.2) were also bounded on the Hardy space H 1 (and in fact on all Hp, 0 < p < oo). Taken together with earlier developments and some later ideas, the real-variable W theory reached its full flowering a few years later. One owes this long-term achievement to the work of G. Weiss, C. Fefferman, Burkholder, Gundy, and Coifman, among others. It became clear after a time that understanding the commutator C\ (and its "higher" analogues) was in fact connected with an old problem that had been an ultimate but unreached goal of the classical theory of singular integrals: the boundedness behavior of the Cauchy integral taken over curves with rninimal regularity. The question involved can be formulated as follows: in the complex plane, for a contour y and a function f defined on it form the Cauchy integral

essentially the Hilbert transform. Also, when y has some regularity (e.g., y is in C(1+e)), the expected properties of C (i.e., L2,LP boundedness, etc.) are easily obtained from the Hilbert transform. The problem was what happened when, say, y was less regular, and here the main issue that presented itself was the behavior of the Cauchy integral when y was a Lipschitz curve. If y is a Lipschitz graph in the plane, y = fx + iA(x), x e R], with A' s L°°, then up to a multiplicative constant, (4.9)

cim--P^[x_y+l{l{x)_m)f(yMyHy where b = 1 + IA'. The formal expansion 1 x - y + i(A(x) - A(y))

then makes clear that the fate of Cauchy integral C is inextricably bound up with that of the commutator C\ and its higher analogues C* given by TT

_J V. x-y

)

x-y

The further study of this problem was begun by Coifman and Meyer in the context of the commutators Q, but the first breakthrough for the Cauchy integral was obtained by Calderon [1977] (using different methods) in the case where the norm || A' ||i» was small. His proof made decisive use of the complex-analytic setting of the problem. It proceeded by an ingenious deformation argument, leading to a nonlinear differential inequality; this nonlinearity accounted for the limitation of small norm for A' in the conclusion. But even with this limitation the conclusion obtained was stunning. The crowning result came in 1982 when Coifman and Meyer, having enlisted the help of Mcintosh and relying on some of their earlier ideas, together proved the desired result without limitation on the size of || A' ||i». The method they used was operator-theoretic, emphasizing the multilinear aspects of the Q, and in distinction to Calderon's approach was not based on complex-analytic techniques.

The major achievement represented by the theory of the Cauchy integral led to a host of other with F holomorphic outside y. Define the mapping results, either by a rather direct exploitation of the f - Of) by C(f) = F+ + F~, where F+ are the lim- conclusions involved or by extensions of the techits of F on y approached from either side. When niques that were used. I will briefly discuss two of y is the unit circle, or real axis, then f - C(f) is these developments. y

OCTOBER 1998

NOTICES OF THE AMS

78 The first was a complete analysis of the I 2 theory of "Calderon-Zygmund operators". By this terminology is meant operators of the form

(4.11)

T(fXx)=JK(x,y)f(y)dy,

initially defined for test functions f e S, with the kernel K a distribution. It is assumed that away from the diagonal, K agrees with a function that satisfies familiar estimates such as

\Kix,y)\
The decisive application of the Cauchy integral to the potential theory of the Laplacian in a Lipschitz domain was In the study of the boundedness of the double-layer potential (and the normal derivative of the single-layer potential). These are n - 1 dimensional operators, and they can be realized by applying the "method of rotations" to the one-dimensional operator (4.9). One should mention that another significant aspect of Laplacians on Lipschitz domains was the understanding brought to light by Dahlberg of the nature of harmonic measure and its relation to Ap weights. These two strands, initially independent, have been linked together, and with the aid of further ideas a rich theory has developed, owing to the added contributions of Jerison, Kenig, and others. Finally, we return to the point where much of this began—the divergence-form equation (4.1). Here the analysis growing out of the Cauchy integral also had its effect. Here I will mention only the usefulness of multilinear analysis in the study of the case of "radially-independent" coefficients, also in the work on the Kato problem: the determination of the domain of VE in the case where the coefficients can be complex-valued. Some Perspectives on Singular Integrals: Past, Present, and Future The modern theory of singular integrals, developed and nurtured by Calderon and Zygmund, has proved to be a very fruitful part of analysis. Beyond the achievements described above, a number of other directions have been cultivated with great success, with work being vigorously pursued up to this rime. In addition, here several interesting open questions present themselves. I want to allude briefly to three of these directions and mention some of the problems that arise. 1. Method of the Calderon-Zygmund lemma. As is well known this method consists of decomposing an integrable function into its "good" and "bad" parts, the latter being supported on a disjoint union of cubes and having mean value zero on each cube. Together with an I 2 bound and estimates of the type (4.12), this leads ultimately to the weaktype (1,1) results, etc. It was recognized quite early that this method allowed substantial extension. The generalizations that were undertaken were not so much pursued for their own sake, but rather were motivated in each case by the interest of the applications. Roughly, in order of appearance, here were some of the main instances: (i) The heat equation and other parabolic equations. This began with the work of F. Jones (1964) for the heat equation, with the Calderon-Zygmund cubes replaced by rectangles whose dimensions reflected the homogeneity of the heat operator. The theory was extended by Fabes, Riviere, and Sadosky to encompass more general singular inteVOLUME 45, NUMBER 9

79 grals respecting "nonisorropic" homogeneity in Euclidean spaces. (ii) Symmetric spaces and semisimple Lie groups. To be succinct, the crucial point was the extension to the setting of nilpotent Lie groups with dilations ("homogeneous groups"), motivated by problems connected with Poisson integrals on symmetric spaces, and construction of intertwining operators. (iii) Several complex variables and subelliptic equations. Here we return again to the source of singular integrals, complex analysis, but now in the setting of several variables. An important conclusion obtained was that for a broad class of domains in C" the Cauchy-Szego projection is a singular integral susceptible to the above methods. This was realized first for strongly pseudo-convex domains, next weakly pseudo-convex domains of finite type in C2, and more recently convex domains of finite type in C". Connected with this is the application of the above ideas to the 3-Neumann problem, and its boundary analogue for certain domains in C , as well as the study solving operators for subelliptic problems, such as Kohn's Laplacian, Hormander's sum of squares, etc. These matters also involved using ideas originating in the study of nilpotent groups as in (ii). The three kinds of extensions mentioned above are prime examples of what one may call "one-parameter" analysis. This terminology refers to the fact that the cubes (or their containing balls) which occur in the standard R" setup have been replaced by a suitable one-parameter family of generalized "balls" associated to each point. While the general one-parameter method clearly has wide applicability, it is not sufficient to resolve the following important question: Problem. Describe the nature of the singular integral operators that are given by Cauchy-Szego projection, as well as those that arise in_connection with the solving operators for the d and 3;, complexes for general smooth finite-type pseudoconvex domains in C". Some speculation about what may be involved in resolving this question can be found below. 2. The method of rotations. The method of rotations is both simple in its conception and farreaching in its consequences. The initial idea was to take the one-dimensional Hilbert transform, induce it on a fixed (subgroup) R1 of K", rotate this R1, and integrate in all directions, obtaining in this way the singular integral (2.2) with odd kernel, which can be written as

(5.1)

7btf)(x) = p . v . J ^ f ( x - y ) d y

where CI is homogeneous of degree 0, integrable in the unit sphere, and odd. OCTOBER 1998

In much the same way the general maximal operator (5.2)

W)(x) = sup4r

\

n(y)f(x-y)dy\

\y\sr

arises from the one-dimensional Hardy-Littlewood maximal function. This method worked very well for V estimates for p > 1, but not for I 1 (since the weak-type L1 "norm" is not subadditive). The question of what happens for I 1 was left unresolved by Calderon and Zygmund. It is now to a large extent answered: we know that both (5.1) and (5.2) are indeed of weak-type (1,1) if CI is in L(logi). This is the achievement of a number of mathematicians, in particular Christ and Rubio de Francia. When the method of rotations is combined with the singular integrals for the heat equation (as in l(i) above), one arrives at the "Hilbert transform on the parabola". Consideration of the Poisson integral on symmetric spaces leads one also to inquire about some analogous maximal functions associated to homogeneous curves. The initial major breakthroughs in this area of research were obtained by Nagel, Riviere, and Wainger. The subject has since developed into a rich and varied theory, beginning with its translation invariant setting on R" (and its reliance on the Fourier transform), and then prompted by several complex variables, to a more general context connected with oscillatory integrals and nilpotent Lie groups, where it was rechristened as the theory of "singular Radon transforms". A common unresolved enigma remains about these two areas that have sprung out of the method of rotations. This is a question that has intrigued workers in the field and whose solution, if positive, would be of great interest. Problem (a) Is there an I 1 theory for (5.1) and (5.2) if Q is merely integrable? 4 (b) Are the singular Radon transforms, and their corresponding maximal functions, of weak-type (1,1)? 3. Product theory and multiparameter analysis. To oversimplify matters, one can say that "product theory" is that part of harmonic analysis in R" which is invariant with respect to the n-fold dilations: X = (X\,X2 X„) - (5lXl,<52X2,...,<5„X„), Sj > 0. Another way of putting it is that its initial concern is with operators that are essentially products of operators acting on each variable separately and then more generally with operators (and associated function spaces) that retain some of these characteristics. Related to this is the multiparameter theory, standing partway between the one-parameter theory discussed above and product theory: here the emphasis is on operators *For (5.1) we assume also that CI is odd. NOTICES OF THE AMS

which are "invariant" (or compatible with) specified subgroups of the group of n-parameter dilations. The product theory of V began with Zygmund's study of the strong maximal function, continued with Marcinkiewicz's proof of his multiplier theorem, and has since branched out in a variety of directions where much interesting work has been done. Among the things achieved are an appropriate H" and BMO theory and the many properties of product (and multiparameter) singular integrals that have came to light. This is due to the work of S. Y. Chang, R. Fefferman, and Journe, to mention only a few of the names. Finally, 1 want to come to an extension of the product theory (more precisely, the induced "multiparameter analysis") in a direction that has particularly interested me recently. Here the point is that the underlying space is no longer Euclidean R", but rather a nilpotent group or another appropriate generalization. On the basis of recent but limited experience, I would hazard the guess that multiparameter analysis in this setting could well turn out to be of great interest in questions related to several complex variables. A first vague hint that this may be so came with the realization that certain boundary operators arising from the 3-Neumann problem (in the model case corresponding to the Heisenberg group) are excellent examples of multiple-parameter singular integrals (in the work of Muller, Ricci, and the author (1995)). A second indication is the description of Cauchy-Szego projections and solving operators for 3j, for a wide class of quadratic surfaces of higher codimension in C" in terms of appropriate quotients of products of Heisenberg groups (in yet unpublished joint work with Nagel and Ricci). And even more suggestive are recent calculations (made jointly with A. Nagel) for such operators in a number of pseudo-convex domains of finite type. All this leads one to hope that a suitable version of multiparameter analysis will provide the missing theory of singular integrals needed for a variety of questions in several complex variables. This is indeed an exciting prospect.

[1965] , Commutators of singular integral operators, Proc. Nat. Acad. Sci. 53, 1092-1099. [1977] , Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad, Sci. U.S.A. 74, 1324-1327. [1952] A. P. CALDERON and A. ZYGMUND, On the existence of certain singular integrals. Acta Math. 88, 85-139. [1956a] , On singular integrals, Amer. J. Math. 78, 289-309.

[1956b] , Algebras of certain singular integrals, Amer. J. Math. 78, 310-320. [1957] , Singular integral operators and differential equations, Amer. J. Math. 79, 901-921. [1985] S. Y. A. CHANG and R. FEFFERMAN, Some recent developments in Fourier analysis and Hp theory on product domains, Bull. Amer. Math. Soc. (N.S.) 12, 1-43. [1994] C. E. KENIG, Harmonic analysis techniques for second order elliptic boundary value problems, CBMS, Regional Conference Series in Mathematics, vol. 83, Amer. Math. Soc, Providence, RL [1939a] J. MARCINKIEWICZ, Sur Interpolation d'operations, C. R. Acad. Sci. Paris 208, 1272-1273. [1939b] , Sur les multiplicateurs des series de Fourier, StudiaMath. 8, 78-91. [1990] Y. MEYER, Ondelettes et Operateurs, vols. I and n, Hermann, Paris. [1991] Y. MEYER and R. R. COIFMAN, Ondelettes et Operateurs, vol. in, Hermann, Paris. [1965] R. PALAIS, Seminar on the Atiyah-Singer index theorem, Ann. of Math. Study, vol. 57, Princeton Univ. Press, Princeton, NJ. [1967] R. T. SEELEY, Elliptic singular integrals, Proc. Sympos. Pure Math., vol. 10, Amer. Math. Soc, Providence, Rl, pp. 308-315. [1970a] E. M. STEIN, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, NJ. [1970b] , Some problems in harmonic analysis suggested by symmetric spaces and semisimple groups, Proc. mternat. Congr. Math., Nice I, Gauthier-Villars, Paris, pp. 173-189. [1982] , TTie development of square functions in the work of A. Zygmund, Bull. Amer. Math. Soc. (N.S.) 7, 359-376. [1983] , Harmonic analysis, Princeton Univ. Press, Princeton, NJ. [1934] A. ZYGMUND, On the differentiability of multiple integrals, Fund. Math. 23, 143-149. [1943] , Complexmethods in the theory of Fourier series, Bull. Amer. Math. Soc. 49, 805-822. References [1956] , On a theorem of Marcinkiewicz concerning [1950a] A. P. CALDERON, On the behaviour of harmonic funcinterpolation of operations, J. Math. Pures Appl. 35, tions at the boundary, Trans. Amer. Math. Soc. 68, 223-248. 47-54. 11959] , Trigonometric series, 2nd ed., Cambridge [1950b] , On a theorem of Marcinkiewicz and ZygUniv. Press. mund, Trans. Amer. Math. Soc. 68, 55-61. [1958] , Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math. 80,16-36. [1962] , Existence and uniqueness theorems for systems of partial differential equations, Proc. Sympos. Univ. Maryland, 1961, Gordon and Breach, New York, pp. 147-195, [1963] , Boundary value problems for elliptic equations (Joint Soviet-American Symposium on Partial Differential Equations, Novosibirsk, 1963), Acad. Sci. USSR Siberian Branch, Moscow, pp. 303-304.

NOTICES OF THE AMS

VOLUME 45, NUMBER 9

UNIQUENESS IN THE CAUCHY PROBLEM FOR PARTIAL DIFFERENTIAL EQUATIONS.* l By A. P . CALDEKON.

1. Introduction. We begin with a few historical remarks. The classical theorem of Cauchy-Kowalewski asserts the existence of analytic solutions of analytic equations or systems of equations with analytic Cauchy data and their uniqueness in the class of analytic functions. In 1901 Holmgren [6] proved that the Cauchy problem has at most one continuous solution with continuous derivatives up to the order of the equations, provided that the latter are linear and have analytic coefficients. The condition on the coefficients is essential in Holmgren's argument and precludes the possibility of generalizing the result to the non-linear case. The first contribution in the direction of removing the assumption of analyticity was made in 1939 by T. Carleman [2], who proved the corresponding result in the case of two independent variables assuming that the characteristics of the equations are non-multiple. In view of the example of E. deGiorgi [3] (see also A. Plis [10]), who exhibited an example with multiple characteristics for which the Cauchy problem has more than one solution, the condition on the characteristics in Carleman's result no longer seems artificial, but, unfortunately, the restriction to two variables is indispensable to the method of proof. In the case of more than two variables, no essential progress was made until 1954 when C. Miiller [8] studied a special second order equation. Later P. Hartman and A. Wintner [4] and E. Heinz [5] settled the case of quasilinear second order equations with the Laplacian as principal part. (See also P. D. Lax [7]). Shortly thereafter, N. Aronszajn [1] extended this result to general second order elliptic ease. For other results on quasilinear equations with principal parts with constant coefficients, see also L. Nirenberg [9]. In the present paper, we shall generalize Carleman's theorem to functions of any number of variables with two exceptions, the case of three variables for equations and three or four variables for systems. The presence of these •Received J u n e 30, 1957. 1 This research was partly supported by the United States Air Force under Contract No. AF18(600)-685 monitored by the Office of Scientific Research. The author is indebted to P. Olum and G. A. Hunt for help with this paper.

16

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T H E CAUCHY TKOBLEM.

17

exceptional cases is due to topological difficulties in our approach. Whether they are really exceptional or not, we do not know. Since the non-multiplicity of the characteristics plays an essential role in our argument and it does not seem to be possible to introduce additional independent variables in a differential problem without sometimes causing multiplicities to appear in the characteristics, we have been unable to reduce the exceptional cases to the other. In special instances, though, such as that of second and third order equations, our result holds for any number of variables. Our method is based on a representation of linear differential operators by means of singular integral operators. More specifically, if Au is a linear differential operator of homogeneous order m defined on all Euclidean space and with bounded coefficients, then Au = Hh.mu, where A is a square root of the negative of the Laplacian and H is a singular integral operator. For operators such as H, there is an approximate functional calculus which allows us to reduce the problem of uniqueness to a simple form. The first section of the paper will be devoted to discussing the operators H and the representation of A briefly described above. For the proof of the basic properties, we shall refer the reader to a paper by A. Zygmund and the author (see [11]). 2. Singular integral operators. We begin this section with an informal discussion. Consider the real fc-dimensional Euclidean space Ek, complex valued functions u{x) on Ek, and the operator (1)

H(u) =a(x)u(x)

-\- I h(x,x —

z)u(z)dz,

where a{x) is complex, bounded and h(x,z) is complex homogeneous of degree — k with respect to z, that is, h(x,\z) = \~kh(x,z) for A > 0, where kz is the point with coordinates \zu Az2, • - • ,\zk, and z = (zu- • • ,Zk), and h(x,z) has mean value zero on the sphere Sk-i — {| z | = (z t 2 -\- • • • + zfc2)* = 1} for each x. Under mild assumptions on h{x,z), the integral above is meaningful as a principal value integral for u(x), say, square integrable over Ek. If a(x) and h(x,z) are actually independent of x, then Hu can be easily computed by taking Fourier transforms. Specifically, if u(x) denotes the Fourier transform of u{x),x* then (2)

Hu=(a-\-%)u, 2

* In contrast with [11] we use here the Fourier integral with a minus sign in the exponent. This accounts for apparent discrepancies in sign.

A. P. CALDERON.

18

where h is the Fourier transform of h (x, z) => h (z), the Fourier integral being taken as a principal value integral. If h(z) is continuous in | z | > 0, then h(z) exists, is a homogeneous function of degree zero, that is, such that h (\z) = ft (z) for every A > 0, and has mean value zero on S^-i = {| z | =• 1}. Conversely, every sufficiently smooth function with these properties is the Fourier transform of some h(z). The function (a-\-h) in (2) is called the symbol of the operator H and we shall denote it by a{H). It is clear from (2) that operators with a and h independent of x can be multiplied and added by simply multiplying and adding their symbols. Let now Au be a linear partial differential operator with constant coefficients of homogeneous order m. Then (3)

4 « = ji(i)i™tt(i),

where PA(X) is the characteristic form of the operator A. the last identity as •\

(4)

Au=[pA(x)\x\-mim]\x

Let us rewrite

\mu(x)

and define the operator A by (5)

/ \ = I x | u(x). Au

Then since the expression in square brackets in (4) is a homogeneous function of degree zero, it is the symbol of some singular integral operator H, and we obtain Au = <j(H)\x\mu(x) and (6)

=
Au = HAmu,

where the symbol
AM =

PA(X) being the

— A2M,

which justifies the assertion that A is a square root of the negative of the Laplacian. Suppose now that the coefficients of A are variable, that is, (7)

Au =

'2aa{x)Dau, a

where Dau runs over all partial derivatives of u of order m. represent each Dau as in (16) (8)

Dau = Ha&.mu,

Let us now

UNIQUENESS IN THE CAUCH.Y PROBLEM.

19

and substituting in (7), we obtain (9)

Au = 2laa{x)HaA.mu=

C2aaHa)Amu^HAmu,

a

where Hu = ^,aaHa.u = '^aa(x) (IIau). Now as is easily seen, since Ha is of the form (1), so are aaHu and H = 2 a Jig.- Thus we have obtained the desired representation of A in the case of variable coefficients. Let us revert to the operators of the form (1). In the general case, that is, when a and h actually depend on x, we shall define the symbol a{H) of the operator H by v{H) =a(x)

-f- h(x, z),

where h\x, z) is the Fourier transform of h(x,z) with respect to z. The symbol is now a function of two arguments, x,z£ E^, which is homogeneous of degree zero with respect to z. One verifies that the symbol of the operator H in (9) is precisely 'Slaa(x)za\ z\'mim, where za = z1aiz2a2- • • zka", and ~^laa(x)za is the characteristic form of the operator A. The operators of the form (1) do not form an algebra under ordinary composition but it is possible to introduce a product Hl o H2 by means of the formula a{H1 °H2) ~a(H1)a(H2). Under mild restrictions on the operators H, this product is always defined, and with this multiplication the class of singular integral operators becomes a commutative algebra which is isomorphic to the function algebra of their symbols. The interest of this kind of multiplication lies in the fact that it is a fairly satisfactory substitute for ordinary composition of operators which, incidentally, is non-commutative. More specifically, the difference between Ht o H2 and H-JI2 has the following property. Our operators II are bounded on L2(Ek) and A is densely defined in L2(Ek) and unbounded, but H1°H2—S1H2 multiplied on the right or the left by A is bounded. Thus, modulo operators with the property of remaining bounded after multiplication by A, ordinary composition is equivalent to the formally simpler multiplication introduced above. We now proceed to formulate precisely the properties of singular integral operators which we shall need. For their proof we refer the reader to [11] -1* By Op, 1 5= /? rfi 2, we shall mean the class of bounded complex valued functions in d with bounded first order derivatives satisfying a uniform Holder condition of order (3 — 1. If O r £ / ? < l , then Cp will stand for the class of bounded continuous functions satisfying a uniform Holder condition of order /?. By CpK, we shall mean the class of functions depending on two or more arguments which are in C with respect to the last argument and whose

20

A. P. CALDEKON.

derivatives of all orders with respect to the coordinates of this are in Cp with respect to all variables. By L2, we shall mean the class of functions on Ek with integrable square, and Lx2 will stand for the class of functions in L2 with first order derivatives (in the sense of Schwartz) in L2. We shall say that the operator (1) is of type C&°° if a{x) £ Cp and h(x,z) € Cp" in | z | 12: 1. The letter c will stand for a constant not necessarily the same in each occurrence. THEOREM 1. Let / ? ^ 0 he given. Consider the class of singular integral operators H of type Cp". Then Hu is well defined as a principal value integral for u € L2 and Hu is in L2. Further, H is a hounded operator on L2. There exists a one-to-one linear mapping a of the class of operators of type Cp" onto the class of functions F(x,z), x,z€Ek homogeneous of degree zero with respect to z and in Op" in \ z | ^ 1, such that, if NH is the least upper hound for the absolute value of F(x,z) =
WHW^CNH,

where || H || is the norm of H as an operator on L2 and c is a constant depending only on k. 2. If u(x) £ Lt2 and u(x) denotes the Fourier transform then u(x)\x\ belongs to L2. Thus the operator A defined by

THEOREM

of u(x),

Au = u(x)\ x | has domain Lx2, range in L2 (see footnote 3) and is symmetric. THEOREM 3. Let H be a singular integral operator of type Cp" with 1 < /J :§ 2. Let MH denote the least upper bound for the absolute values of a(H) =F(x,z), its derivatives with respect to coordinates of z of order 2k, the first order derivatives of these and of F(x,z) with respect to coordinates of x, and the Holder constants of the latter in \ z | 2 : 1 . Let H* denote the adjoint of H and let H*, Hi o H2 bto defined by (11)

a(H*)=a(H),

a^oH^^a^^a^,).

Then H* and HX°H2 are singular integral operators of type Cp". Further, if H is of type Cp" and u £ Lx2, then Hu and H*u are in Lx2, so that the operators AH, HA, H*A, AH* are defined on Lx2. Further, if u£ Lx. then 2

The function a(E) here is the symbol of H. The definition of the operator A given here is different from the one given in [11]. They are, nevertheless, readily seen to be equivalent. 3

UNIQUENESS IN THE CAUCHY PEOBLEM.

\\(AH — HA)u II ^ cMH || u ||,

\\\(H*~H*)u

21

|| ^ cMfl || « ||,

\\(H*—H*)\u\\^cMH\\u\\, (12

|| K{HxoH2-HiH2)u

|| ^ CiY^Ma, || u ||, \\(HxoH2 — HlHI)Ku

|| ^ c . ¥ f l l 3 / H , || « ||,

where c is a constant depending on k only and the norms are L2-norms. Consequently, the operators in (12) can be extended continuously to bounded operators on L2. THEOREM 4. If Hi and H2 are in Cp™, jSSO and the functions
5. Let A = ~2,aa(x)Da,

where a = ( a 1 , a 2 , - • •,<**), »x-\-<*2

a

+ ' " • + «*-="», and Da = dai*a'*+a*/dxia*dx2a'- • -dx*"* be a linear homogeneous differential operator of homogeneous order m with coefficients aa(x) in Cp, fis^O. If u(x) is in L2 and has derivatives of orders less than or equal to m in L2, then Amw is defined and Au === H\mu, where H is a singular integral operator of type Cp" and (13)

a(H)==

where za = z^z^'

im^aa(x)za\z\-m,

• • • 2&°*- In particular, if u € Lx2, then

(14)

du/dXj = iRjXu = iARjit,

where a(Rf) = Z/ | z \'1.

Conversely, if u$.L2,

(15)

then

iXu = 2 Rjdu/dxf = 2 dRjU/dxf.

In what follows, we shall consider singular integral operators depending on a parameter t and function u(t) depending on t and with values in L2. We shall say that such a function is continuous if ||u(tf) —«(*<>) (I tends to zero as t tends to t0 for all t0. We shall say that u(t) is difEerentiable if there exists a function du/dt*=v(t) of the same type such that \\(u(t)—u(to))/(t

—

to)—v(t0)\\

tends to zero as t tends to t0, for all t0. Our results on uniqueness are essentially applications of the following LEMMA

1.

Let «(<)> 0 = ^ = ^> have values in Lx2, be continuously

22

A. P. CALDERON.

differentiate and be such that Au(t) is continuous. Let P{t) and Q(t) be singular integral operators such that a(P)—F1(t,x,z) and 1 in \ z \ 2 ; l. 4 Assume that P(t) has a two sided inverse for each t or else vanishes identically, and let n(t) = (t-^-X/n)'". Then if M(O) = 0 and (16)

(Hn2\\du/dt+(P-\-iQ)Au\\*dt^c

fn2\\u\

• dt

for some constant c and arbitrarily large n, u(t) = 0 in a neighborhood of * = 0. Proof. We will first derive inequality (23) and then show that (16) and (23) are incompatible for sufficiently large n, unless u(t)—0 in a neighborhood of t = 0. We shall write u' for du/dt etc. and shall omit displaying supscript n of <^,. We have [u'-\- {P + iQ)Au] = [(<#>«)' + iQA$u] + [PA<j>u — 'u]. Squaring, we obtain *• || u'+ (P + iQ)Au r^\\(<j>uy (17)

+ iQA4>u ||* + || PAfu — t'u ||2

— [('u] — [4>'u, (u)'] + [(*u)', PAu] + [PAfu, (4>u)']

-+- i[QA<}>u, PAu] —i[PA4>u, QA<j>u] —i[QA4>u, 'u,QAu], where the brackets denote inner products of the functions involved. Now we integrate between 0 and h and estimate the integrals of the last eight terms. First, we have [(<, V| + [*'«, (<#,«)'] = 2«J>'2 || u ||2 + * * ' [ ( « ' , u ) + ( u , u ' ) L and integrating between 0 and h we obtain

J [ { [(<*>«)', <*>'«] + [*'«, {
Now one verifies readily that 4.4." + '2 = 2<#/2 + <j>'2/n * Note t h a t both Fx and J1, are assumed to be in Cg* as functions of all their variables, t included.

UNIQUENESS IN THE CAUCHY 1'KOBLEM.

and that <j>(h)'{h) \\ u ||2 :§ 0. (18)

23

Consequently, it follows that

( [(u)','u]dt+['u, {4>u)']dt^

— n-1 C <$>'* || u \\* dt.

J 0

J 0

Next we estimate the integral of the terms [(u)',PAu] -f- [PA<£u, (u)']. Let us write v = <j>u and P' for the operator defined by a{P') = (d/M)a(P) (i.e., P' is the derivative of P(t)). Since a(P) € Cfi°°, fi > 1 in | z | S: 1, using (10), one verifies readily that Pv is differentiable and that (Pv)' — P'v-iz-Pv'. Since v is in Lt2, it follows from Theorem 3 that P ^ is in 2 Lt and APv is denned. Further, since Av = A« is continuous, writing «> for Pv, we have {|>(< +

S),A«;(<+

* ) ] - [ * ( < ) > Aw(<)]}/ S

= [Av(t + s), ( » ( * + *) — » ( 0 )/*] + [(»(< + *') — f(0)/«» A w ( 0 ] , and as s tends to zero, the right hand side tends towards (Av, w') -f- (v'} Aw) • that is, (o,Aui) is differentiable and its derivative is equal to the expression above. More explicitly, we have (u, AP<£u)' = [Au,P(uy] + [AM] + [(<£«)', APu]. Consequently, we can write [(u)',PAu] + [PA«, (u)'] = [u, AP<£u]' — [A.£u,P'<£u] _j_ [(P_P*)A4>u, (*«)'] + [ ( * « ) ' ,

(PA~AP)u].

Since P = P*, according to (12), the operators ( P — P*)A and PA — AP are bounded and their norms, as functions of t, are bounded. Also, according to (10), the norm of P 7 is bounded. So if we replace (u)' by '« + u' above and integrate, we obtain J [(<}>u)',PA<}>u]dl -f- [PAu)']df (19)

^*'(fc)[«(fc),AP(*)u(fc)] - c ^ V

II Aw || || u ||

+ | * * ' I I I « P + *,II«IIII«'||]
Next we have

[QA<j>u,PAu] — [PAcj>u, QAu] — [ ( P * 0 — «, Aw] (20)

— [(P*Q — Q#P)Au, A0u] + [A(P* — P#) Q A
24

A. P. CALDERON.

Now, according to (12), the operators A(P*— P*) and A(Q*— Q*) are bounded and their norms, as functions of t, are bounded. Further, since P = P* and Q = Q* and PoQ^QoP w e have (P*Q — Q#P)A=(PQ

— PoQ)x—(QP

—

Q°P)A,

and according to (12), the operators on the right are bounded and their norms, as function of t, are bounded. Consequently, integrating (20), we obtain (21)

f {%[QA<j>u,PAu] — i[PA<£u, QAu]}dt^—c J 0

C tf || u || || AM || dt.

Jo

Finally, %['«, QAu]—i[QAu,'u\ =i$'[(AQ* — QA)u,u], and since Q = Q # , it follows again that AQ*—QA is a bounded operator and has bounded norm. Thus, integrating, we obtain (22)

j {i[>'w, QAu, '«] }dt^ Jo

— c j | 'dt.| || u Jo

Now we integrate (17) and using (18), (19), (21), and (22) and observing that (&) :S h'n, we obtain (23)

f 4>2\\u'+ (P + iQ)\u\\2dt^ Jo

C"\\PA$u — 'u\\2dt Jo

+ n- f >

II « II2 dt-c

f \ \ W | || u ||* + 4,' || u || || Au ||

+ **||u||!|«'II ]««—<*-*», where c is a constant independent of n. Now we will show that this last inequality is incompatible with (161 for sufficiently large n, unless u(t) = 0 in a neighborhood of t = 0. Assume first that P(t) does not vanish identically and set (24)

|%'2l|w||2<« = F ; Jo

C"2\\PAu\\2dt = p2P. Jo

'Then one verifies readily that, unless u(t) = 0 in a neighborhood of t = 0, we have

<25) and h2nP/n2-*ao (26)

r V l | w | | 2 ^ = o(iy« 2 ) as re tends to infinity, which implies that 7r2» = o ( i y n 2 ) .

Since P(t) has a two sided inverse P(t)-1 for each t and is continuous in the norm topology for operators on L2 (this is an immediate consequence of

UNIQUENESS IN THE CAUCHY PROBLEM.

25

(10) and our assumptions on P(t)), it follows that P(t)'1 has bounded norm and, consequently, || Au || fg c || Phu ||, where c is independent of t, which in turn implies that r V l | A « | | 2 d < = 0(P2/2).

(27)

Since P and Q are bounded operators and have bounded norms, for those n for which (16) holds, we have from (16), (25) and (27) r V l K I I 2 < f t = 0(f>/ + / / n ) 2

(28)

Finally, for the same n from (24), (25), (27), and (28), we obtain C || PX4,u -
Jo

44' | 11| u \\*dt - o (P/n), J0

f V || u || || Au || dt = o(pP/n),

Jo

fH4>2 II u || || u' U dt-o(PP/n

Jo

+ P/n2).

Now the right-hand side of (16) is larger than or equal to the right-hand side of (23) for all n for which (16) holds. Hence, for such n, from the estimates (24) to (29), we obtain o(P/n2)

^ (P — 1 ) 2 / 2 + P/n — o(P/n + PP/n + P/n*) —

Now, neglecting the higher order terms o(I2/n) n and dividing by pP we get

and o(P/n2)

o(P/n2).

multiplying by

o(l)^(n(/,-l)2 + l)/P^((/>-l)2 + l)/P>i which is, of course, impossible for sufficiently large n. Jf P(t) = 0 identically, then (23) reduces to f i 2 1 | «' + iQ\u |» dt^

(1 + 1/n) f V 2 II « II2 — c f*| H' I I « II2 dt

Jo

Jo

Jo

and this combined with (16), (24) and (25) gives o(7 2 /n 2 ) ^ (1 + l/n)P

—

o(P/n)

which again is readily seen to be impossible for sufficiently large n. Lemma 1 i8 thus established. LEMMA

differentiable.

(30)

2. Let u(t), Then

0 5 | < ^ f c have values in L* and be continuously

C^Wdu/dtfdt^ Jo

fV'2IMI'<"^»2(i +ft)-2fVII»!'<«• Jo

Jo

26

A. P. CALDERON.

Proof. To obtain the first inequality, we merely set P = Q = 0 in (17), integrate and use (18). The second inequality follows from the fact that ^ , ' 2 ^ 7 i 2 ( l + A)-2<#.„2 in O^t^h. 3. Uniqueness in the case of a single linear equation. Our problem consists in showing that if u € Cm is a solution of a linear homogeneous partial differential equation A (u) = 0 of order m, and u and its derivatives of order less than or equal to m vanish on a non-characteristic manifold M, then u must vanish in a neighborhood of M. The problem is local, and we need only consider what happens in an arbitrary neighborhood of a given point of M. We shall consider the function u only on one side of M, and, for this purpose, it will be convenient to replace u by zero on the other, which will neither disturb the fact that u is a solution of A (u) = 0 nor destroy the assumed continuity properties of u. Further, and without loss of generality, we shall assume that M is of class Cm. The differential operator A will be assumed to have non-multiple characteristics, coefficients of highest order derivatives in Cp, j8 > 1, and remaining ones measurable and bounded. We shall first bring our differential equation to a suitable form and then replace by it an inequality involving singular operators to which, after some manipulation, we shall apply Lemma 1. We shall denote by k -\- 1 the number of independent variables. By introducing suitable coordinates t, xlf x2)- • •, xk, we can assume that M k

coincides with the paraboloid t = | x | 2 — [ 2 a;/] 1 and that M = 0 in t f= | x | 2 , i

and we shall show that u = 0 for t^h, for some h > 0. 2 Since u = 0 in t ^ | x | , we may freely modify the coefficients of A there, and, in fact, we shall replace A by (31)

A' = g0A + {l—

g0)A0,

where g0 is a function in Cx such that, 0 :S g0 ^ 1, g0 = 1 in t 2 : | x | 2 and <7o = 0 in tS? \x\2 — S, 8 > 0, and A0 is the operator obtained from A by replacing its coefficients by their values at i = a; = 0. We note that A' coincides with A at t = x = 0 regardless of the value of 8. By taking S sufficiently small and by restricting our attention to a sufficiently small interval 0 ^ t f= h, we may assume that the oscillation of the coefficients of A' is less than any preassigned number e. Since t= | x \2 is non-characteristic, the hyperplane t = 0 is non-characteristic at t = x = 0. Consequently, we may assume that the coefficient of dum/dtm does not vanish in

U N I Q U E N E S S I N T H E CAUCHY

PROBLEM.

27

a neighborhood of t = x = 0. Hence, the coefficient of dmu/dtm in A' does not vanish for 0 :£ t :£ h, we may write our equation in the form m

(32)

dmu/dtm + 2 A)dum-*/dtm-> + B{u) = 0,

where .A,- is a linear partial differential operator of homogeneous order / involving only differentiations with respect to xu • • • xk, and B(u) is linear homogeneous and of order less than m. Our previous considerations make it clear that the characteristics of (32) can be assumed to be nowhere multiple and the oscillation of the coefficients of the Aj arbitrarily small. We shall regard the function u(t,x) in (32) as a function «(<) with values in L2(Ek). Since u{t,x) has continuous derivatives up to order m, and vanishes for x outside a fixed bounded set, the functions (33)

V;{t) = (iA)m~W-lu/dti-\

;' = 1 , V • -,m,

have values in L^, are continuously differentiable and the At>y are continuous. Now we represent the operators Aj in (32) as in Theorem 5; that is, we set Aj = i'HjA.i, where Hj is the singular integral operator defined by

P)(t,x,z)

being the characteristic form of Aj.

(34)

dvn/dt + 2 HjiAv^j+t

B (u ),

Thus (32) can be rewritten as dvj/dt — iAvi+l = 0,

i

the last equation being an immediate consequence of (33). Now we shall write these equations in matrix form. For this purpose we shall consider m X.m matrices of singular integral operators and regard them as operators acting on m-vectors with functions in L2(Ek) as components. By the norm of such a vector, we shall mean the square root of the sum of the squares of the norms of its components. The meaning of the norm of such a matrix operator is clear, and the way in which the properties of singular integral operators extend to matrix operators is also apparent. Thus, if we write v(t) for the vector of components M<),*>*(<),• • -,vm(t), and w(t) for [0,0,- • -, — £ ( « ) ] , and if »(t) denotes the matrix operator of entries Htj = — J (/ being the identity operator) if j = I -f 1, Hmj = H«_^+i, and remaining ones equal to zero, we may write (34) as (35)

dv/dt +

9l(t)iS.v*~w{t).

Now we will estimate || w ||2 in terms of || v \\2. Since B is a sum of linear

28

A. P .

CALDERON.

homogeneous partial differential operators, each of homogeneous order, applied to the derivatives d'u/dt', representing each of these operators as in Theorem 5, we obtain \\Bu\\2^c

2

|| A»#u/«tt>p —c

2

\\

di\lu/dt'\\2,

and a repeated application of Lemma 2 to each of the terms of the last sum gives f V, 2 1| Bu ||2 ^ c 2

Jo

j

f 4>™2 II dm-i-lA'u/dtm-i-x

Jo

\\*dt,

or, according to (33), f ^ . „ 2 | | w | | 2 ^ ^ c f <j>n21| v ||2 dt. ./o Jo (35), it follows that

From this and

Chn2\\dv/dt + M(t)i\v\\*dt^c f 4n2\\ v \\* dt. Jo Jo In order to apply Lemma 2, we shall replace this inequality by a similar one involving a diagonal matrix instead of 91(t). Consider the matrix a(9l) of entries <j(Htj) and the characteristic polynomial det[Xl-\-a(9t)] of —a(94). Then a simple computation gives (36)

det[A/ + a ( 3 0 ] = \m + Pl(t,x,z)\z\-1\'"~1 +

+ -••+ pm_1(t,x,z)\z\-m+1\ pm{t,x,z)\z\-m,

where pj(t,x,z) is the characteristic form of the operator A, in (32). But as is readily seen, this function coincides on | z \ = 1 with the characteristic form of the equation (32) which was assumed to be non-multiple. Consequently, for each (t,x,z), | z \ > 0, the matrix a(St) has distinct characteristic values Ai,A2,• • -,Am. Let us postpone the discussion of the cases h — 1 and h = 2, and assume temporarily that 1 ^ 3 . If k jS 3, the set of points (t, x, z), \ z | > 0, is simply connected and the principle of monodromy implies that there exist single-value continuous functions \r(t,x,z),k2(t,x,z),• • ,Xm(t,x,z) coinciding at each point with a characteristic value of a(91). Further, since the entries of
UNIQUENESS IN THE CAUCHY PROBLEM.

29

to 2, the same holds for the Ay(<, x, z). Further, the oscillation of the entries of a ( S O can be assumed to be arbitrarily small for each fixed z, and consequently, the same holds for the \){t, x, z). Define now the matrix Ni(t,x,z) by N1(t,x,z) = (aij(t,x,z)), where , aJ a 1 «tf~ A/('> > )'" - Then, since the Xf are all distinct, Nt is nonsingular and has an inverse. Set N = N1-1 and denote by D the diagonal matrix with entries — \ f { t , x , z ) in the main diagonal. Then o(9l)N1'=N1D, and consequently, Na(Sl) =*DN. Let No—>N(0,0,z) and define the matrix singular integral operators, 9l, 9l 0 and 2>, by o-(9l) =N, CT(910) =-No, a (2)) = D . Since N0 is nonsingular, it follows from Theorem 4 that the operator 9l 0 is invertible. Since the entries of o-(9l—9l 0 ) and their partial derivatives with respect to coordinates of z of order 2k can be assumed to be arbitrarily small,5 it follows from (10) that the norm of 9 l — 9 t » can be assumed to be arbitrarily small, and since 9l 0 has an inverse 9 l = 9 l ( i ) has an inverse with bounded norm. Consequently, f fli.2 || tldv/dt J* and J n2lv\\2dt^c (37)

+ 9l MiAv ||2 dt ^ c f V. 2 II dv/dt + 9li\v Jo

f dt

f <*>n21| 9lv \\2 dt, and from (36), we obtain

f tf I 9l di>/<« + HlStiKv \\2dt^c

f V 2 1 | 9lv ||2 d*.

Now 1 in | « | §: 1. Consequently, if we define dft/dt by o^dftl/dt) ^dai^D/dt, then rf9l/d< has bounded norm and (38)

c

hdv/dt = d(chv)/dt—(dcn/dt)v=d(chv)/dt

+ v',

where || 1/ || :S c || v ||, and consequently, also || v' || Sjj c || 9li> ||. Further, since o-(9l)o-(SO =N
= -D2V — o-(2))o-(9l),

we have 9l o # = 2) o 9l and 9 l # t A - = 2>tA9l + 2>t(9lA — A9l) + (2) o9i _ 2 > 9 l ) t A + ( 9 l # — 91 o # ) { A . But according to Theorem 3, the last three terms on the right represent bounded operators, and consequently, we may write (39)

tlSliW

= 2)tA9lt; + v",

* Note that the oscillations of the entries of this matrix and of its derivatives with respect to coordinates of z can be estimated in terms of the oscillation of the coefficients of the equation (32).

30

A. P. CALDERON.

where || v" || ^ c || v \\, and therefore, also || v" || :g c \\ tflv ||. Substituting (38) and (39) in (37), we obtain

(40)

C\n1\\d(cnv)/dt + 3)i\(ctlv)\\2dt^c f ^ ./o ^o

|| 9lv ||2d<

+c r v i ^ I2 M+c r V 2 1V I2 <« ^c r v i %»i2 #. »/0

•'O

»/0

Let now Wi,w2,- • • ,wm be the components of 9lv, and F t -+- i d , ^2 + iQi-, • • -,Pm-\-iQm the entries in the main diagonal if i2), where —fr(Qj) and
(41)

f V

II dwj/dt + (P} + iQ,)Lw, ||2 dt ^ e £ f
Let now ;', 1 5= ;' 5§ w, be an index such that

(42)

r v 1 1 «>, »• * ^ i / » r <^»2 211 «>* 112 <«

for infinitely many n. Then for such an index, (41) yields

f V II dw,/»2 II »/ II2 dt for infinitely many n. Now, as is readily verified, the remaining assumptions of Lemma 1 are satisfied, and thus Wj(t)=0 in a neighborhood of < = 0. But then (42) implies that all the other components of w = 9tv must vanish in the same interval. Consequently, 9lv = 0 for t sufficiently small, whence v = 0 and u = 0 for the same values of t. Thus uniqueness is established, provided k ^ 3. To complete the proof in the cases h = 1 andfc= 2, we still have to construct the functions A,(<, ar, z) coinciding at each point with a characteristic value of 0, has two simply connected components. On each of these, we can construct the functions \j(t, x, z) as we did in the case fc=2 3, and they will turn out to be either real or else have a non-vanishing imaginary part. Then we pair the functions on each component in such a way that X./(t,x,z) = — Xj{t,x,— z)

UNIQUENESS IN THE CAUCHY PROBLEM.

31

and this will yield the functions Ay(i, x, z) over the whole set (t, x, z), \z\ > 0, with the desired property of either being real or else having a non-vanishing imaginary part. If fc = 2, the set of points (t, x, z), \ z | > 0, is no longer simply connected; i.e., the principle of monodromy does not hold, and indeed, it is possible to construct a differential operator for which no continuous single valued functions Xj(t, x, z) exist. For such an operator, our argument can be carried no further and the question of uniqueness remains open. Yet, in a special case which is certainly not devoid of interest, the functions Kj(t, x, z) still exist. This is when there exist at most two complex characteristic values of a-(Si) at each point (t,x,z), | z j > 0. To show the existence of the Xj(t, x, z), we first construct the functions kj(t,x,y) on the universal covering space of the set (t,x, z), \ z | > 0. The same argument used previously shows that either Xs(t,x,y) is real at all points or else its imaginary part does not vanish. Let us denote by Xt and A2 the two complex A,- and assume that at (t0, x0, y0), Im[Ai] > 0,Im[A 2 ] < 0, A3 < A4 < • • • < Am. Since the Ay are distinct at each point, these inequalities hold everywhere on the covering space. Now if {tuxuyx) and (t2,x2,y2) are two points above the same point (t, x, z), then the sets of values of the A,- at these points coincide. Since the inequalities above hold at both (t^x^y^) and (£2, rr2, y2)> it follows that Ay( 0, which proves our assertion. To conclude, we note that there will be at most two complex Ay if the order of the equation does not exceed 3. Consequently, there is uniqueness of solutions of the Cauchy problem for second and third order equations with non-multiple characteristics, regardless of the number of independent variables. Our results on a single equation can be summarized in the following THEOKEM 6. Let A(u) = 0 be a linear partial differential equation of order m in a function u of k variables. Let the coefficients of derivatives of order m be real and in 'C^, /} > 1, and the remaining ones measurable and bounded. Then if the characteristics of the equation are non-multiple,** solutions of the Cauchy problem are unique in Cm, provided that ky^=3 or m<;3. 6

* Although not strictly necessary, here as well as in Theorem 7 we assume the characteristic form to be free of multiplicities in the complex domain.

32

A. P. CALDEKON.

4. Uniqueness in the case of a system of linear equations. With a few changes, the method used in the preceding section can be applied to a system of equations. Assume that we have a system of r equations in r functions u1} u2,- • •, ur. Then we regard the functions us as the components of a vector function u and the system of equations as a single equation in u with r X T matrices as coefficients. After bringing the equation to the form (32), we can proceed as before. The vf in (33) will now be r-vectors and the v(t) in (35) an mr-vector. The matrix 94-(t) in (35) will become an m X m matrix with f X f matrices as entries, that is, an ordinary mr X mr matrix. The characteristic polynomial of — 0, there corresponds a one dimensional subspace of the rar-dimensional complex Euclidean space, namely, the corresponding eigenspaee of the matrix a(91). Our problem is then to select at each point (t,x,z), | 2 | > 0, a non-zero vector from this eigenspace so as to obtain a continuous eigenvector function corresponding to the characteristic value function Xj(t, x, z). Since the entries of a(9l) as functions of (t, x, z) are homogeneous of degree zero with respect to z, it will be sufficient to carry out the construction for | z | = 1. Further, since the oscillation of the coeffi-

U N I Q U E N E S S I N T H E CAUCHY

PROBLEM.

33

cients of the equations can be assumed to be arbitrarily small it will be enough to construct the eigenvector function, denote it momentarily by Wj(t, x, z), for (t, x, z) = (0,0,2), I z I = 1 , and then extend it to all (t, x, z), I 2 I = 1, by imposing the condition that the Hermitian inner product of •Wj(t, x, z) and w,-(0,0,2) be real and positive. Consider the rm-dimensional complex Euclidean space, its unit sphere S, and the collection Prm^\ of all its one dimensional subspaces. Then Prm-i can be regarded as the (rm — 1)-dimensional complex projective space, and S can be thought of as a fibre bundle over Prm^. The fibre over each point of Prm-i is the intersection of the corresponding subspace of our Euclidean space with S, i. e., a 1-sphere; the group is the complexes of absolute value 1 (see [12], page 106). We are given a continuous mapping <j> of the set of points (0,0,2), I 2 I = 1, that is, of S^, into Prm^ and we want to select continuously a vector in S belonging for each (0,0,2) to the onedimensional space corresponding to (0,0,2). For this purpose, consider the bundle over Sk-i induced by the mapping : Sk^ —»Prm-i (see [12], page 47). This induced bundle is a 1-sphere bundle over <S^i with the complexes of absolute value 1 as group. But then, if k — 1 ^ 2 , there exists a cross section (see [12], page 99) ; that is, there exists a continuous mapping 1^(0,0,2) of Sic-i into S such that for each 2 the unit vector ^(0,0,2) is contained in the one dimensional space corresponding to (0,0,2). This mapping can be assumed, in addition, to be in C°° (see [12], page 25). Once the eigenvector functions of the matrix a (SI) corresponding to each characteristic value function Xj(t, x, z) have been constructed, we can obtain without difficulty the matrix iv\ in (43), and from here on the argument proceeds exactly as in the preceding section. If the equations in the. system have sufficiently differentiable coefficients and the functions uuu2,- • •,«r are also sufficiently differentiable it is possible to avoid the topological difficulty discussed above, and consequently, drop the additional restriction k=^=3. We note that if the functions uuu2,- • •,ur and their derivatives of orders less than or equal to m — 1 vanish on a non-characteristic manifold M and can be further differentiated continuously n times, and if the coefficients of the equations have continuous derivatives up to order n — 1, then all the derivatives of the Uj of orders less than or equal to m -\-n — 1 vanish on M. This means that if we replace our system of equations by another obtained from this one by differentiation, the Cauchy data of the new system will still vanish. Consider a linear differential operator A(u) of order m with r X r

34

A. P. CALDEEON.

matrices as coefficients and thought of as acting on vector valued functions M = («!,u 2) - • • ,ur) in the variables x1}x2,• • • ,xk. With the notation used in Theorem 5, let aa be the coefficient of Da, and let ir(A) be the r X r matrix ir(A) =^laaza, where za runs over all monomials in zx, z2, • • • ,zk a

of degree m. This matrix we shall call the characteristic matrix of A. The polynomial in z with functions of x = (xu x2, • • • ,xk) as coefficients given by p(A) = d e t [ 2 < M ° ] coincides with the characteristic form of .4. a

Let now A± and A2 be two operators as above. Then v{A1-\-A2) = TT(.A1) -\-TT(A2) =7^=0 unless Ax-\-A2 is of order less than m, in which case, ir(Ai) + *(A2) == 0. Further, if AXA2 is defined, •7r(A1A2)=ir(A1)ir(A2). If det[Ai — ir(A)] = Ar-f- &iAr"1 + • • • + 6 r -iA+ &r is the characteristic polynomial of the matrix TT(A), then the bj are homogeneous polynomials in 2 of degrees jm with functions of x as coefficients. Since a matrix satisfies its own characteristic equation, we have (44)

rr(A)r + b1*(Ay-* + - • • + br-^(A)

+ brI = 0.

Let now Bs be the differential operator of homogeneous order mj with characteristic matrix bjl (I being the unit matrix). Then (44) implies that

A' +

'ZBjAr-t^R i

is of order less than mr. (45)

Br(u)=R(u)

Consequently, if A (u) = 0, then — {A'-1 +rj?lBjAr-i-1)A(u)

=

R(u).

i

The matrix Br is diagonal and its entries on the main diagonal are all identical and equal to an ordinary (in contrast with the matrix differential operators) partial differential operator, denote it by B, which is of homogeneous order rm and whose characteristic form is d e t [ i r ( 4 ) ] , that is, coincides with the characteristic form of A. In terms of components, equation (45) can be written as B(uf) =Rj(u1>- • -,M r ), ; = 1,2,• • • ,r, where B is of order mr and has non-multiple characteristics and the Rj are of lower order. Now the reader will have little difficulty in verifying that the argument used in the preceding section applies also to systems in which the principal part of each equation contains just one function uh as is the case with the system above. We omit further details and summarize our results in the following

100 UNIQUENESS IN THE CAUCHY PROBLEM.

35

THEOREM 7. Let A(u) be a system of r linear partial differential equations of order m in r functions Uj of k variables. Then if the characteristics of the system are non-multiple:5* i) solutions of the Cauchy problem are unique'in Cm, provided that the coefficients of derivatives of order m are real and in Cp, (1 > 1, the remaining ones are measurable and bounded, and fc = 2 or fc>4; ii) solutions of the Cauchy problem are unique in Crm, provided that the coefficients of derivatives of order m are real and in Om(r-i), the remaining ones are in Cm{r-D, and mr^3 or k=^=3.

5. Uniqueness in the non-linear case. The argument by which one extends the results on uniqueness to the non-linear case is well known. We shall briefly discuss the case of a single equation. Systems are treated in exactly the same way. Let F(x, u, ua) = 0 be a partial differential equation, x denoting the point xx, x2,- • •, xk, and ua running over the derivatives of the function u of order less than or equal to m, and assume that F is of class C3. Let u and v be two solutions with the property that u — v and its derivatives of order less than or equal to m — 1 vanish on a non-characteristic manifold M, and assume that u and v are of class Cm+2. Denote by Fu and Fa the partial derivatives of F with respect to u and u„ respectively. Then F(x, u, ua) =F(x, v, va) = 0 and by the mean value theorem, F(x, u, ua) — F(x, v, va) = (u — v) I Fu [x, v -\- (u — v)s,va + (ua — t>„)s] ds + 1,(ua — va) |

Fa[x,x + (u — v)s,va-\-(ua

— v a )s]ds = 0.

If we now replace u, v, ua and va by their values as functions of x in the integrals on the right, we find that u — v satisfies a linear differential equation for which M is non-characteristic. Further, given the differentiability properties of F, u, and v, one verifies readily that the coefficients of this equation are in C2. From our previous results and assuming non-multiplicity of the characteristics of the equation, it follows that u — v = 0 in a neighborhood of M. M A S S A C H U S E T T S I N S T I T U T E OF TECHNOLOGY. CORNELL UNIVERSITY.

36

A. P .

CALDERON.

REFERENCES.

[1] N. Aronszajn, " Sur l'unicite du prolongement des solutions des equations aux d e r i v e s partielles elliptiques du second ordre," Comptes Rendus, vol. 242 (1956), pp. 723-725. [2] T. Carleman, " Sur un problem d'unicite pour les systemes d'equations aux derivees partielles a deux variables independentes," Arkiv for Mathematik, vol. 20B, no. 17 (1939), pp. 1-9. [3] E. de Giorgi, " Un esempio di non-unicita della soluzione del problema di Cauchy, relativo ad una equazione differenziale lineare a derivate parziali di tipo parabolico," Rendiconti di Matematica e della une Applicazioni, ser. 5, vol. 14 (1955), pp. 382-387. [4] P. H a r t m a n and A. Wintner, " On the local behaviour of solutions of non-parabolic partial differential equations. III. Approximations by spherical harmonics," American Journal of Mathematics, vol. 77 (1955), pp. 453483. [5] E. Heinz, " Ueber die Eindeutigkeit beim Cauchyschen Anfangswertproblem einer elliptischen Differentialgleiehung zweiter Ordnung," Ifachrichten der Akademie der Wissenschaften in Gottingen, no. 1 (1955), pp. 1-12. [6] E. Holmgren, " Ueber Systeme von linearen partiellen Differentialgleichungen," Ofversigt af Kongl. Tetenskaps Akamemiens Forhandlingar, vol. 58 (1901), pp. 91-103. [7] P . D. Lax, "A stability theorem for solutions of abstract differential equations, and its application to the study of the local behaviour of solutions of elliptic equations," Communications on Pure and Applied Mathematics, vol. 8 (1956), pp. . [8] C. Muller, " On the behaviour of the solutions of the differential equations u' = F(x, u) in the neighborhood of a point," Communications on Pure and Applied Mathematics, vol. 7 (1954), pp. 505-515. [9] L. Nirenberg, " Uniqueness for the Cauchy problem for differential equations with constant leading coefficients," to appear in the Communications on Pure and Applied Mathematics. [10] A. Plis, " T h e problem of uniqueness for the solution of a system of partial differential equations," Bulletin de V Aca.dem.ie Polonaise des Sciences, vol. 2 (1954), pp. 55-57. [11] A. P. Calderon and A. Zygmund, " S i n g u l a r integral operators and differential equations," American Journal of Mathematics, vol. 79 (1957), pp. 901-921. [12] N. Steenrod, The topology of fibre bundles, Princeton, 1951.

102 COMMUTATORS

OF SINGULAR

INTEGRAL

OPERATORS*

B Y A. P. CALDERON D E P A R T M E N T O F MATHEMATICS, U N I V E R S I T Y O F CHICAGO

Let

Communicated by A. Adrian Albert, March 26, 1965

A(f) = lim f <—0

k{x -

y)f(y)dy,

J\x-y\>c

where x, y are points in n-dimensional Euclidean space Rn and k(x) is a homogeneous function of degree — n with mean value zero on \x\ = 1, and let B(J) = b(x)f(x). I t is well known (see ref. 1) that if A; and b are sufficiently smooth and b is bounded, then (AB — BA)(d/dx,) and (d/dxj)(AB — BA) are bounded operators in V, 1 < p < °°. The purpose of the present note is to extend and strengthen the preceding result and establish some related facts of independent interest. These are stated in Theorems 2 and 3 below. THEOREM 1. Let k(x) have locally integrable first-order derivatives in \x\ > 0, and suppose that the partials of k(x) + k( —x) belong locally to L log + L in \x\ > 0. Let b(x) havefirst-orderderivatives in Lr, 1 < r < <». Then ifl, g _ 1 = p _ 1 + r~l andf is continuously differentiable and has compact support, we have \\(ABBA)^f\\,
Furthermore, (AB — BA)f has first-order derivatives

(AB -

BA)f

\aXj

< cf\P,

(b)

where, again, c is independent off. THEOREM 2. Let h(x) be homogeneous of degree —n — 1 and locally integrable in\x\ > 0. Let b(x) have first-order derivatives in U, 1
Hx-

y)[b(x) -

b(y)]f(y)dy.

Ct maps L" continuously into L" and \\C(J)\\t < c|jgrad &||r||/||j, S \h(x)\dv, where the integral is extended over \x\ = I, dv denotes the surface area of \x\ = 1, and c depends on p and r but not on e. Furthermore, as e tends to zero Cf(f) 'converges in norm in L". A similar result holds if h(x) is odd provided that it belongs locally to L log + L in \x\ > 0 and that the functions Xj h(x), j = 1,2, ..., n, have mean value zero on \x\ = 1. This, however, will not be proved in the present note. THEOREM 3. Let F(t + is) be analytic in s > 0 and belong to Hv, 0 < p < ». Let S(F)(t) = {f x(t — u,s)\F'(u + is) \ 2du ds ]v', where x(t,s) is the characteristic 1092

103 VOL. 53, 1965

MATHEMATICS: A. P. CALDER6N

1093

function of the set s > 0, \t\ < s. Then there exist two positive constants C\ and c2 depending on p only, such that ci\\F(t)\\v < \\S(F)\\„ < cz\\F(t)\\p, where F(t) = Km «-M)

F(t + is). The novelty in the preceding statement is the first inequality for p < 1. A similar result for the function g of Littlewood and Paley when F has no zeros was proved by T. M. Flett (ref. 3), whose method we borrow partially. Actually, only the case p ^ 1 will be needed in this note, but its proof is no less laborious than that of the general case. Proof of Theorem 8: We will assume first that F(t + is) is analytic in s ^ 0 and that \ F\ (i2 + s2)k-* 0 a,s (t2 + s2) — » for every k > 0. Then, of course, F belongs to H" for every p > 0. We introduce now some notation. For a function G defined on the real line we write

M„(G) = J

G" dt\ ,

p > 0.

If G is also denned in the upper half-plane, we write m(G) = sup x(t - u,s)\G(u,s)\,

S(G) = [fx(t

- M,s)|grad G\ Hu ds]''*,

where x(t,$) is the characteristic function of the set s > 0, 11\ < s. By integration we obtain M22[S(G) ] = 2 J" s\ grad G\ 2dt ds. Now if 5 is any positive number, we set G = 1^1*, then a simple calculation gives A(G2) = 4|grad(?j 2 and an application of Green's formula yields M22(G) = 4 f

(0)

4

s\ grad Q\ 2dt ds = 2 M22[S(G)].

(1)

On account of the definition of G and the analyticity of F, we have the following well-known inequality Mv[m(G)} < c MP(G),

0 < p < «>.

(2)

Now let p J? 1, then S(G")2 = f

x(t

-

u , s ) | p G p - ' grad G\2du ds <

p2m{G)2p-2S{G)2,

that is, S(G") < pm(Gy~lS{G),

1 < p < c°.

Now let ccfi > 0, 0 < cr < 1, a<j + /9(1 -
Then

S(G)2 = f

f

x(t

~ u,s)|grad G\ 2du ds = c-fp-w-*)

(3)

( x |grad Ga\ 2)° {x\^&AG^\2Y-'duds,

whence from Holder's inequality we obtain

S(G) < [ i 5 ( G - ) ] ^ SiG^Y".

(4)

104 1094

MATHEMATICS:

A. P. CALDER6N

PROC. N. A. S.

Let us assume now that we have the inequality cMT(G)>MT[S(G)] for some r, r > 0. Let 0 < q < r and p = r/q.

(5) ilP

Then (3) applied to G

S(G) < pro(G1'1,),,~1<S(G!1/,,) =

gives

pm(G)i'-wS(&l»),

whence, applying Holder's inequality, we get < p*MilmWv-vi'SiG1")*]

Mt'[S(G)]

< p*M r/f [5(G l ")»]M r /( f ^,)[m(G)«»-»'»] = p« Mr<'[S(G1'p)]Mq[m(G)]

and from the last expression, (2), and (5) applied to GllP it follows that Ma<[S(G)] < cpq Mr«[G1»']Mg<'ip-»'p(G) = cp* M9"p(G)M"'(G) or

M,[S(G)] < cgMQ(G).

(6)

On account of (1), (5) holds with r = 2. Hence the preceding inequality holds for 0 < q < 2. Now we will show that (6) holds for 0 < q < a>. Since (5) implies (6) with q < r, it is enough to show that (6) holds for q > 4. Let h(t) > 0 be any bounded function with compact support. Then S(G)2hdt = |

f J

— at

h(t) f

x(t

~ M,s)jgrad G\Hu ds dt

%J — co

/

+ "> h(t)x(t - u,s) dt du ds.

Now we observe that if P(t,s) denotes the Poisson kernel for the half-plane, then x(M) — c sP(.t,$) a n d consequently I m/

h(t)x(t - u,s) dt < c\ — CD

«/

h(t)sP(t - u,s)dt
where H(t,s) is the Poisson integral of h(t).

r.

H(u,s),

— CD

Thus,

•+0O

S(G)2h dt < c f

| grad G\ 2s H(t,s)dt ds.

Now, from (0) we have A(G*H) = HAG2 + 2(grad G2) • (grad H) = 4H\ grad G| 2 + 2G(grad G) • (grad H) ^ 4H| grad G\2 - 2G\ grad G\ | grad H\ and

r. •+

S(G)2h dt<-f

sA(G*H)dt ds + '-\f sG\ grad G\ | grad H\ dt ds

105 VOL. 53, 1965

MATHEMATICS:

A. P. CALDER6N

1095

and applying Green's formula to the first term on the right 4 G2h
S(G)*hdt< — oo

rr U

— oo

+ - I 4

<

- f

dt S x(t — u,s) G|grad G\ (grad H\du ds

%/ — oo

G2/i d/ + - f

4 J-m

m(G) S(G) S(H)dt.

4 J -co

Now we set p = q/(q — 1) and apply the three-term Holder inequality with exponents 2q, 2q, p to the preceding integrals and get 4 f

S(G)*hdt < cM2q*(G)Mp(h) + cM2q[m(G)]M2q[S(G)]Mp[S(H)].

(7)

• / — oo

Since H is harmonic and 1 < p < m, we have MP[S(H)] < cpMP(h), and since 4 < g < o>; we also have M2q[m(G)] < cqM2q(G). Substituting in the preceding inequality, setting Mp{h) = 1, and taking the supremum of the left-hand side over all such h, we find that Mq[S(Gy] = M2q2[5(G)] < c M2q(G)[M2q(G) + M2qS(G)], and this implies that M2q[S(G)] < c' M2q(G) provided that M2q[S(G)] < » . To see that this is the case we observe that since m(G) is bounded, (7) holds with M„[m(G)] replacing M2q[m(G)] and Mq[S(G)] replacing M2q[S(G)] and from this, arguing as above, we obtain M2tnS(G)]

< cM2q\G)

+

cM„[m(G)]Mq[S(G)].

Since the right-hand side is finite for q = 2, it follows by induction that the lefthand side is finite for arbitrarily large q and hence for all q > 2. Thus (6) is established for 0 < q < °°. Now we prove the converse inequality. Let q > 0. Then (1) and (4) give Mq«(G) = ilf,*(G»/*) = 2M2iS(G"2)

< cM,^^")1'^^")'"-')],

where a = 2q/(q + 2), 0 = 2/q,
Mq<(G) < But

= M(5+2>/s2"[S((?""2)]

M^+1)lq[S{G"^y} MQ+1[S(G)W-^]

= M t 2 <'-" [5(G)].

Applying (6) to the right-hand side of the first of the preceding identities, and observing that M(q+2)iq[Ga"n} = Mqaqn(G), substitution in the preceding inequality yields Mq"{G) <

cqMqa°<(G)Mq™-°)[S(G)].

Since q — aaq = 2(1 —
(8)

106 1096

MATHEMATICS:

A. P. CALDER6N

PKOC. N. A. S.

To obtain (6) and (8) for F we set G = \F\ and observe that |grad G| = | F ' | . Finally, we must remove the conditions we imposed on F at the beginning of the proof. If F(z), z = t + is, is analytic in the upper half-plane and belongs to H", then F{z + i/n) = F„(z) is bounded there. Let now em(z) = exp( — z"m), where 0 < a < 1/4 and arg(z") is between 0 and ir/4. Then a simple calculation shows that

1

\em'(t + is)\2dtds

< c2a,

where c is independent of m. Consequently, S(em)2 < c2a. inequalities can be readily verified: S(Fnemy

< 2{S(Fn)2 +

< 2{S{Fny + m(FnyS(emy] S(Fn emY < 2"[S(Fny

Now, the following c*m(Fnya]

2

+

c"m{Fnya"i ].

Integrating we get M/[S(Fnen)] Since M/(Fn)

= lim M/(Fn

< 2>[M,'[S(Fn)]

+

em) and by (8), M/(Fn

Ca"i2M/[m{Fn)]}. em) < c / M/[S(Fn

em)\ from

m

the inequality above we obtain M/{Fn)

+ c p f t "/W/[m(F B )]],

< c / 2"[M/[S(Fri)}

and letting a tend to zero Mp{Fn) <

cp2Mp[S(Fn)].

Finally, as n tends to infinity, Mp(Fn) converges to MP(F) and S(Fn) increases and converges to S(F). Thus we can pass to the limit in the preceding inequality and obtain half of the desired result. To obtain the other half we observe that, since (^n em)' converges to Fn', we have S(Fn) = limm inf S(Fn em). Thus from (6) applied to Fn em and Fatou's lemma we get M,[S(Fn)]


and a passage to the limit completes the proof of the theorem. Proof of Theorem 2: We begin with the one-dimensional case. becomes simply x~2, and the proof reduces to estimate f

Ct(J)gdx = f

(x-

y)-2[b{x) - b(y)]g(x)f(y)dx

Here h{x)

dy

in terms of the norms of/, g, and b'. For this purpose there is no loss of generality in assuming that these functions are infinitely differentiable and have compact support. Let e(x) be the characteristic function of x > 0 and x(x) that of | x\ > e. Then b(x) = 1

e(x -

t)b'{t)dt,

and substituting, the integral above becomes

X

b'(t) f

(x - y)-2x(\x

- y\)[e(x - t) - e(y - t)]g(x)f(y)dx dy dt

107 VOL. 53, 1965

MATHEMATICS:

A. P. CALDER6N

1097

and the problem reduces to studying the class of the function represented by the inner integral. For this purpose we let z be a complex variable and set /*(*) = 1T- (

—'—/(*)<&,

3 = 1 if Im(a) > 0,

j = 2 if Im(z) < 0,

and define similarly gj(z). Then we have/(a:) = fi{x) — f2(x) and similarly for g. Furthermore, the f, belong to H", 1 < p < °°, in the corresponding half-planes and, with the notation of the preceding proof, we have Mp(fj)
l
(9)

Corresponding relations hold also for g and g,. We will study the contribution of /i to the integral in question, an analogous argument being applicable to f2. Let us introduce the following kernels K0(x,y,t) = (x - y)-*x(\x

- y\)[e(x - t) - e(y - t)] 2

Ky(x,y,t) = (x - y - ic)- [e(:r - f) - e(y - t)] K2{x,y,t) = l(x - t)* + {y - 0* + e 2 r' A
KJ(x,y,t)g(x)f1(y)dxdy

k2(t) = / K2{x,y,t)\g(x) fi(y)\dx

dy.

We are interested in estimating ko. On account of the inequality between the Kj stated above, we have \k0\ < | fci| + ck2 and thus it will suffice to estimate fci and k2. On account of the analyticity of fx(y) if x > t we have + •»

/

rt

(x - y - ie)-2fi(y) dy

-r

Ki(x,y,t)fx(y)

dy = I

— no

« / — oo

[(Z + is) -

(x -

*'e)]-2/i(< + is) d(is).

J s= 0

As readily seen, for x < t the integral on the left above is also given by this last expression. Thus, fa(0 = -

f


J —a>

J

[(* + is) -

(x -

H>]-2/I(<

+ is) rf(is),

S=0

and interchanging the order of integration we get hit) = - f

/i(< + is) I

J s =0

[(< + is) - (a; - ie)]~2 g(x)

dxd(is).

J —a>

Since gr(x) = gri(a;) — gr2(aO and 92(2) is analytic in Im(z) < 0, its contribution to the inner integral above is zero and the value of this reduces to 2irig\'(t + is + it). Thus we have fci(0 = - 2xt I /x(f + is) £d'(f + is + te) d(is). J«=o Let us introduce now

108 1098

MATHEMATICS:

F{z) = - 2T* f

A. P. CALDERdN

PHOC. N. A. S.

hiz + is)gt'(z + is + u)d(is).

Js = 0

Then we have fci(t) = F(i). Furthermore, since / i and 3 / are bounded and 0(z~l) and 0{z~2), respectively, F(z) belongs to Hp, p > 1, and with the notation of the preceding proof we have (2T)- 1 S(F) < mVdSigAz and if q~l = p ~ l + r - - 1 ,1 < p, q< °=>, r < Mr,^{h)

= MTh^{F)

< cMHr^[S{F)}

+ U)) < ro(/i)<S(ffi) ro

, then by Theorem 3 and (9) we have

<

cM,[m(fd]M
< c Mp(fi)Ml/q-i(g{)

< c MP(f)Mg/^(g).

(10)

Now we estimate k2. We have f

K,(x,y,t) |/(2/) I dy< t[(x - t)* + e 2 ]" 1 sup <S2 f

"[(y - r)2

+ 5 2 r V ' | / ( 2 / ) | * / < c e[(x - t)* +

**]-%),

where / is the maximal function of Hardy and Littlewood associated with | / | . Consequently, \Mt)\

< ef(t) sup ef 6

Mr/^ih)

~[(x-

t)* + a 2 ]- 1 \g(x)\ dx < cf(t) g(«).

• / — 00

< cMv(])Mqli^{g)

<

cMv(l)MQlt^(g).

This combined with (10) shows that Mr/^&o) < c MV(J) M„it-i(g) where c depends on p and r but not on e. As readily seen, this implies that Mt[Ct(j)\ < cMTQ>')

M,(f). We now pass to discuss the n-dimensional case. As before, we assume that / and the partial derivatives bj of b are infinitely differentiable and have compact support. We denote by v a unit vector in Rn and by E its orthogonal complement and fix e, e > 0. Let s be a real variable and k(x, v) = I

h(v) s-2[b(x)

- b(x + vs)]f(x + vs) ds.

Jls\>e

Then setting y = x + vs, integration in polar coordinates shows that C.(/) = 'A f

k(x,y) dv,

(11)

where dv denotes the surface area element of the unit sphere in Rn. We now fix v and set a; = z + vt, where zeE. Then from the inequality for the one-dimensional case established above we get f

k(z + vt,vydt < c\ I

jgrad b (z +

vt,v)\TdtX''

X

u:>

\f(z + vt,v)\* dt

iip,

\h{v)\.

Integrating with respect to 2 over E and applying Holder's inequality to the righthand side, we obtain

109 VOL. 53, 1965

MATHEMATICS:

A. P. CALDERON

c [f\&adb\r

[f\k(x,v)\°dxyi*<

dxY"[f\f(x)\'

1099

dx]v> \h(v)\.

From this and Minkowski's integral inequality applied to (11) we obtain I | a ( / ) | | 5 < C | | g r a d b | | r | | / | | p / | f t W | dv, where c depends on p, q, and r but not on «. Concerning the convergence of Ct(J) as e tends to zero we merely observe that our assertion obviously holds if/and the bj are assumed to be infinitely differentiable and have compact support, whence the general case follows from the inequality above by approximation. Proof of Theorem 1: Since (b) can readily be obtained from (a) by duality, we shall only prove the latter. Let us consider first the case when k(x) is an odd function. There will be no loss in generality in assuming that k(x) is infinitely differentiable in | x\ > 0 and that / and the b, are infinitely differentiable and have compact support. Let fh bit and kj denote the jth. partial derivatives of /, b, and k, respectively. Then integration by parts yields f

k(x - y) [b(x) - b(y) ]fj(y) dy = +

f J

f

k{x - y) b,(y) f(y)

d-y

k,(x - y)[b(x) - b(y)]f(y) dy \x-y\>t

- f

k(ve)[b(x) - b{x + ve)]f{x + v^t"-1

dv,

where v5 denotes the j\h component of the unit vector v and dv denotes the surface area element of the unit sphere in Rn. Now, the first term on the right represents an ordinary truncated singular integral and its norm in Lq can be estimated in terms of the norms of bj and / . To estimate the norm of the second term we use Theorem 2, and in the last term we replace b(x)— b(x + ee) by »i

Jo

2ibj(x + tve)vj€ dt

and apply Minkowski's integral inequality to the resulting integral. Collecting results and letting e tend to zero, (a) follows. In the case when k(x) is even, the operator A can be represented as a finite sum of operators of the form AiA2 where A\ and A% have odd kernels and satisfy the hypothesis of the theorem (see ref. 2). Since d/dXj commutes with A-i, we have {AxAtB - BArAt) ~ = Ar(A2B - BA2) ~ + (A1B - BAJ — At, ax,ax, ax, since Ai and A2 are bounded in L" for every p, 1 < p < oo} the desired result follows. * This research was partly supported by the NSF grant GP-3984. 1 Calderon, A. P., and A. Zygmund, "Singular integral operators and differential equations," Am. J. Math., 79, 901-921 (1957). *Ibid., "On singular integrals," 78, 289-309 (1956). 3 Flett, T. M., "On some theorems of Littlewood and Paley," / . London Math. Soc, 31, 336-344 (1956). 4 To obtain Green's formula for the half-plane under our assumptions we apply it to n cos(n_1i) sin(ra-1s) and the function Gl or G2H over the square — » | ^ t fs »f, 0 < s < rnr, and let n tend to infinity.

110 Proc. Nail. Acad. Sci. USA Vol. 74, No. 4, pp. 1324-1327, April 1977

Cauchy integrals on Lipschitz curves and related operators (commutators/singular integrals/weighted inequalities) A. P. C A L D E R O N Department of Mathematics, University of Chicago, Chicago, Illinois 60637

Contributed by A. P. Calderon, January 12, 1977 ABSTRACT In this note, we establish certain properties of the Cauchy integral on Lipschitz curves and prove the Lp~ boundedness of some related operators. In particular, we obtain the recent results of R. R. Coifman and Y. Meyer [(1976) "Commutateurs d'integrales singulieres:" Analyse harmonique d'Orsay n° 211, Universite Paris XI] on the continuity of the so-called commutator operators.

uniformly in A. Expressing k in terms of h, and using the uniform boundedness in L2 of the truncated Hilbert transform and Minkowski's integral inequality, we obtain the desired result. Our goal is to estimate the norm of B(\) in terms of A(A) and M, that is, a bound for v'{t). This in conjunction with Eq. 1 will give us an estimate for the norm of A(A) in terms of M alone. Let 0\ and Oi be the open subsets of the complex plane consisting of the points lying above and below the curve V, respectively. With a f u n c t i o n / ( / ) in Co™, we associate the functions Fi(io) and F2{w), analytic in 0\ and O2, respectively, given by

1. T h e C a u c h y integral T H E O R E M 1. Let T be a curve in the complex plane given by the equation z(t) = t + itp(t), where tp(t) is a real-valued function on the real line with a bounded derivative, and let

AVJ:

M

-dz(s), (>0 z(s)-z(t)" Then there exists a positive number a such that || ip' || „ < a implies that the operator sup(\ A^J\ is of weak type (1,1) and bounded in LP, 1 < p < °°, and that lim t —oA^f exists pointwise almost everywhere for f in L p , 1 < p < °°. Proof: We shall first consider the case in which
2xi Jr\*-t\>< \,-

Ax«/ = A(X)/ = - M i m

/(»)

f

/|.-(|>.z*(s)-zx(0 2x1 ,—0 J|s-!|>< where z x (() = 1 + iX^((), 0 < X < 1, and it), 0 < X < I)-*')] ;~6 J |fi - s | > . L [[z -» (^s ) - z * ( ( ) F B(Xtf = 2xi -i-lim

F)(w)

I

B(s)fds,

f e

Co-

[I +

/<*)

•dz(s), z(s) — w

wGOj.

[2]

[3]

/ ( ( ) = F,(z(t))

+

FMt))

!|F/(z(«))ll2s(i+l|A(l)ll)l|/||2.

(4)

We now introduce the operator

Jhrtt)- ds)} 2xt

J-~

{z(s)-z{t)-ibfJ

-JMdz(s), & > 0,

/ G C0"

As 8 —- 0, C 5 / converges in the mean of order 2 to

i -lim ,. rf

fMQ-Ws)],fMdz(s)

>,_0 J|«-i|>< \z{S)-z(t)f

2xj

[s)dzx(s).

-^'(Oz'tO-'/W so that B ( l ) / = l\m CJ + j-o 2

-^(t)t'(tr'f(t) + A{\W(t)z'(t)-\j(t).

[5]

To estimate the norm of the operator Cf = lim^ „oC?>f, we consider the bilinear form

il]

On the other hand, A(A) is uniformly bounded in L . To see this, we write the kernel of A (A) as 1

J-™

Consequently we have

2

s- r

r

F/(z(0) = | / ( t ) ± A ( l ) / .

They are well-defined, at least f o r / E Co™, and in this case A(X)/ and B(\)f are continuous functions of t. The operator B(A) is obtained as the formal derivative of A(A) with respect to A, but since for ( > 0 the integral in the definition of 6(A) is the derivative of the one in the definition of A (A) and they converge uniformly as « —*• 0, we have indeed A(A)/ = A ( 0 ) / +

2-rri

It is not difficult to see that these functions extend as C * functions to the curve V and that

d*M,

(zx(s) - zx( t))zx>(s)\n'

. (-D'" +1

Lij,g)= C + ~gCfdz(t) = Um C+"gCJd>'z(t), g £

k(\,s,t))

Setting

where, as is readily verified, Jr(\,s,() is infinitely differentiate and has a double Fourier transform h(\u,v) which is integrable

•At)1324

x:

e(t — u)(ff(u)du

Co".

Ill Proc. Natl, Acad. Sci. USA 74 (1977)

Mathematics: Calderon where e(t) is the characteristic function of t =i 0, substituting in the expression for C&f above and interchanging the order of integration we obtain

[z(s) - z(t) - ib}2 To calculate the inner double integral, we consider the functions Fj(w) and Gy(w) associated with / and g as in Eq. 2 and replace / and g by

1325

Evidently, the same inequality holds for 4>'{z) l, and multiplying we find that

[ § |0'(OIA][ f; |0'(OI-'^] < *2\i\m + M2) that is, the function | 0 ' ( O | belongs to the class A 2 of Muckenhoupt with constant TT2(M2 + 1). Returning to the function Hi, let us consider the functions Cl(0(3)), Fi((j>(z)), H\{(t>{z)) in the upper halfplane a n d the associated maximal and Lusin functions m(t) = s u p |Gi(<Mi + « + to))| |ll|
/ ( s ) = F 1 (z(*)) + F 2 (z(5)) g(t) = G1(z(t))

+

/-•

S(')

G2(z(t)).

Observing that

X

—

I W

= I J\u\
12

h - F i ( < M ( + « + 10)) rfudo \dz I

S(02 = f [-r J\u\
1 [2(s)-2(()-t«r

W

i<^( + " +

iv))\dudv. I

Because _ J2iriF,'(z(() + ii) 10

ifj = l if j = 2

and similarly for C ; , and denoting by T u t h e p a t h z = * + i ^ ( 0 , f > u, we obtain

^-H1(0(«)) = G1(«(z))^-F,W2)) dz dz we have S(r) < m(t)S(t)

and

f_*"s(0k'(*)l
L(/,g) = lim

-[xr m
X £ * " •
-ib))dzdu.

Because , we can pass to the limit under the integral signs above. Thus, denoting by H\{z) and H2(z) the analytic functions in 0\ and O z , respectively, such that H i ' = F / G i , H 2 ' = F2G2', and tending to zero at infinity, we find that L{f,g)

2

= -i £ *

But the function | 0 ' ( O | is in the class A2 and consequently (see ref. 5)

f*J

\HMt))\\*'(*)\dt = fl~ \Hd'(t)\dt '(t)\dt

J*_*" m(t)*\<M)\dt < CM J I" \GdMt))\2Wt)\dt

V(«)[FI(2(M))C2(Z(U))

= CM f

+

"' \GMtW\z'(t)\dt

+ #1(2(11)) 4- H 2 (a(u))]rfu. [6] We proceed now to estimate the functions H,. For this purpose we let 0 conformally onto 0\ and such that <£(z) ^ z a s z - ^ °°. Because the boundary of Oi is a C™ curve, 0{z) and <£'(z) extend continuously to R(z) > Oand 0'(z) does not vanish in R(z) > 0 and is bounded there. Furthermore, the vector '(t) is tangent to the curve T at the point 4>{t) and therefore

f

+

~ S(O2|0'(O|A i cM f

+

~ |F,M0)| 2 k'(0|d«

=cM

Xr |Fi(z(,))|2|z ' (f)| ' i '

in which CM is a constant depending on M and, as above, z(t) = t + \
|arg0'(f)| ^ arctanM.

\\HMt))\\&cM\\FMt))U\GMt))h,

Taking the logarithm of 4>'{z) and using the m a x i m u m principle, we find that the preceding inequality is satisfied at all points of R(z) > 0. As is readily seen, this implies that

where c « is another constant depending only on M. Clearly, a similar inequality is valid for H 2 . Thus Eqs. 4 and 6 yield

0 < R(*'(z)) < \4>'{z)\ < R('(z))(\ + M 2 ) 1 ' 2 Now let I be an interval (s —a,s + a). Then, since fl(
R((f>'(t))dt < 7rfl(0'( s + ia)\

and this combined with the preceding inequality gives |yy

f( \*'{t)\dt

< n\4,'(s

+ ia)\(l

+ M2)'/2.

|i.(/,g)|sc„(U|Aa)«)2«/i!2flg«2, which implies that IIC||
112 1326

Mathematics: Ca\6erbn

Proc. Natl. Acad. Sci. USA 74 (1977)

Evidently, the preceding argument and this last inequality are valid for A(\) and B(\), 0 < X < 1. Thus, Eq. 1 implies that

ce(t)<

|5(t)| < c - } t ( 0

l|Ati)/||p
[10]

Kp<<»

[11]

|A< 2 >/-A< 3 >/|
U(X)I < \\A(0)l + cM J ^ (^ + |Uwr)
[7]

where, as above, the norms are norms of operators in L 2 . But A(0) is just half the ordinary Hilbert transform so that ||.A(0)J| = lk- Consequently, the function | A ( X ) | , 0 < X < 1, is majorized by the solution of the differential equation /l

\2

[12]

where the constants c in Eqs. 9, 10, and 12 can be taken so as to depend only on the function TJ, and the one in Eq. 11 so as to depend on this function and on ll^'IU, and where m(J) is the ordinary Hardy-Little wood maximal function of / . Furthermore, we have

UMf - A<»flp < cWl.

1

1

1131

with c depending only on p and the function rj(t). To see this, we write

that is ||A(X)| ^ ( l - C i i X ) - 1 - ^ ,

0<X
(A">/-A<3'/)(«)=

1.

Let now a = supMcM ~', where the s u p r e m u m is taken over all M ^ 0 and the corresponding constants c ^ , CM > 1, for which Eq. 7 holds with M„ < M. Then if M j „ < a, and JI^'JL < M c M ~ 1 , setting X = | | ^ | U M - 1 < C M _ 1 < l a n d ^ = X<^, the preceding inequality becomes llAyll < (1 - c „ X ) - ' - i = (1 - U ^ I L M - ' C M ) -

i,

and letting M c M ~ J tend to a we finally obtain llA^U S ( l - l l ^ l U a - ' ) " 1 - | ,

ll?ll-
[81

To complete the proof of our theorem, we shall henceforth consider only functions tp with j|
X,-,o|>S |A ' (/A ' )(,)l '" Scll/ " 1 Hence, from Eq. 8 and Theorem that IU^(//z')llp
1 in ref. 1 we conclude K p < 2 ,

A " » / = J"8(f)-'!)(« - s M l ) - ' ) ( A j A ' ) ( s ) d 2 ( S ) A'2)f= S{1 - 7((r - s)r\t))\(z(s) z(t))-
,

, (z(s) -

z(t)rlf(s)ds,

where JJ(/) is an even non-negative function in Co00 which equals 1 near the origin, e(t) is an arbitrary positive measurable function of t and

s«)= J\(«-sMf)-')
X-.o|>J A(2,/|di - dl/L provided that J / d ( = 0 and / ( ( ) = 0 for 11 - 10| > - p

k(t,u) = SHt)-'v((t =

-

s)t(t)-i)(z(u) -z(S))-'dz(s)-(z(u)-z(t))-» j5(()-',(((-sW()-i)((Z(U) - z(s))-< - (z(u) z(t))-i)dz(s).

Because \z'\ < 1 + a and a is finite (see the remark below), this last integral is readily seen to be majorized by cc(t){t — u)~2 with c depending only on the function rj. If on the other hand, t(t) <\t-u\< 2e(r), then

k(t,u)= f

XtrlM(t - sW)-*)

,

%J \s—t\

<2i{t)

-vdt-uMtr^izi^-zit^dzis) + 6(t)~Wt-uHt)-i)\im

f 6—0

X (z(u)~

|91

J&<\s-t\<2t{t)

zis^dzis)

+ (z(u) -

z(t))~l

and estimating the first integral in terms of rf and taking Eq. 10 into account, we find that in this case k(t,u) is majorized by c t ( ( ) _ 1 - Finally, if \t — u\ < t(t) we have the same preceding expression for k(t,u) with the last term omitted so that the same estimate as in the preceding case holds. From all this there follows |fc
with c depending only on p and li^LU. Now, the standard duality argument shows that the same result holds for 2 < p < °°. Consequently, A^ is bounded in LP, 1 < p < °°. Next consider the following operators with ip still in CQ°:

A™/ =

Sk(t,u)f(u)du

and for simplicity assume that TJ(() = 0 for J(j > 1 and TJ(/) = l f o r \t\ < % Then if |( - u\ > 2e((),

((f) 2 )- 1 ,

which implies that |AU>/-A<3>/|


where m(f) is the Hardy-Littlewood maximal function of/, and this in turn clearly implies Eq. 13. We are now near the completion of our proof. From Eqs. 11, 12, and 13 there follows that A ( 2 ) is bounded in LP, 1 < p < » with a norm that can be estimated in terms of p and ||i/IL only, and this combined with Eq. 9 and Theorem 1 in ref. 1 implies that A ! 2 ' is also of weak type (1,1). But then Eq. 12 allows us to conclude that the same result holds for A131. So far we have assumed that ^r is a function in Co™, but since all preceding estimates depend on JI^'IU only, as far as their dependence on \p is concerned, a passage to the limit shows that the same results and estimates hold for operators involving general functions tp with I ^ I U < «• Furthermore, the estimates for A ( 3 ' are independent of the function t(t), which is positive measurable but otherwise arbitrary, and this implies that sup | A ^ , / | is of weak type (1,1) and strong type (p,p), 1 < p < CD, whenever

113 Proc. Natl. Acad. Sci. USA 74 (1977)

Mathematics: Calderon

1327

Finally, to prove the pointwise existence of \im(^*oA^J, we observe that this limit clearly exists at (Q if / is in Co™ and /(*o) = 0, or if / is in Co™, / is constant near to and <^'(/o) exists. From this we conclude that if / is in Co™ then lim(—oA^J" exists almost everywhere, whence the general result follows from the fact that sup \A^J\ is of weak type (1,1) and strong type (p,p), 1 iMcM-1 = or = <*>. Our method, however, cannot possibly yield this result. Indeed, the inequality

where p is sufficiently close to fl, we have

THEOREM 2. Let F(z) be analytic in the disc | z| < R and ^ a real Lipschitz function on the real line such that \\
This research was supported by National Science Foundation Grant MCS75-05567.

(W)(0-

f

F(2)(A z , ( /)(0-.

[15]

1

If z" — u + it), setting 1 = t ~ utp{t), s ~ s — u ||B(0)|| >cM,c> 0. as in the last part of the proof of Theorem 1. For additional references see refs. 2-4. 2. Related integral operators

L

J=S,

, -V(*^)/<.)*-

*J\s-t\>tS

—t

\

S —t

/

Then the operator supt\Lti\ is of weak type (1,1) and strong type (p,p), 1 < p < <», and limt^Q (L
AZJ = f

,

[s-t-z-Ws)

J\s-t\>* - ^m-'fWs,

\z\=p
[14]

1.

Benedek, A., Calderon, A. P. & Panzone, R. (1962) "Convolution operators on Banach space valued functions," Proc. Natl. Acad. Sci. USA 48, 356-365. 2. Calderon, A. P. (1965) "Commutators of singular integral operators," Proc, Natl. Acad. Sci. USA 53,1092-1099. 3. Coif man, R. R. & Meyer, Y. (1975) "On commutators of singular integrals and bilinear singular integrals," Trans. Am. Math. Soc. 212.315-331. 4. Coifman, R. R. & Meyer, Y. (1976) "Commutateurs d'integrales singulieres:" Analyse harmonique d'Orsay, n° 211., University Paris XL 5. Gundy, R. P. & Wheeden, R. L. (1975) "Weighted integral inequalities for ihe non-tangential maximal Junction, Lusin area integral, and Walsh-Paiey series," Studia Math. 49,107-124.

ON THE EXISTENCE OF CERTAIN SINGULAR INTEGRALS. By

A. P. CALDERON and A. ZYGMUND Dedicated to Professor MARCEL RIESZ, on the occasion of his 65th birthday

Introduction. Let / (x) and K (x) be two functions integrable over the interval (—oo, — oo). It is very well known that their composition + 00

f

j(t)K(x-t)dl

-00

exists, as an absolutely convergent integral, for almost every x. The integral can, however, exist almost everywhere even if K is not absolutely integrable. The most interesting special case is that of K (x) = 1/x. Let us set

31 J X — t -oo

The function / is called the conjugate of / (or the Hilbert transform of /). It exists for almost every value of x in the Priacipal Value sense:

-oc

x+e

Moreover it is known (See [9] or [7], p. 317) to satisfy the M. Riesz inequality

a)

[J\f)rdzY'zA,[f\frd*r, - oc

where Av The limit by dF(t), (— oo, + graphical

I
- oo

depends on p only. There are substitute result for p = 1 and p = oo. / exists almost everywhere also in the case when / (t)dt is replaced there wheie F (t) is any function of bounded variation over the whole interval oo). (For all this, see e.g. [7], Chapters VII and XI, where also biblioreferences can be found).

115 86

A. P. Calderon and A. Zygmund.

The corresponding problems for functions of several variables have been little investigated, and it is the purpose of this paper to obtain some results in this direction. To indicate the problems we are going to discuss let us consider two classical examples. Let / (s, t) be a function integrable over the whole plane, and let us consider in the half-space z > 0 the Newtonian potential u (x, y, z) of the masses with density / (g, t). Thus u (x, y, z) = fff

(s, t) ^ - ,

R* = (x - sf -r (y - tf + z\

the integration being extended over the whole plane. Let us also consider the partial derivatives

zj l / ( « > 0 ^ 3 - ,

ux = — \

\f(s,t)^^-dsdt.

Here — (4 TI)'1 M2 is the Poisson integral of /, and it is a classical fact that it tends to / (x, y) as 2 -> 0, at every point (x, y) at which / is the derivative of its indefinite integral. On the other hand, by formally replacing 2 by 0 in the formula for uz we obtain the singular integral

/ / It can be written in the form (2)

jjf(s,t)K(x

— s,y —

t)dsdt,

with (3)

K(x,y)

=

{x + V.

It is a simple matter to show that at every point (x0, y0) at which / is the derivative of its indefinite integral the existence of the integral is equivalent to the existence of lim MX, as the point (xr y, z) approaches (x0, y0, 0) non-tangentially (and that both expressions have the same value), but neither fact seems to have been established unconditionally. Here again the integral (2) is taken in the principal value sense, which in two dimensions means that first the integral is taken over the exterior of the circle with center (x„, y0) and radius e, and then s is made to tend to 0. Another example, of a somewhat similar nature, arises from considering in the plane the logarithmic potential « of masses with density / (s, t). Hence

116 On the Existence of Certain Singular Integrals. u(x,y)

= \ I f(s,t) log- dsdt;

87

r2 = (x — sf + {y — tf.

If in order to avoid unnecessary complications we assume that / vanishes in a neighborhood of infinity, then in any finite circle u is the convolution of two integrable functions, and so the integral converges absolutely almost everywhere. The integral obtained by formal differentiation, say with respect to x, is

and so, as a convolution of two integrable functions, again converges absolutely almost everywhere and represents a function integrable over any finite portion of the plane. Using this fact one proves without difficulty (see [1]) that the integral actually represents ux. Thus ux and w„ exist almost everywhere. Let us, however, differentiate the integral (4) formally once more, with respect to x and with respect to y. We get the integrals of type (2) with (5)

K ( X t y )

. ^ ^

t

K{Xty)__*M_,

respectively. These two kernels are not essentially different, since one is obtained from the other through a rotation of the axes by 45°. It may also be of interest to observe that they appear respectively as the real and imaginary parts of

1 _ 22

1 (x + i yf

The existence almost everywhere of the integrals (2) in the cases (5) has been established by Lichtenstein for functions / which are continuous (or, slightly more generally, Riemann integrable). This result seems not to have been superseded so far, though the existence almost everywhere of uxi, uvv, uly together with the relation uzz — uyy = — 2 T I / was established by Lichtenstein [6] (see also [2], [8]) in the much more general case of / quadratically integrable. The kernels (3) and (5) have one feature in common: they are of the form QMS'2'

x = qcos(p,

y = Q sin
where g ( (actually a trigonometric polynomial) whose mean value over (0, 2 n) is zero. Several examples of kernels of this type could be considered, but we shall now state the problem in a more general form.

117 88

A. P. Calderon and A. Zygmund.

Suppose we have a function / (xx, x2, . . . xn) integrable over the whole w-dimensional space, and a kernel K (xl, x2, . . . xn) = o~n Q (aj, a 2 , . . ., a„), where Xj = o cos a7 for all j , and a x , x 2 , . . ., an are the direction angles. What can be said about the existence and the properties of the integral (6)

f(xl,xi,...,

x„) = / /fo , s 2 , . . . , s„) K (xl — sx, . . ., xn — sn) ds! . . . dsn ?

An answer to this problem is our main object here. This is the plan of the paper. In Chapter I it will be shown that, if / 6 £ " , 1 < p < o o , then the integral (6) converges, in the metric L", to a function f 6 L", provided a) the mean value of Q over the unit sphere is zero, b) the function Q(a.l, a 2 , . . ., a„) satisfies a smoothness condition (See Chapter II). (In the case p = 2 condition b) can be considerably relaxed). The function / satisfies the condition analogous to (1). The cases p = 1 and p = oo are also investigated. The main result of Chapter II is that under conditions a) and b) the integral (6) exists almost everywhere not only for p> 1, but also for p= 1. The result holds, if f ds-L ... dsn is replaced by d[i, where (u is an arbitrary mass distribution with finite total mass. If / € i p , p> 1, the partial integrals of the integral (6), that is the integrals over the exterior of the sphere of radius e and center (xlt xit ... xn) are majorized by a function of L", independent of e. Chapter III is devoted to some applications of the results previously obtained to the problem of the differentiability of the potential. Other problems connected with our main topic will be considered in an another paper.

W o l f P r i z e in M a t h e m a t i c s , V o l . 2 ( p p . 1 19—1 6 4 )

eds. S. S. Chern and F. Hirzebruch © 2001 World Scientific Publishing Co.

A. N. KOLMOGOROV V. M. Tikhomirov Moscow State University Life Story Andrei Nikolaevich Kolmogorov, one of the greatest mathematicians of the twentieth century and one of the most outstanding scholars in the history of Russian science, was born on April 25, 1903. His mother Maria Yakovlevna Kolmogorova died in childbirth. The responsibility for his upbringing was taken on by his mother's sister, his aunt Vera Yakovlevna Kolmogorova (1863-1950). Maria Yakovlevna and Vera Yakovlevna were the daughters of Yakov Stepanovich Kolmogorov, a nobleman and rich landowner. Kolmogorov's first years were spent in the Tunoshna estate owned by his grandfather. The estate was situated near the old Russian town of Yaroslavl and the greatest Russian river, the Volga. Kolmogorov always said that his childhood was happy. Andrei was surrounded with love, kindness, attention and care. His relative's goal was to cultivate the child's curiosity and an interest in books and nature. Vera Kolmogorova, together with a good friend, organized a small school for children living near the estate. At this school the teaching was carried out "according to the latest educationals theories". a School children, with the help of their teachers, published the journal "The Swallow of Spring". The five-year-old boy "edited" the mathematical section of the journal. Kolmogorov wrote sixty years later: "The first impression of mathematical 'discovery' I experienced at the age of five or six, noticing the regularity in the identities 1 = l 2 , 1 + 3 = 2 2 , 1 + 3 + 5 = 3 2 , 1 + 3 + 5 + 7 = 4 2 , and so on." In the journal he had published this "discovery", and it could be considered his first "scientific publication". In 1910, when Kolmogorov reached the age of seven, he and his aunt moved to Moscow. There he entered one of the most progressive grammar schools (gymnasiums) of the time. It was organized by a group of radically inclined intelligentsia. Unlike the majority of grammar schools of that time, it was co-educational (this took place only at two grammar schools in Moscow) and practised many interesting experimental teaching methods. In the school, the spirit of freedom reigned and the teachers tried to support every sign of talent. Kolmogorov's range of interests at the age 10-14 was extremely broad. He took a serious interest in biology and physics, and by the age of 14 he was already familiar with higher mathematics. He was fond of chess, and fascinated by social problems. In particular, he thought about a constitution for a community, where the principles of higher justice were to be put into practice. He actually drafted the text of such a constitution. He took part in the elections for the Constituent Assembly in autumn 1917. Equally, his interest in history was especially profound. "The first scientific report which I made at the age of 17 at the University of Moscow," he recalled a

If words in quotation marks are not followed by the name of the author, they are from A. N. Kolmogorov.

120 later, "was a report to the seminar of Professor S. V. Bakhrushin about the history of Novgorod in the fifteenth century." Kolmogorov then made a small discovery, which was recognized by S. V. Bakhrushin, one of the leading Russian historians of that time. Kolmogorov asked the professor whether or not he should publish his result. The reply Bakhrushin gave him was: "Certainly not! You have found only one proof. That is very little for a historian. You need at least five proofs." The disappointment he felt at that moment might have influenced his future destiny, and he began to study mathematics, where just one proof suffices.b Nevertheless, he did not immediately decide to devote his life to mathematics. While he was studying at the University of Moscow in 1920, he enrolled in the Faculty of Metallurgy at the Institute for Chemical Technology, because he had intended to study practical engineering. But very soon his interest in mathematics outweighed "his doubts about the relevance of the profession of mathematician". He continued his studies only at the University, and from then on his whole life was linked to the University of Moscow. He immediately succumbed to the University's atmosphere of creative enthusiasm that had then existed there. The outstanding figure among the mathematicians at that time was Nikolai Nikolaevich Luzin. Kolmogorov came into lively scientific contact with some of Luzin's students, Pavel Sergeevich Alexsandroff and Pavel Samuilovich Uryson, who later played an important part in the development of topology. Once, in 1920, at Luzin's lecture, the 18-year-old freshman disproved one of the lecturer's hypothetical assertions. He presented his argument with confidence to the group of mathematicians present, and for the first time he attracted some attention. This resulted in P. S. Uryson inviting Kolmogorov to become his student. But it was the problems put forward by Luzin that drew the young man's attention. He began to construct a general theory of operations on sets, wishing to advance beyond the boundaries at which P. S. Alexsandroff and M. Ya. Suslin had stopped. Kolmogorov also attended V. V. Stepanov's seminar on trigonometrical series. There he solved one of the problems posed by Luzin, and after that Luzin "with some solemnity" (as Kolmogorov wrote) suggested that he be his student. "In relation to his pupils," Kolmogorov wrote not long before his death, "Luzin had a definite idea of which of them was destined for working on 'metrical' or 'descriptive' problems. My predestination was to deal with 'metrical' one(s)..." In the summer of 1922 (the work was dated 2 June 1922), Kolmogorov obtained an outstanding result: He constructed a Fourier series divergent almost everywhere, and this immediately brought him international repute. This was the beginning of his incomparable creative career. But earlier, in 1920-21, Kolmogorov realized his ideas on descriptive set theory. He wrote a long paper on this subject, dated February 1922. But this paper was devoted "to a direction that was not at all envisaged by Luzin." It was the reason that the first part of the paper was published in 1928 and the second part of it wasn't printed until 1987. Kolmogorov's manuscript devoted to history of Novgorod land-ownership in the 15th century was found in his papers after his death. It was published in 1994 and was highly appreciated by specialists. In the manuscript, Kolmogorov used some statistical methods for his historical conclusions.

121 In 1925, Kolmogorov graduated from the University of Moscow and became Luzin's research student. The same year saw the beginning of his work on probability theory (in collaboration with A. Ya. Khinchin). In 1929 he set out on a long boat journey along the Volga with Pavel Sergeevich Alexsandroff, and this marked the beginning of a friendship which lasted until Alexsandroff died on 16 November 1982. In 1935, Kolmogorov and Alexsandroff acquired a house not far from Moscow on the bank of the river Klyaz'ma, and it was in that house that most of their creative life continued. For more than 50 years, this house was the center of mathematical activity for several generations of Soviet mathematicians as well as many guests from abroad. Kolmogorov had accumulated many accomplishments in his field in his lifetime. By 1931, Kolmogorov became a Professor at Moscow University, and founded several departments. In 1939, he was elected full Member of the Academy of Sciences of the USSR. From 1954 to 1958 he was the Dean of the Faculty of Mechanics and Mathematics at the University of Moscow. From 1964 to 1966 and from 1973 to 1985 he was the President of the Moscow Mathematical Society. Between June 1930 and March 1931, Kolmogorov went on academic journeys abroad (Gottingen, Munich, Paris). There he met Hilbert, Caratheodory, Landau, Levy, Frechet, Lebesgue, Borel, Weyl, and other outstanding mathematicians. Kolmogorov often spoke of the exceptional significance of being able to do his research work in such a scholarly atmosphere. He felt privileged to have spent this time abroad, and to have established personal contacts with these scholars from different countries. In 1954, he spent two months as Professor at the Humboldt University in Berlin. In 1958, he was Professor at the University of Paris, for the spring term. Kolmogorov attended the International Congresses of Mathematicians in Amsterdam (1954), Stockholm (1962), Moscow (1966) and Nice (1970). At the Amsterdam Congress, Kolmogorov was invited to conclude the scientific program of the Congress with a lecture on dynamical systems (a lecture by J. von Neumann had opened the scientific program). More than twenty scientific organizations have elected Kolmogorov to be a Honorary Member. He was elected Member of the Royal Netherlands Academy of Sciences (1953), of the Royal Society of London (1964), of the National Academy of Sciences of the USA (1967), of the Academie des Sciences of Paris (1968), Honorary Member of the Roumanian Academy of Sciences (1965) (he had been a Corresponding Member since 1957), Foreign Member of the Polish Academy of Sciences (1956), Honorary Member of the Hungarian Academy of Sciences (1965). He was a Doctor Honoris Causa of the Universities of Paris, Stockholm, Warsaw, Budapest, of the Calcutta Mathematical Society, of the Royal Statistical Society of London, etc. Kolmogorov was awarded the State Prize of the USSR (jointly with A. Ya. Khinchin, 1941), the P. L. Chebyshev Prize of the Academy of Sciences of the USSR (jointly with B. V. Gnedenko, 1949), the highest Prize of Soviet Government (jointly with V. I. Arnol'd, 1965), the N. I. Lobachevskii Prize (1986), and the following international prizes: The Balzan Prize (1963), the Prize was awarded simultaneously but in other nominations to Pope John XXIII, the historian S. Morrison, the biologist K. Frisch, the composer P. Hindemith), the Wolf prize, the Helmholtz Gold Medal, and many others.

122 Kolmogorov devoted the last two and a half decades of his life mainly to the problem of instruction and teaching in secondary schools. Scientific Legacy There are different ways of speaking about the scientific achievements of a great scholar: We can give a chronological account, we can consider separately each scientific direction, or we can classify the papers according to their importance, and so on. I prefer to follow a different way. In the original mathematical work of an outstanding scientist, it is possible to differentiate several components: results (solutions to difficult problems, finding new formulas, etc.), ideas (introduction of new definitions, concepts and interpretation of old ones, enunciation of problems, working out the beginning of new scientific directions), theories (where the aim is to explain a group of phenomena), methods and conceptions. All these options are present in Kolmogorov's work. And we shall investigate what makes up each of these components. In his article "The Architecture of Mathematics" that describes his program, Nicolas Bourbaki notes with regret that there is no mathematician, even among those with the broadest erudition, who would not feel to be a stranger in certain fields of mathematics. The exceptions to this rule are the geniuses like Poincare and Hilbert, who left their mark in almost every mathematical area. In my opinion, there is complete justification to add the name of Andrei Nikolaevich Kolmogorov to this short list. The breadth of Kolmogorov's scientific range of interests is unique. Trigonometrical and orthogonal series, descriptive set theory, mathematical logic, the theory of measure and integral, topology, the theory of approximations, geometry, celestial mechanics, cohomology theory, turbulence, ergodic theory, superposition of functions, functional analysis, ... . And in addition to all this, the vast field of probability (Markov chains and processes, statistics, stationary processes, etc. — Kolmogorov transformed probability theory completely, starting from its foundations); information theory; the theory of automata; and still more — numerous works on applications to biology, geophysics, production control, the theory of marksmanship, the theory of versification, etc. And in almost every one of these fields, we owe to Kolmogorov fundamental theories, ideas, methods and outstanding results. First and foremost, it is necessary to mention the main conceptual line of his legacy. When the Academy of Sciences of the USSR decided to publish his collected papers, it was assumed that this publication would consist of two volumes. The question of how to divide the material into two parts was, naturally, passed on to the author himself to solve. Kolmogorov suggested dividing them into one volume of the papers on mathematics and mechanics and, in the other, papers on probability theory and information theory. This division reflected the fact that enormous world of mathematics is in fact itself divided into two parts, as it were into two kingdoms. In one, deterministic phenomena are studied, and in the other, random ones. A century ago the Kingdom of Chance occupied a very modest territory in our world of mathematics. Until the end of the century there was still no fundamental role for random phenomena in the natural sciences (although the foundations of

123 thermodynamics were already laid). Then the Kingdom of Chance began to colonize greater areas and became comparable to the rest of the mathematical world. (And as Kolmogorov once remarked, the Congress of the Bernoulli Society, are comparable to the International Congress of Mathematicians.) Kolmogorov was a leader of progress in both kingdoms, and he discovered many previously uncharted regions. At the end of his life he put forward a grandiose program of simultaneous and parallel study on the complexity of deterministic phenomena and the statistical determination of random phenomena. The basic idea is that the Kingdom of Order and the Kingdom of Chance have no real boundaries, that our mathematical world is one, and in principle, indivisible. The attempt to reveal the essence of the concepts of "order" and "chaos" crowns Kolmogorov's original work. In this concept, all the ideas of probability theory, the foundations of mathematical logic and the theory of algorithms, the ideas and methods of information theory, the results of ergodic theory and of the theory of dynamical systems, and intense speculations about the natural sciences are encompassed. But many of Kolmogorov's first works (on the theory of functions, on set theory, geometry, etc.) can be interpreted as a prologue to the realization of this immense program. And now let us turn to the concrete results Kolmogorov obtained. It is impossible to describe in one article all of his contributions to science with any totality, but whilst inevitably restricting myself, I would like to present Kolmogorov's original work as a whole, picking out the basic components and linking them together. The clarity and wide importance of the goals which Kolmogorov set for himself are striking. It is difficult to deny oneself the pleasure of trying to explain the significance of the results he obtained to the average reader, and not be disrespectful to a certain degree to his or her field of interest. We have already spoken of the main concept of his life. Now we shall speak of theories. At the very outset of his creative work, Kolmogorov was obsessed by "a vague desire to study mathematics, which has links with physics and the natural sciences". This led him to perceive the deep and mysterious secrets of nature. Here we must first focus on his papers on classical mechanics. Can the Solar system last forever? This is probably the central problem of astronomy. Is perpetual motion possible in simpler planetary systems, consisting, say, of only three bodies? Or, will perhaps the evolution of a planetary system always (on a set of full measure) end in catastrophe? It was natural to try to consider first the motion of systems close to integrable ones. Problems of this kind date back to Newton and Laplace, and Poincare, who called the problem of describing the evolution of Hamiltonian systems for small perturbations "the fundamental problem of dynamics". In his papers Kolmogorov solved the fundamental problem of dynamics for the three-body problem for the majority of initial conditions in the case of general position. As a direct consequence of Kolmogorov's theorems, it was found that a satellite of Jupiter moving in the plane of the circular orbit of Jupiter along an elliptical orbit, perturbed by Jupiter but not perturbing it, always remains in an elliptical orbit. Kolmogorov's theory proved to be applicable to a large number of problems in mechanics and physics (the problem of the stability of rapid rotation of an asymmetric rigid body around a fixed point; the problem of magnetic surfaces in systems of tokamak type, and so on). Kolmogorov gave a survey lecture on his results, to which we have referred, at the International Congress of Mathematicians

124 in Amsterdam. The ideas of Kolmogorov, developed in the works of V. I. Arnold and J. Moser, were called KAM theory (the theory of Kolmogorov, Arnold and Moser), the name now used by almost every mathematician. Kolmogorov made a significant contribution to the creation of the theory of stochastic processes. In 1931, he published his paper "Analytic methods in probability theory". The initial methodological position is remarkable for its simplicity and depth. Deterministic processes, in which initial position determines their further evolution, are replaced by processes where "the state x of the system at some moment of time t stipulates only a known probability for the occurrence of a possible state y at some later moment t > 0." This consideration led Kolmogorov to a definition of a Markov processes. He wrote it out in the general form of an integral equation, which had been found for special cases by Smoluhovskii. From this integral equation one can deduce direct equations (already seen in the work of such powerful physicists as Planck, Einstein, Fokker, Smoluhovskii) and inverse equations, which were not known to the physicists. In his work Kolmogorov combined the theory of heat, due to Fourier, the theory of Brownian motion, due to Einstein and Smoluhovskii, and a description of probabilistic random walks, on which Markov and his followers had worked, and also the ideas of Bachelier and Wiener, who constructed the first examples of stochastic processes. In their article, written on the occasion of Kolmogorov's fiftieth birthday, P. S. Alexsandroff and A. Ya. Khinchin wrote: "In the whole of probability theory in the twentieth century, it is difficult to name any other piece of research that has turned out to be so fundamental for the further development of science and its applications than this paper by Kolmogorov." In the course of his work on the theory of Markov processes, Kolmogorov began a fruitful collaboration with known physicists (S. I. Vavilov, M. A. Leontovich and others). Kolmogorov delighted in affirming that in his joint paper with Leontovich, the "physical" part was due to him and the mathematical part to Leontovich. Physical intuition often lead Kolmogorov, even in his research along purely abstract lines. According to him, cohomology theory occurred to him from visualizing descriptions of a flow of a liquid over a manifold. A clear understanding of the physical picture of undamped oscillations with a continuous spectrum served as a lodestar for the creation of the theory of stationary processes. Wiener was very proud of his construction of the theory of interpolation and filtration of stationary stochastic processes. It was with some bitterness that he acknowledged Kolmogorov's priority in this area. Stationary processes, being processes whose probability characteristics remain constant in time, are the idealization of a great number of stochastic natural phenomena (in the atmosphere, in the ocean, and so on); they occur constantly in technical applications, for example in radio engineering. One of the most important problems is the prediction of the future by observing the process over a period of time. It was this theory that Kolmogorov and Wiener worked out. It discovered innumerable applications in different fields of science and technology. The theory of stationary processes led Kolmogorov to consideration of turbulence problems. In 1941, he became interested in this field, which was absolutely new to him, and which belongs, strictly speaking, to the domain of physics, or, rather, hydro- and aerodynamics, rather than mathematics. There was no mathematical theory of turbulence before 1941, but there were several outstanding schol-

125 ars who tried to give phenomenological explanations of turbulence. Among these distinguished people were Jeffrey Taylor and von Karman. Kolmogorov had been thinking about turbulence for about half a year when War World II started, he then had to switch to other problems (he was engaged in correcting the artillery fire and bombing and radically reformed this science as well). Kolmogorov published only three articles on the theory of turbulence in "Doklady" at the Russian Academy. Kolmogorov's papers on turbulence had an enormous influence on the whole further development of this important field of the natural sciences. ... Fifty-eight years passed, and one of Kolmogorov's students, Akiva Jaglom, won a grant for a series of books on Kolmogorov's theory of turbulence. The science has been extended drastically. "How long are you going to write this book", I asked him. "You have probably heard about the recent sentence in an American court of law. The criminal was sentenced to three life terms and thirty more years. So, three lives and yet another thirty years might be enough," he replied. The motion of fluids and gases is subject to deterministic laws; however the very character of the motion turns out to be so complex that it calls to mind a stochastic process. Among Kolmogorov's concrete results one can point to the widely known "two-thirds law", which has the characteristic of a law of nature. In a turbulent flow (under certain conditions) the mean square of the difference of the velocities at two points, at a distance r (of mean sizes), is proportional tO r 2 / 3 . Kolmogorov made a very significant contribution to the natural sciences in his research inspired by biological problems. His work in biology led him to the solution of a series of problems in mathematical biology and to remarkable results in pure mathematics. In a joint paper with I. G. Petrovskii and N. S. Piskunov, which was stimulated by their work on biological problems, they were the first to construct a mathematical theory of the stability of solutions of travelling wave type for the diffusion equation with a nonlinear right-hand side. (The list of papers on this subject extends at present to some thousand items.) Exemplified on the physical side, the formulisation and description of the qualitative picture of the phenomenon is due to Kolmogorov. ("I had noticed how a Bickford fuse burns," he said of this, although the work arose from attempts to describe the propagation of a gene.) Biological problems led Kolmogorov to formulate a theory of branching processes. Had Kolmogorov done nothing more than create the theory of stochastic processes, the theory of turbulence and laid the foundations of KAM theory, he would still have been one of the greatest stars in the world of science. In his monograph "Limit distributions for sums of independent random variables" , Kolmogorov presented the results of the initial stage of the summation of random variables, which had begun in the work of Jacques Bernoulli, de Moivre and Laplace, and had been continued by Poisson, Tchebyshev, Markov, Bernstein and others. During the war Kolmogorov made an important contribution to the theory of ballistics, and according to the experts, his results completely transformed this field. Reduced to tables, graphs and nomograms, they helped the Allies to achieve victory in the World War II.

126 And finally we must mention his contribution to information theory, where (with Khinchin and others) he laid the mathematical foundations of the theory and obtained some interesting formulae and theorems. What are the fundamental results obtained by Kolmogorov? Let us begin with the metric theory of functions. The best known work from the first period of his scientific activity was his example of a Fourier series divergent everywhere. If x(-) is integrable on [-7r,7r] (i.e. x(-) E Li([-7r,7r])), then from it one can construct a Fourier series: J2kezxi*eik', where xk = (27r) _1 JTx(t)e~ihtdt. Fourier series date back to the eighteenth century, but the theory of their summation began to develop in the middle of the nineteenth century. The following result, discovered by the 19-year-old Kolmogorov, caused a sensation. Theorem 1. There is a function integrable on [—7r,7r] whose Fourier series diverges almost everywhere. In 1926, Kolmogorov constructed an example of a series that diverges everywhere. The question of whether for every square integrable function (i.e. a;(-) € I<2([7r,7r])) the Fourier series converges almost everywhere was raised by N. N. Luzin in 1915, was solved by Carleson in 1966, and was completed by Hunt (for any function x(-) in Lp([—n, n]) for p > 1, its Fourier series converges to x(-) almost everywhere). In particular, it was not known until Carleson's theory whether it is possible to replace £i([—rr,7r]) in Theorem 1 by C{[—7r,7r]), the set of continuous functions on [—7r,7r]. In the field of trigonometric series, in which Weyl, Hardy, Luzin, Men'shov, Bari and many other outstanding mathematicians worked, two results stand out by reason of their power and completeness: Kolmogorov's example and the Carleson-Hunt theorem. In connection with Luzin's problem it is appropriate to mention another remarkable result which Kolmogorov found jointly with D. E. Men'shov. Theorem 2. There is an orthogonal system on the unit interval consisting of functions of unit modulus and a square integrable function on the same interval such that the Fourier series of this function, with respect to the given system diverges everywhere. In the same note Kolmogorov stated the following result: There is a function in L2([—7r,7r]) such that the Fourier series with respect to a rearranged trigonometric system diverges almost everywhere (so that Carleson's theorem reflects a very deep fact concerning harmonics in their natural order). Kolmogorov did not publish his proof of this theorem, and this result was proved in 1960 by the Polish mathematician Zahorski. The next theorem, like the two previous ones, is very simple in its formulation, but its significance is not immediately obvious to anyone who is not connected with harmonic analysis. One of the most actively developing branches of harmonic analysis studies are problems on the convergence of Fourier sums and of other important integral operators in various function spaces. One of the most important operators is the Hilbert transform H, associating to a function x(-) the function z(-) that is conjugate to x(-) in the sense of complex analysis. We

127 have the formula Hx(t) = x(t) = - ( 2 7 r ) - 1 J x(t + T) cot(r/2)dr. Various generalizations of the Hilbert transform (singular integral operators) play a very important part in classical analysis. The Hilbert transform maps Lp([—7r,7r]) into LP([-TT, n]) for p > 1 (M. Riesz), or, as we say, this is a transformation of strong type (p,p) for p > 1. For p = 1, it is not a transformation of strong type (1,1). However, we have the following result. Theorem 3. Let x(-) be a function integrable on [—TT,TT]. Then the measure of those points where the Hilbert transform Hx(-) is greater than a in absolute value satisfies the inequality meas{t\ \Hx(t)\ > a} < Ca _1 ||o;(-)|| z , l([ _ 7ri7r]) . In modern language, this inequality means that the operator H is of weak type (1,1). Theorem 3 proved to be one of the starting points for a very broad field of mathematics, with which the names of Riesz, Marcinkiewicz, Zygmund, Calderon, Hardy, Littlewood, Paley, and others are associated. The very concept of weak type was very important in the theory of singular operators. It is very difficult to know which result to choose when talking of Kolmogorov's work on probability. Kolmogorov deemed this field to be his "narrow speciality", and in it he was unquestionably a world leader. His input here was enormous. At the beginning of his work in this field he found some fundamental theorems crowning the research of Chebyshev, Markov, Bernstein and others. In classical probability theory there were two fundamental themes: The law of large numbers and limit theorems. In this century, a third one has been added: The law of the iterated logarithm discovered in the simplest situations by Khinchin. In particular, Kolmogorov found a necessary and sufficient condition for the law of large numbers, and very significantly extended the limits of applicability of the law of the iterated logarithm. He made other fundamental discoveries. We shall mention only his definitive results related to the strong law of large numbers. We owe the concept of strong law to Borel. A sequence of independent random variables satisfies the strong law of large numbers if its sequence of arithmetic means converges almost certainly to some number (unlike the law of large numbers, where it converges in a weaker sense — in the sense of measure). Theorem 4. Let {{„} n gN be a sequence of independent equally distributed variables. Then the condition of finiteness of the expectation of |£i| (and hence of all |£ n |) is necessary and sufficient for it to satisfy the strong law of large numbers. One of the basic and natural questions is: What distributions are limits of sums of mutually independent normalized terms? It turned out that under natural assumptions the answer was only the infinitely divisible distributions introduced by Bruno de Finetti. A random variable subordinate to such a law can be represented as sum of n independent equally distributed terms, for any n. The problem then was how to describe infinitely divisible laws.

128 Theorem 5. Let $(•) be a characteristic function of an infinitely divisible distribution with finite dispersion, (£)|. Then we have the formula ip(t) = i/3t+ f (eitx - 1 -

itx)d\/x2,

JR

where (3 £ R, and dX is the Borel measure on R. Later, using a completely different method, P. Levy removed the requirement of finite dispersion (replacing the last term by {eitx - 1 - itx) i i ^ L - dX{x))

J£'

and then Khinchin showed that Levy's general result could be easily obtained by Kolmogorov's method. "In this way," Khinchin wrote, "this method gives the simplest and clearest substantiation to-date of the canonical form of infinitely divisible laws, making their totality easy to visualize." We spoke earlier of the theory of stochastic processes, and here we shall mention only two of Kolmogorov's remarkable results, concerning stationary random sequences: A criterion for the regularity of a sequence (i.e. a criterion that observation of a sequence implies the influx of new information) and a formula for the error of prediction. The question of regularity is primary in the theory of stationary sequences and processes. M. G. Krein solved it for processes. In mathematical statistics, Kolmogorov proved one of the most fundamental results in this discipline. Given a random variable £, whose distribution function is F(-), let {xj}™=1 be the result of n independent observations of this variable arranged in increasing order. The function 0

Fn(x) = { k/n 1

if x < x\, if Xk < x < Xk+i, if x > xn

is called the empirical distribution function. One of the fundamental questions in statistics then arises: To what extent does the empirical distribution function Fn(-) "resemble" the function F(-) itself? We then ask: Is it true that P(sup \F(x) — Fn(x)\ < e) —¥ 1 as n —> oo? Glivenko gave the answer to this question, and in the same issue of the journal that he did so, Kolmogorov not only answered this question, but he also gave the limit distribution of the variable Dn = -y^nsup \Fn{x) — F(x)\. More precisely, the following theorem holds: Theorem 6. Let F(-) be a continuous distribution function. $(A) = P{Dn(x) < A} as n —• oo uniformly in A tends to $(A) = ^ ( - l ) f c e x p ( - 2 A ; 2 A 2 ) fcez [Kolmogorov's

distribution).

Then the probability

129 The question on how to measure the degree of approximation of Fn(-) to F(-) occurred to such great scholars as von Mises and Cramer, but they were far from answering it. It is only natural that Kolmogorov's test is found in all textbooks on mathematical statistics. There is one result in topology of which Kolmogorov spoke with satisfaction and of which he was, I think, proud. A map / : X —> Y of a topological space X onto a topological space Y is called open if the image of any open set under this map is open. "The problem of whether the dimension can increase under an open map was of very great interest to P. S. Alexsandroff. For some time we worked together on proving the nonexistence of an open dimension-raising map. During these attempts the reasons for our failure were gradually revealed. This analysis of our failures led in the end to a counterexample." Thus they proved the following result. Theorem 7. There exists an open map of a one-dimensional compactum onto a two-dimensional one. This work acted as a stimulus in other areas of research, among which we must first mention the work of L. V. Keldysh. In particular, by a very complicated procedure, she constructed an open map of a one-dimensional compactum onto a square. Many years later, I. M. Kozlovskii constructed an open map from a onedimensional compactum onto a square by a very transparent and simple method, using Kolmogorov's construction as a fundamental element. In 1956, Kolmogorov obtained a result on the summation of random variables, of which he had been dreaming since the thirties. He was clearly satisfied when he reported on this work at a seminar. We have already spoken of infinitely divisible distributions. We denote the totality of such distributions by D. Let F(-) be the distribution function of the random variable £ and <&„(•, .F) be the distribution function of the sum of n independent random variables that have the distribution F(-). We recall that only infinitely divisible distributions can be limits of sums of independent equally distributed random variables. Is it true that the distributions of sums of equal independent terms are uniformly close to D? The answer is given by the following theorem. Theorem 8. There is a constant C such that for any distribution F(-) and for any n G N there is a \t(-) £ D such that

I^F)-^)!^^-1/5. At once the question arose about the exponent 1/5: Can it be improved? Among those working on this were Prokhorov, Meshalkin, and Kolmogorov himself. The definitive solution was found by T. V. Arak in 1981-1982; the optimal exponent turned out to be 2/3. Do functions of several variables exist? That would seem to be a pointless question; of course they exist. Let us phrase this question more precisely. Some functions of three variables can be expressed in terms of functions of two variables, say, fi(x,y,z) = (
130 The conviction that functions of three variables are not reducible to functions of two variables (in the sense that not all are representable as superpositions of functions of two variables) was so great that Hilbert (Hilbert again) indicated a concrete analytic function of three variables which (as he thought) could not be represented as a superposition of continuous functions of two variables (Hilbert's thirteenth problem). Alas! this expectation was not to be fulfilled. It turned out that any continuous function of any number of variables can be represented as a superposition of continuous functions by using only one function of two variables, namely s(x, y) = x + y, and the remaining functions are of one variable. Here is the story of this discovery. In 1956, Kolmogorov proved in an article that any continuous function of four variables can be represented as a superposition of continuous functions of three variables. Refuting Hilbert's conjecture was reduced to a concrete problem about the representation of functions defined on universal trees in R3. In Kolmogorov's opinion, this paper of 1956 was his most complicated technical achievement, requiring the longest period of uninterrupted thought in his life. Once again he left the final step to his successors. In the spring of the following year a third-year undergraduate student named V. I. Arnold obtained the necessary result about functions on universal trees, and thus Hilbert's thirteenth problem was completely solved (i.e. Hilbert's conjecture was disproved). Soon after that Kolmogorov devised a comparatively simple, natural and very beautiful construction, which led to the following remarkable theorem, which we have already mentioned. Theorem 9. For any integer n > 2 there are continuous functions ipij(-) defined on I — [0,1] such that any function f continuous on In can be represented in the form 2n+l

f(xu...

,xn) = ^2 Xil

/ n

^2^ij(xj)

where the functions x(-) ore continuous on R. We conclude this section devoted to results with a theorem relating to the ergodic theory of dynamical systems, in which a solution was given to a problem raised by von Neumann and dated back more than 25 years. A dynamical system is a pair (T,A), where T = (T, E, fx) is a measure space, and A : T —> T is a measure-preserving map. In the space L2(T) the map A generates a natural unitary operator Ux(t) = x(At). The spectrum of this operator is called the spectrum of the dynamical system. The problem of the classification of dynamical systems up to an isomorphism was studied for many years as the basic problem in ergodic theory. At the beginning of 1930, von Neumann proved that two dynamical systems with a common point spectrum are isomorphic. It was assumed to be plausible that the spectrum always uniquely determines the dynamical system. "Bernouilli schemes" provide an example of dynamical systems with a special kind of spectrum. We suppose that independent tests have been carried out, for which 1 occurs with probability p, and 0 with probability 1— p, 0 < p < 1. Such a procedure gives a measure on the set of all sequences of 0's and l's, and a shift by one step in such sequences generates a dynamical system, known as a Bernoulli scheme. Theorem 10. Bernoulli schemes for different p are nonisomorphic.

131 This led to the solution of von Neumann's problem and signaled the beginning of a period of rapid growth in this theory. We have cited only ten results from the rich collection of Kolmogorov's theorems. Again we must express our astonishment at the number of fundamental original problems (therefore so simple in their formulation) that he managed to solve. Now it is time to turn to the ideological aspects (ideas) of Kolmogorov's creativity. In his early period, Kolmogorov spent a lot of time on the interpretation (within the framework of the axiomatic method) of the limits of the fundamental concepts of mathematics. We give his answers to questions about the most primary structures. What is an integral? Recall the evolution of this concept: Archimedes, Newton, Leibnitz, Cauchy, Riemann; then the Lebesgue integral, then the integrals of Stieltjes, Hellinger and others; later the integration of vector-valued functions (in infinite-dimensional Frechet space). Kolmogorov proposed the following simple definition of integral, which covers all the above. Let T be some set and M. a distinguished system of subsets of T such that A £ M, B £ M. ^> An B £ A4. Further, assume given a generally speaking multivalued function with values in a vector space, defined on any A £ M.. We define the integral of / on the set T for a given system M. as the limit (in the sense of Shatunovskii-Moore-Smith) of the Riemann sums

(Rf)r(T) = £

f(Tn),

where T = {Tn} is a partition of T, i.e. T = UnTn, % l~l Tj = $, i ^ j . We denote this integral by M. = J fdT. This construction subsumes all the concepts of integration we spoke of earlier (Riemann, Lebesgue, Stieltjes, and so on). It implies, in particular, the following most unusual fact: The Lebesgue integral is a limit of Riemann (not Lebesgue!) sums. Kolmogorov had a high opinion of his paper "Study of the concept of the integral" and often expressed regret that its definition had not yet taken root. Is it possible to extend substantially the concepts of derivative, integral and summation of divergent series? This was the problem considered by Kolmogorov in a paper of 1925. It turned out that the search for effective methods of differentiation, integration or summation of series came up against the same difficulty as the effective construction of nonmeasurable functions. For example, there is a method of summing divergent series that satisfies the conditions

(a) ^ Xn = Xi + Y^ Xn+1 , n€N

n€N

(b) ^2 «6N

aX

n = a XI

Xnj

ngN

and if by using this it is possible to define the sum of the series of the form ^ n g N s i n 3 n x , then it is possible to construct an effective example of a function that is nonmeasurable in the Lebesgue sense. What is the ^-dimensional measure of a set situated in n-dimensional Euclidean space? In his paper of 1933, Kolmogorov approached the problem from an axiomatic point of view. The following result holds. Definition. Let fi be a set function defined on all Suslin sets E £ R " (<^> E £ Sus(R™)) and satisfying the following axioms:

132 (1) E C U £ n , (E,En e Sus(R")) => n{E) < £ n e N / i ( £ „ ) , (2) Et G Sus(R"), ^ n £,- = $, u? = 1 Ei C £ =*• n = i M(#0 < M ( # ) , (3) M (J") = 1, where Jfc = [0, l] fc cube, denotes the k-dimensional Euclidean cube, (4) E' := /x(E') < /i(£) (i.e. if E' is a "nonexpanded" image of E, then its measure is not greater than that of E). Then we say that p is a /c-dimensional measure. At the time when the paper referred to appeared, many different A;-dimensional measures had been constructed (lengths of curves, Hausdorff measures, Caratheodory measures, and so on), satisfying the axioms (l)-(4). In this paper Kolmogorov obtained the following fundamental result: For any natural number k < n there are two special k-dimensional measures, the maximal measure JIk and the minimal measure [ik such that any k-dimensional measure /i of any Suslin set E lies between ~pk(E) and fik: £fc (E) < fi(E) < ~p,k (E). Both measures /xfc and ~pk are defined in a completely natural way. Let us turn to descriptive set theory. We shall not discuss it in detail. This field used to be at the epicenter of the interests of the Moscow Mathematical School, but relatively few are interested in it now. In 1916 P. S. Alexsandroff in Russia and F. Hausdorff in Germany independently solved the continuum problem for Borel sets, proving that any Borel set is either countable or has the power of the continuum. For this Alexsandroff and Hausdorff used the property that any Borel set can be obtained by means of some constructive procedure. A clear description of this procedure was given by Suslin: he called it A-operation. Given a sequence of sets £ = {ESl„,Sn}, numbered "by processions" { s i , . . . ,sn},Si £ N, we construct the set A{E) according to the following rule:

A{E) = u(£ S l n ESlS2 n ESlS2S3 n • • • n £„...,„), where the union is carried out over all the sequences {s„} n 6 N. Then we say that A{£) is obtained from £ by A-operation. In 1916, the same year, Suslin obtained a result that made a great impression on his contemporaries. He proved that there are sets obtained from closed subsets of an interval by means of an A-operator that are not Borel sets. Sets obtained from closed ones by means of an ^-operator were called Suslin sets. Thus the class of Suslin sets (or A-sets) proved to be broader than that of Borel sets (B-sets) and it was soon discovered that many natural procedures of analysis lead to A-sets and j4-measurable functions that are not 5-measurable. In the autumn of 1921, Alexsandroff gave a course of lectures at the University of Moscow on descriptive set theory. Apparently he spoke in this course of the reoperation he had introduced — an operation "complementary" to the A-operation. Kolmogorov attended this course. He immediately tried to work out a maximal general scheme of set-theoretic operations (tf5-operations). Every map / : N ->• N generates a sequence {/(n)}„ 6 N- Let T = {/} be some family of such maps and £ = { E „ } „ 6 N some sequence of sets (for definitenesssubsets of the interval I — [0,1]). Definition. If the set $?(£) is formed by the rule $jr(5) = U/ejr ( n „ 6 N E / ( „ ) ) ,

133 then we say that it is obtained from £ by means of the Ss-operation generated byT. An A-operation is obviously a ^-operation. Kolmogorov defined in a general form a complementary ^-operation, of which AlexsandrofF's T-operation is a special case. Kolmogorov's first journey into mathematical research was devoted to these questions. The work was completed in January 1922, while he was a second-year undergraduate student. The paper of that particular research contains the germ of the general theory of operations on sets, which was later developed by Hausdorff, Kantorovich, Lyapunov and others. In the paper he proved the following fundamental result: There is a set of the form &?(£) the complement of which is not a set of the form $^r(5') for any sequence £' = {E'n}neN of (closed) sets E'n. Two of Kolmogorov's papers, from 1925 and 1932, are on mathematical logic. The 1925 paper is, in fact, the first paper on mathematical logic by a Russian mathematician. It was of great significance; witness its inclusion in Heijenoort's famous book on mathematical logic From Frege to Godel. From the beginning of the century there was no end to arguments about the essence of mathematics or the applicability of logical laws or paradoxes in set theory. All this led to a new line, "intuitionism". This recognized the invalidity of the application of "tertium non datur" (the law of the excluded middle) in the field of transfinite induction. In his paper "Tertium non datur", Kolmogorov proposed an interpretation of classical mathematics in which all of its propositions are converted into intuitionistic propositions. Thus, Kolmogorov proved that all finite conclusions obtained by means of transfinite application of the "tertium non datur" principle are true and can be proved without its help. In the 1932 paper, Kolmogorov proposed investigating "the logic of the solution of problems" (say, problems on construction) side-by-side with "the logic of proof. To the principle of syllogism: "if b follows from a, and c follows from b, then c follows from a", there corresponds "if the solution of problem a reduces to a solution of problem b, and the solution of problem b reduces to a solution of problem c, then the solution of problem a reduces to a solution of c". Kolmogorov introduced the corresponding symbolism and constructed a calculus of problems. The form of this calculus coincided with the system that was formulated slightly earlier by Heyting, of the axioms of intuitionistic logic. This gave another interpretation of intuitionistic logic. Kolmogorov published two papers on geometry. One of them ("On the foundations of projective geometry", 1932) reflects his youthful love of projective geometry "which was old-fashioned, but on which Aleksei Konstantinovich Vlasovc lectured in a truly talented way" (Kolmogorov always listed Vlasov among his teachers). In the early 1930s "synuhetic structures" were being developed extensively, when algebra united with topology (recall the representation theory of compact groups). Kolmogorov was at the beginning of "topological geometry". How beautiful and unexpected the links between geometry and topology can be is shown by the following result of Kolmogorov.

Professor of Moscow University; Kolmogorov heard his lectures devoted to projective geometry, when he was s student.

134

Let three systems of elements (called points, straight lines and planes) satisfy the projective axioms of combination, where each of the systems is a connected compactum and the relations of combination are continuous.01 Then a geometry that has the three given systems as objects is isomorphic to either R P 3 , C P 3 or H P , i.e. either real, or complex, or quaternion projective geometry. Here the most important thing is the idea of combining geometry and topology, along with the aesthetic element included in the formulation. The main point of the proof is the reference to Pontryagin's theorem (proved on Kolmogorov's suggestion) giving a description of all the connected locally compact topological skew-fields with countable base (there are only three: R, C and H). These facts undoubtedly belong to the golden treasure-house of topological algebra. This fully justifies the thought repeatedly expressed by Kolmogorov: Aesthetic motives not infrequently have interesting consequences. Four papers by Kolmogorov are concerned with functional analysis. In each of them there are important ideological elements. In a paper of 1931, Kolmogorov gave the first criterion for compactness in Lp. Later M. Riesz gave a similar criterion to the Arzela criterion — more usual and natural, but Kolmogorov's result remains significant, since it can easily be extended to any measure space, whereas Riesz's cannot. The best-known of these papers on functional analysis is undoubtedly a paper from 1934. This paper contains the definitions of a linear topological space and a bounded set in it (in his historical survey, Bourbaki singles out the importance of these concepts for the further development of functional analysis) and in which the following classical criterion of normability is proved: for a linear topological space to admit a norm it is necessary and sufficient that it has a bounded convex neighborhood of zero. At the source of the theory of normed rings created later by I. M. Gel'fand, is their joint paper of 1935 which contains the following beautiful result: Let T\ and T2 be two topological spaces (satisfying the first axiom of countability). For Ti and T2 to be homeomorphic it is necessary and sufficient that the rings C(T\) and CiT?) (of continuous functions on T\ and T2, respectively) or the rings Cb(T\) and Cb(T2) (of continuous bounded functions) be algebraically isomorphic; that is, an algebraic isomorphism of the algebras of continuous functions determines a topological homeomorphism of the compacta themselves. In Kolmogorov's papers on approximation theory, the formulation of the problem of revealing the connection between the concepts of approximation and smoothness is central. Kolmogorov owned the origination for the very important concepts of n-diameter and £-entropy. Kolmogorov made a fundamental contribution to algebraic topology. To him we owe the introduction of what is perhaps the central concept in the whole of this field — that of cohomology, one of the concepts fundamental to the whole of modern mathematics. Kolmogorov also defined the most significant operation in cohomology groups: the product operation, giving a ring structure. The American topologist Alexander had the same idea at the same time. Kolmogorov completed his own theory with the remarkable discovery of the duality We say that a straight line passing through two points depends continuously on these two points, and so on.

135 for closed sets in compact spaces. In the mid-1950s, Kolmogorov turned to the theory of algorithms and the theory of automata and set himself a task close in spirit to the one we discussed in connection with the theory of operations on sets. It is very well expressed in the notes to his 1958 joint paper with V. A. Uspenski: "Authors striving to interpret the concepts of 'a computable function' and 'an algorithm' have tried to survey the existing variants of the definitions in the literature and finally convinced themselves that there is no hidden possibility of extending the range of the concept of computable function'. The result of this work is the broadest (apparently, of all possible) definitions of the basic objects of the theory-algorithms and automata." In ergodic theory, Kolmogorov introduced two fundamental concepts — metric entropy (Kolmogorov's idea was definitively formulated by Ya. G. Sinai) and a quasiregular dynamical system. In almost every field to which he turned his attention, Kolmogorov initiated one or several principal scientific lines. From his research, tens or hundreds of research workers began to work, survey articles, monographs began to appear and scientific conferences were arranged. For example, in 1930s in the field of probability alone he laid the foundations of the theory of probability and of stochastic processes; he began to develop Markov processes, the theory of Markov chains with a countable number of states; in statistics, the theory of statistical criteria and (jointly with Khinchin) the theory of stationary processes. These directions appear as separate chapters in modern courses on probability theory, mathematical statistics and stochastic processes, and if we look at the content of these courses, we get the impression that the sum total of these new directions created by one man in the course of less than ten years can be compared with the sum total of the new directions that have emerged in these sciences over the last fifty years. Kolmogorov's ideas always had a remarkable vitality. They were quickly taken up, and eventually overtaken by successors, and were interwoven with others, forming scientific directions and schools. There is a multitude of methods, constructions and lemmas that Kolmogorov devised to solve individual problems, and they are constantly being used in further research. He developed particularly many methods in the theory of probability and stochastic processes (starting, say, with the well-known Kolmogorov inequality); he introduced many constructive ideas into the theory of functions, topology and other fields. For example, the number of uses of the comparison theory — an auxiliary result, which he obtained when proving the theorem on the intermediate derivative — is exceptionally large. The same applies to the principle of introducing a new invariant of entropy type to prove the nonisomorphism of various aggregates. But what had the greatest effect was his method of solving functional equations with small denominators — a special kind of implicit function theorem — which is the basis of KAM-theory. Peculiarity of the Creative Manner The creative power of a scholar can be seen in his achievements in the arts, whether in his originality as a writer, as a painter, or as a poet. The legacy of a great scholar, as any cultural phenomenon, has its own history, its own architecture, and possibly its own undeciphered plan. It is linked to profound conceptions, with consequences in culture in general.

136

At a superficial glance, Kolmogorov's genius may seen to be a pile of isolated and unrelated parts. Kolmogorov himself, it would seem, sometimes confirmed this verdict. "For me, prone to lack singleness of purpose ..." he wrote once to his teacher N. N. Luzin, and he repeated something similar many times. But if we consider his creativity more intently, we see that his efforts were all subject to one general plan and the diversity of what was planned and achieved was the result of the infinite number of goals he set for himself. Observing the creativity of great men brings us closer to discovering the nature of man. There are two opinions about the nature of genius. Thomas Mann — Kolmogorov's favorite author — dedicated his last work "Doktor Faustus" to one of them. For Mann, a genius is a person who, in order to achieve his goal which requires surmounting impossible difficulties, is compelled to make a pact with dark, demonic forces. In the novel the hero, Faust, consciously infects himself with a fatal disease, so that his inflamed brain can reflect the transcendental images and ideas. But there is another point of view. In Russian culture it is most clearly expressed in Pushkin's creative work (indeed in his own personality). In Pushkin's short tragedy 'Mozart and Salieri', Mozart was the shining chosen one who was given the gift of hearing the divine voice. It was most likely that Pushkin did not know the words written by Mozart in one of his letters, but he had a profound understanding of the essence of the very phenomenon of Mozart's genius. Mozart wrote, "Ideas come to me in quantities and with unusual facility. Where do they come from? I know nothing at all of this ... The work grows, I hear it more and more distinctly ... Then I comprehend it in one glance." To a large extent, the great discoveries made by Kolmogorov saw the light of day in a similar way. Kolmogorov was undoubtedly one of the "shining" geniuses. Over and over again one and the same motif crops up in his publications and discussions. Once, the remarkable Soviet geometer Boris Delone, appeared before participants of the School of the Mathematical Olympiad, and had expressed the idea that the only difference between a great scientific discovery and a good Olympiad problem is that solving the Olympiad problem requires five hours, but finding a powerful scientific result requires an investment of 5,000 hours. (Apparently this was the way Delone himself worked, and this was the manner in which many others such as Hilbert worked — we shall compare further the creativities of Hilbert and Kolmogorov.) Kolmogorov often recalled those words of Delone. But each time the talk turned to these famous 5,000 hours, he'd say with a feeling of some awkwardness and even of irritation, that he himself was unable to concentrate his thoughts for such a long time on one and the same problem. In an interview, Kolmogorov was asked the question "How do you work?" He said, "You read some books, you prepare your own lectures with some new variations or other, and suddenly, from the soil of this everyday work, some unexpected idea emerges, and, vaguely as yet, some completely different approach can be seen. Then having worked it out, almost everything else is neglected — and one thinks, thinks endlessly along the lines which have just appeared. Fortunately, I usually had the opportunity to do this, but in the whole history of my scientific discoveries such complete oblivion, cut off from everything else, might last for a week, sometimes possibly for two — not more." And then the sudden inspiration would come. Thus it was with the discovery of the trigonometrical series divergent almost everywhere, which was preceded by

137 three days of uninterrupted reflection and complete concentration; thus it was the final result, connected with Hilbert's thirteenth problem, and with the entropy invariant for dynamical systems and many others. After concentrating enormous energy there suddenly came the moment when the whole picture of the phenomenon appeared in its complete form. And then what? Wilhelm Ostwald's classification of scholars is well known. He divides them into classicists and romantics. "The first," Ostwald wrote, "can be compared to a mill, carefully grinding with logical millstones the initial ideas, in order to extract from them the most far-reaching possible consequences and to develop the theory to the limits of possible completion and perfection. The latter one must rather equate with generators of new ideas. Having stated the idea, they quickly lose interest in it, however brilliant it may have been, and take no part in the further development of it." This opinion was often quoted. I am not completely satisfied with Ostwald's classification, but the label "romantic" suits Kolmogorov very well. This accumulation of energy over a short span of time, which we spoke of earlier, led to a powerful explosion that caused gaping holes in what one would think of as impregnable bastions. Suddenly, there rushed in tens and hundreds of followers. Usually, Kolmogorov did not pursue the matter further; it was as though a creative tiredness would set in and he already had in his mind another goal in view. Something like that happened to him almost all the time. That was the case with the solution of Hilbert's thirteenth problem. When he had almost reached his goal, he declared that he would leave the completion of the work to his successors; and it was the same with problems in classical mechanics, ergodic theory and many other subjects. Kolmogorov was very much interested in the essence of creative talent. He distinguished algorithmic, purely geometric and logical faculties in man. Now, a division into two groups is more common (analytic and descriptive geometric), corresponding to the different functions of the two hemispheres of the brain. I find it difficult to say which of the qualities mentioned above were the most prominent in Kolmogorov's case. Analytic papers with an abundance of calculations and transformations exist side-by-side with his papers where "the art of a consistent, correctly partitioned logical argument" plays a fundamental role. And besides, in his research in function theory and topology, for example, there are geometric constructions which stand out by their beauty. By the way, Kolmogorov was left-handed from birth, and as a child he taught himself to use his right hand, which he did well. We cannot rule out that he may be one of that rare breed of men, each of whose hemispheres fulfills both functions, so that asymmetry is really nonexistent. An essential part of an intellect is its quickness of thought. P. S. Alexsandroff once spoke jokingly of "quick" and "slow" geniuses, and he included Hilbert among the latter. Kolmogorov was undoubtedly one of the "quick" geniuses, but what stuck me personally most of all about him was not so much the quickness of his thinking as the quickness of his grasp and his perception. I was often a witness of the way scarcely perceptible emanations or very vague rough drafts which came to his attention, were immediately incorporated into an ordered and complete system. (Sometimes one had the impression that Kolmogorov was one of those people who

138 have as it were "a priori" knowledge — it seemed he knew everything, although it was not quite clear when he had learned all of it.) And it was unusually interesting to observe how suddenly, at the right moment, things surfaced in his conciousness such as the universal tree of Menger and Kronrod's constructions, or the ideas of Poincare—Bogoliubov-Kryolv and the NewtonKantorovich method, the Pontryagin-Shnirei'man designs and Shannon's theory. It is natural to raise the questions: Who had the greatest influence on Kolmogorov, on "whose shoulders" did he stand, on what is his creativity mainly based? This subject is still awaiting a more careful study, and we shall only touch upon it. In the first volume of Kolmogorov's selected works there are 60 articles. In none of them does he express any thanks to his teachers or colleagues (for help, for useful advice, for suggesting the problems, or anything else). In 15 of these articles there is no bibliography at all, and in the others the work of 93 mathematicians is mentioned. Of scholars of former ages, only Aristotle and Leibniz are cited, of modern times (and in a not very important context) only Grothendieck is mentioned. All others are famous scientists from a preceding generation and from his own generation. Of his predecessors the most significant references are to Hadamard, Birkhoff, Borel, Brouwer, Hilbert, Caratheodory, Lebesgue, Luzin, Taylor, von Karman, Hardy and Hausdorff; in other volumes to Chebyshev, Bernstein, von Mises and Fisher. I think that is precisely these individuals who must be included among those of his predecessors that he was most indebted to. Somewhat surprising is the absence of references to Poincare. This is largely because Kolmogorov learned of Poincare's ideas by reading the works of Chazy and Chariier. The other mathematicians to whom Kolmogorov refers to were part of the current scientific scene. Here we must mention the great influence the works of Krylov-Bogoliubov and de Rham had on him. More than once in this article there has appeared the name of one of the greatest mathematicians of all time, David Hilbert. Here I would like to draw some comparisons between the genius of Hilbert and that of Kolmogorov. Intellectual achievements cannot be completely classified (they make up "a partially ordered set"), and geniuses cannot be compared. My comparison of them arises not from a wish to contrast one with the other or to elevate the one at the expense of the other, but from an attempt to demonstrate how far creative styles can differ and by what different paths they can achieve their great aims. Hilbert had a determination not matched I dare say by anyone else's. His scientific biography is sharply divided into periods, dedicated to work in one field. Hilbert's first creative period was devoted to algebra and arithmetic (the theory of invariants 1885-1893, algebraic number theory 1893-1898); in the first decade of the century in the course of two summers (1908-1909) he solved Waring's problem. Kolmogorov did not write any papers on algebra or arithmetic, just as Hilbert did not work in the field of random phenomena. The next stage of Hilbert's interests was geometry (1989-1902). It ended with his classic book Grundlagen der Geometrie. Kolmogorov did work in the field of geometry, in topology and algebraic topology in the 1930s, while working at the same time in the field of analysis, probability theory, function theory and many others. Then came Hilbert's "analytic" period (1900-1910), when he developed the calculus of variations and infinite-dimensional analysis (in particular, the theory

139 of integral equations). Kolmogorov turned to problems of mathematical analysis and the theory of functions in two main periods of his life (trigonometrical and orthogonal series, the theory of approximation, functional analysis in the 1930s, dynamical systems, superpositions, ergodic theory in the 1950s). Then Hilbert went through the period when he solved mathematical problems in the natural sciences, mainly in mathematical physics (1910-1922). Kolmogorov studied many problems in natural sciences: Biology and physics in the 1930s, turbulence, geology and meteorology in the 1940s, celestial mechanics in the 1950s, and oceangraphy in the 1970s. Hilbert's last period was devoted to the logical foundations of mathematics. Kolmogorov was interested in logic both in the 1930s and 1950s, which was in the very last period of his creativity. He, like Hilbert, concluded his creative life with work on logic. As we said, Hilbert did no work on random phenomena. The theory of probability, on the contrary, was a constant concern of Kolmogorov's. He was involved in it in from 1920s to 1950s, and moreover, he was not just involved in classical problems, but also in random processes, statistics, information theory and numerous applications. Hilbert could only work on one subject at a time. Kolmogorov, at the height of his creativity, could work down to the foundations, the roots, the very essence of the subject; Kolmogorov was striving to go forward, new goals beckoned him, he tried hard to conquer unattainable heights, leaving it to others to incorporate the new territories. But both achieved brilliant results, they introduced fundamental ideas, worked out deep theories, created fruitful methods and basic concepts. Furthermore, they were both great teachers. The number of mathematicians who they influenced cannot be counted. But, the number of their actual students can be compared. The list of Hilbert's disciples is given in his collected works. The specialities of the students correspond exactly to the various interests of their teacher at the given time. And now I would like to present a list of Kolmogorov's pupils with whom he had worked during the decade 1953-1963: V. M. Alekseev (classical mechanics), V. I. Arnol'd (superpositions, classical mechanics), G. I. Barenblatt (hydrodynamics), L. A. Bassalygo (information theory), Yu. K. Belyaev (stochastic processes), L. N. Bol'shev (mathematical statistics), A. A. Borovkov (probability theory), R. L. Dobrushin (probability theory), E. B. Dynkin (stochastic processes), V. D. Erokhin (approximation theory), S. V. Fomin (ergodic theory), I. M. Gel'fand, B. V. Gnedenko (probability theory), L. A. Levin (complexity), A. I. Mal'tsev (mathematical logic), P. Martin-Lef (complexity), Yu. T. Medvedev (mathematical logic), L. D. Meshalkin (probability theory, ergodic theory), V. S. Mikhalevich (probability theory), A. S. Monin (oceanology, turbulence), S. M. Nikol'skii (approximation theory), A. M. Obukhov (atmospheric physics, turbulence), M. S. Pinsker (information theory), Yu. V. Prokhorov (probability theory), Yu. A. Rozanov (stochastic processes), B. A. Sevast'yanov (stochastic processes), A. N. Shiryaev (stochastic processes), Ya. G. Sinai (ergodic theory), S. Kh. Sirazhdinov (probability theory), V. M. Tikhomirov (approximation theory), V. A. Uspenskii (mathematical logic), A. M. Yaglom (turbulence) and V. M. Zolotarev (probability theory). Kolmogorov did most of his research work in his house in Komarovka, outside Moscow. He loved this house very much and there he played the role of gardener. As a result of his fifty years of work in the Komarovka garden, it had huge birches,

140

larches, maples, crab-apples, and the whole garden was full of flowers-lilacs, roses, and peonies. Whatever Kolmogorov planted flourished. Once, many years ago, he gave a colleague a bush of sweet briar. This colleague was no longer alive and now little remains of the once rich and well-tended garden. Only the sweet briar, given by Kolmogorov, to this day does not cease to flower and bear fruit. This luck of a gardener illuminated both his original research and his work as a teacher; everything he touched flowered and bore fruit. Let me go back to where I started. The heritage of a great man is rich beyond price; the comprehension of which confirms our faith in mankind, and the personality of a genius sets a moral example and let us discover directions in life. What is conducive to a genius? What sets him in motion? How splendidly and happily were combined in the fate of Andrei Nikolaevich Kolmogorov a cheerful childhood, a remarkable secondary school, an unusual time resulting in an atmosphere of creative elan during his youth, and the outstanding School of Mathematics at the University of Moscow, early recognition, and an unusual gift of originality ... But let us not forget that without which even the combination of all these qualities would not lead to such a beneficent outcome: the presence of inner stimuli in the final analysis on the fundamental moral perseverance of the individual. Kolmogorov himself once said on this subject: "It is absolutely essential in science (as in poetry, music, etc.) that the man who has the right moral qualities should accept his work as a crucial duty." And here, it seems to me, the great commandment: the greatest stimulus in life is the voluntary acceptance of one's duty. In answer to the question: "By whom were you guided in life?" Kolmogorov replied thus: "I always considered that the most important thing is truth." Striving for truth is one of the basic aims in the life of a genius. Kolmogorov is one of those incomparable people who embellish life by the simple fact of their existence. The mere awareness of the fact that somewhere on earth, beats the heart of a man who is endowed with such perfect wisdom and such a pure soul gave wings, gave joy, gave the strength to live, and protecting us from evil and inspiring us to good deeds.

141 On Normability of a General Topological Linear Space 1

Our discussion is based on the following definition of topological linear space. A set E is a topological linear space if (a) for the elements of E the operations of addition and multiplication by real numbers are defined which satisfy the axioms for a linear space 2 ; (b) for any subset A of E its closure A C E is defined, which satisfies three axioms for a topological space 3 ; (c) the operations of addition and multiplication are continuous. Open sets of a topological space in the sense of footnote 3, regarded as neighborhoods of the points of these sets, satisfy in general the first three Hausdorff axioms 4 and the first axiom of separation. However, in our case of topological linear spaces as well as in the general case of topological groups, the second and third axioms of separation are necessarily fulfilled, i.e. the space is regular. This fact will be proved in §1. In this connection it seems quite natural that the general theory of linear functionals and operators should be developed in particular for topological linear spaces. A considerable part of this theory, however, has been developed only for the case of normed spaces, i.e. spaces where each element is associated with a non-negative number \x\ satisfying the following conditions: \ax\ = \a\ \x\, \x + y\<\x\

(1)

+ \y\;

(2)

and the topology of the space is determined by the distance between two elements: p{x,y) = \x-y\.

(3)

The question arises: which topological linear spaces can be normed? In other words, what conditions should be imposed so that a topological linear space can be endowed with a norm satisfying conditions (1) and (2) and determining the a priori topological relations in this space? In order to answer this question we will use the following definition.5 1 "Zur Normierbarkeit eines allgemeinen topologischen linearen Raumes", Stud. Math. 5 (1934), 29-33. 2 See S. Banach, Theorie des operations lineaires, Warszawa, 1931, Chap. 2. 3 See, for example, P. Alexandroff, "Uber stetige Abbildungen kompakter Raume", Math. Ann. 96 (1926), 555. These axioms read as follows:

1. A subset A C E consisting of at most one element coincides with its closure. 2. A = A for any set Ac E. 3. MUN = MUN. 4

See F. Hausdorff, Grundziige der Mengenlehre, Berlin, 1927, 227-229. S . Mazur and W. Orlicz, "Uber Folgen linearer operationen", Stud. Math. particularly p. 152. 5

4 (1933), 152-157,

142 Definition. A set A C E is said to be bounded if for any sequence {an} of real numbers and for any sequence {xn} of elements of A the condition an —>• 0 implies a-n^n —• 0 (where 0 in the latter relation denotes the origin of the space E). This definition enables us to state the following theorem: T h e o r e m . For a space E to be normable it is necessary and sufficient that in E there exist at least one bounded convex neighborhood of zero. Here the set A C E is said to be convex if x £ A, y £ A, A > 0 and fi>0

^±™£A.

imply

(4)

This theorem will be proved in §2. §1. In this section, E is an arbitrary topological group, i.e. an arbitrary topological space on which an operation of addition is defined satisfying all the group axioms, and the addition and subtraction are continuous. We will prove that in this case E is regular or, in other words, that for any neighborhood U(XQ) of an element XQ £ E there exists a neighborhood V(xo) which belongs to U(XQ) together with its closure. Using the transformation x' = x — XQ we can reduce this problem to the case XQ = 0. Thus, let a neighborhood U of zero be given. Since 0 + 0 = 0, we can, by the continuity of addition, find a neighborhood of zero V such that x' £ V, x" £ V implies x' + x" £ U. Now let x £ V. Since x — x — 0 and subtraction is continuous, there exists a neighborhood W(x) of x such that x' £ W(x), x" G W(x) implies x' — x" £ V. We now find a point x* that belongs simultaneously to W(x) and V (x* exists since x £V). Since x and x* belong to W(x), it follows that x — x* £V. Further, since x — x* and x* belong to V, it follows that x = (x — x*) + x* belongs to U, whence VCU, (5) and so on. §2. The necessity of the conditions of the theorem is obvious since the unit ball, consisting of all points with \x\ < 1, is a convex bounded neighborhood of zero. It remains to show that the conditions of the theorem are sufficient. Let U be a convex bounded neighborhood of zero. We denote by aU the set of points x = ax' such that x' £ U. For any a ^ 0 the set aU is a convex bounded neighborhood of zero. We define the norm by |a;| = sup \a\,

x £ E\ aU ,

(6)

where the supremum extends over all a satisfying the condition x £ E\ aU. It can be seen immediately that the norm defined by (6) satisfies condition (1). We now show that condition (2) also holds. To this end we first note that x £ aU and y £ aU imply Xx

+ ™£aU X+H for all A > 0 and /x > 0 and therefore it follows from |x| < a and \y\ < a that Xx + fiy —;

S ct.

143 Putting |a:| = A, \y\ = p., X + fx = a, a j\x\ \x + y\

y' = -y we obtain i 'i

= a,

a

\

i

\y\ = - W = a >

Xx' + fiy'
+ \y\

It remains to show that the distance p(x,y) = \x — y\ determines the same limiting relations as in the space E. Let us consider the point 0 and prove that the sets all, a > 0 form a complete system of neighborhoods of zero. Let W be an arbitrary neighborhood of zero. We have to find a > 0 such that all is contained in W. If this were impossible, there would exist sequences an —• 0 and xn G U such that the points anxn would lie outside W for all n and therefore the sequence anxn would not tend to zero, which contradicts the boundedness of U. Therefore the sets all form a complete system of neighborhoods of zero. It can be seen immediately that the ball |a;| < a is contained in the set aU. Therefore, if we prove that 0 is an interior point of every ball \x\ < a, a > 0, this will imply that the system of balls |ai| < a, a > 0, is equivalent to the complete system of neighborhoods all of zero. But each ball \x\ < a, a > 0 contains a neighborhood of zero, coinciding with the intersection 6 of ^alf and —^all. Indeed, if x e \aU and x £ — \aU, then x lies in every neighborhood bU with |6| > a/2, whence it follows that |a;| < ka < a. Using the transformation x' = x — Xo we can see that the system of balls with center at XQ is equivalent to the complete system of neighborhoods at this point. Thus, the theorem is proved completely. 26 May 1934

144

LIST OF W O R K S B Y A. N. KOLMOGOROV 1923 1. Une serie de Fourier-Lebesgue divergente presque partout, Fund. Math. 4, 324-328. 2. Sur l'ordre de grandeur des coefficients de la serie de Fourier-Lebesgue, Bull. Acad. Pol. Sci. A, 83-86. 1924 3. Une contribution a l'etude de la convergence des series de Fourier, Fund. Math. 5, 96-97. 4. (with G. A. Sehverstov) Sur la convergence des series de Fourier, C.R. Acad. Sci. Paris 178, 303-306. 1925 5. La definition axiomatique de 1'integrale, C.R. Acad. Sci. Paris 180, 110-111. 6. Sur les bornes de la generalisation de l'integrale. (Paper No. 6 in the Selected Works of A. N. Kohnogorov, Vol. 1, Kluwer, 1991.) 7. Sur la possibility de la definition generale de la derivee, de l'integrale et de la sommation des series divergentes, C.R. Acad. Sci. Paris 180, 362-364. 8. Sur les fonctions harmoniques conjuguees et les series de Fourier, Fund. Math. 7, 24-29. 9. On the tertium non datur principle, Mat. Sb. 32:4, 646-667 (in Russian). 10. (with A. Ya. Khinchin) Uber Konvergenz von Reihen, deren Glieder durch den Zufall bestimmt werden, Mat. Sb. 32:4, 668-677. 1926 11. (with G. A. Sehverstov) Sur la convergence des series de Fourier, Atii Accad. Naz. Lincei. Rend. 3, 307-310. 12. Une serie de Fourier-Lebesgue divergente partout, C.R. Acad. Sci. Paris 183, 1327-1329. 1927 13. Sur la loi des grands nombres, C.R. Acad. Sci. Paris 185, 917-919. 14. (with D. E. Men'shov) Sur la convergence des series de fonctions orthogonales, Math. Z. 26:2/3, 421-441. 1928 15. On operations on sets, Mat. Sb. 35:3/4, 414-422 (in Russian). 16. Sur une formule limite de M. A. Khinchine, C.R. Acad. Sci. Paris 186, 824-825. 17. Sur un procede d'integration de M. Denjoy, Fund. Math. 11, 27-28. 18. Uber die Summen durch den Zufall bestimmer unabhangiger Grossen, Math. Ann. 99, 309-319.

145 1929 19. Bemerkungen zu meiner Arbeit "Uber die Summen zufalliger Grossen", Math. Ann. 102, 484-488. 20. General measure theory and calculus of probabilities, in Proc. Communist Academy Math. Sect. Vol. I, 8-21 (in Russian). 21. Contemporary dispute on the nature of mathematics, Nauch. Slo. No. 6, 41-54 (in Russian). 22. Uber das Gesetz des iterierten Logarithmus, Math. Ann. 101, 126-135. 23. Sur la loi des grand nombres, Atti Accad. Naz. Lincei. Rend. 9:6, 470-474. 1930 24. Sur la loi forte des grands nombre, C.R. Acad. Sci. Paris 191, 910-912. 25. Zur topologish-gruppentheoretishen Begriindung der Geometrie, Nachr. Ges. Wiss. Gottingen. Fachgr. 1 (Mathematik) 8, 208-210. 26. Untersuchungen iiber den Integralbegriff, Math. Ann. 103, 654-696. 27. Sur la notion de la moyenne, Atti Accad. Naz. Lincei. Rend. 12:9, 388-391. 1931 28. Uber die analitishen Wahrscheinlichkeitsrechnung, Math. Ann. 104, 415-458. 29. Sur le probleme d'attente, Mat. Sb. 38:1/2, 101-106. 30. The method of medians in the theory of errors, Mat. Sb. 38:3/4, 47-50 (in Russian). 31. Eine Verallgemeinerung des Laplace-Liapounoffshen Satzes, Izv. Akad. Nauk SSSR OMEN, 959-962. 32. Uber Kompaktheit der Funktionenmengen bei der Konvergenz im Mittel, Nachr. Ges. Wiss. Gottingen 9, 60-63. 1932 33. Theory of functions of a real variable, in Mathematics in the USSR During 15 Years, GTTI, Moscow, Leningrad, pp. 37-48 (in Russian). 34. Sulla forma generale di un processo stocastico omogeneo. (Un problema di Bruno de Finetti), Atti Accad. Naz. Lincei. Rend. 15, 805-808. 35. Ancora sull forma generale di un processo stocastico omogeneo, Atti Accad. Naz. Lincei. Rend. 15, 866-869. 36. Zur Deutung der intuitionistischen Logik, Math. Z. 35, 58-65. 37. Zur Begriindung der projektiven Geometrie, Ann. Math. 33, 175-176. 38. (with P. S. Aleksandrov) Introduction to the Theory of Functions of a Real Variable (GTTI, Moscow, Leningrad) (in Russian).

39. 40. 41. 42. 43.

1933 (with P. S. Aleksandrov) Introduction to the Theory of Functions of a Real Variable (2nd ed.) (GTTI, Moscow, Leningrad). Grundbegriffe der Wahrscheinlichkeitsrechnung (Springer-Verlag). Beitrage zur Masstheorie, Math. Ann. 107, 351-366. (with M. Leontovich) Zur Berechnung der mittleren Brounschen Flache, Phys. Z. Sow. 4 : 1 , 1-13. Sulla determinazione empirica di una legge di distrebuzione, G. 1st. Ital. Attuar. 4, 83-91.

44. Uber die Grenzwersatze der WahrscheinUchkeitsrechnung, Izv. Akad. Nauk SSSR OMEN, 366-372. 45. Zur Theorie der stetigen zufalligen Prozesse, Math. Ann. 108, 149—160. 46. Sur la determination empirique d'une loi de distribution, Uch. Zap. MGU 1, 9-10. 47. On the question of the applicability of prognosis formulas found in a statistical manner, Zh. Geofiz. 3:1, 78-82 (in Russian). 1934 48. (with I. Ya. Verchenko) On points of discontinuity of functions of two variables, Dokl. Akad. Nauk SSSR 1:3, 105-106. 49. Zur Normierbarkeit eines allgemeinen topologischen linearen Raumes, Stud. Math. 5, 29-33. 50. (with I. Ya. Verchenko) Continuation of the study of points of discontinuity of functions of two variables, Dokl. Akad. Nauk SSSR 4:7, 361-362 (in Russian). 51. On convergence of series in orthogonal polynomials, Dokl. Akad. Nauk SSSR 1:6, 291-294 (in Russian). 52. Quelques remarques sur l'approximation des fonctions continues, Mat. Sb. 41:1, 99-103. 53. On certain new trends in probability theory, in Bull. USSR Second Congress of Mathematicians in Leningrad, June 24-30, 1934 (USSR Academy of Sciences Press) 8 (in Russian). 54. Modern mathematics, Front Nauki Tekh. No. 5/6, 25-28 (in Russian). 55. The Institute of Math, and Mech. at MGU, Front Nauki Tekh., No. 5/6, 75-78 (in Russian). 56. Zufallige Bewegungen (Zur Theorie der Brownschen Bewegung), Ann. Math. 35, 116-117.

57.

58. 59. 60.

61. 62.

63. 64.

1935 On certain modern trends in probability theory, in Proc. USSR Second Congress of Mathematicians in Leningrad, June 24-30, 1934, Vol. 1 (USSR Academy of Sciences Press), 349-358 (in Russian). Deviation from a formula of Hardy for partial isolation, Dokl. Akad. Nauk SSSR 3:3, 129-132 (in Russian). La transformation de Laplace dans les espaces lineaires, C.R. Acad. Set. Paris 200, 1717-1718. Zur Grossenordnung des restgliedes Fourierschen Reihen differenzzierbarer Funktionen, Ann. Math. 36, 521-526. 1936 Uber die beste Annaherung von Funktionen einer gegebenen Funktionenklasse, Ann. Math. 37, 107-110. Basic Notions of Probability Theory (ONTI, Moscow) (in Russian). Translated from German: Grundbegriffe der WahrscheinUchkeitsrechnung (SpringerVerlag, 1933). Equations, in BSE 56, 163-165 (in Russian). Modern mathematics, in Collection of Papers on the Philosophy of Mathematics (ONTI, Moscow), pp. 7-13 (in Russian).

147 65. Theory and practice in mathematics, Front Nauki Tekh. No. 5, 39-42 (in Russian). 66. Uber die Dualitat im Aufbau der kombinatorischen Topologie, Mat. Sb. 1, 97-102. 67. Anfangs griinde der Theorie der Markoffschen Ketten mit unendlich vielen moglichen Zustanden, Mat. Sb. 1, 607-610. 68. Homologierung des Komplexes mid des lokalbikompakten Raumes, Mat. Sb. 1, 701-706. 69. Zur Theorie der Markoffschen Ketten, Math. Ann. 112, 155-160. 70. On Plessner's condition for the law of large numbers, Mat. Sb. 1, 847-849 (in Russian). 71. (with P. S. Aleksandrov) Endliche Uberdeckungen topologischer Raumes, Fund. Math. 26, 267-271. 72. Les groupes de Betti des espaces localement bicompacts, C.R. Acad. Sci. Paris 202,1144-1147. 73. Proprietes des groupes de Betti des espaces localement bicompacts, C.R. Acad. Sci. Paris 202, 1325-1327. 74. Les groupes de Betti des espaces metriques, C.R. Acad. Sci. Paris 202, 1558-1560. 75. Cycles relatifs. Theoreme de dualite de M. Alexander, C.R. Acad. Sci. Paris 202,1641-1643. 76. Foreword to the Russian translation of the book by A. Heyting: Review of Studies on Foundations of Mathematics. Introduction to Proof Theory (ONTI, Moscow) (in Russian). 77. Sulla teoria di Volterra della lotta per l'esistenca, G. 1st. Ital. Attuar. 7, 74-80. 1937 78. Uber offene Abbildungen, Ann. Math. 38, 36-38. 79. Skew-symmetric quantities and topological invariants, in Proc. Seminar Vector and Tensor Anal, and Appl. to Geometry, Mechanics, and Physics, 4, (GONTI, Moscow), pp. 342-347 (in Russian). 80. Markov chains with countable set of states, Bull. MGU Mat. Mekh. 1:3, 1-16 (in Russian). 81. (with I. G. Petrovskii and N. S. Piskunov) Studies on the diffusion equation, combined with increase in the amount of matter and its application to a problem in biology, Bull. MGU Mat. Mekh. 1:6, 1-26 (in Russian). 82. On the statistical theory of crystallization of metals, Izv. Akad. Nauk SSSR Ser. Mat. No. 3, 355-360 (in Russian). 83. Ein vereinfachter Beweis der Birkhoff-Khintchineschen Ergodensatzes, Mat. Sb. 2, 367-368. 84. Zur Umkehrbarkeit der statistischen Naturgesetze, Math. Ann. 113, 766-772. 85. (with P. S. Aleksandrov) Editing and supplementing the translation of the book by F. Hausdorff: Set Theory (GONTI, Moscow) (in Russian). 86. The continuum, in BSE 34, 139-140. 1938 87. (with P. S. Aleksandrov) Introduction to the Theory of Functions of a Real Variable (3rded. (revised)), (GONTI, Moscow) (in Russian) (lsted.: Moscow, 1932).

148 88. 89. 90. 91. 92. 93. 94. 95. 96. 97.

98. 99. 100. 101.

102.

A. A. Markov, in BSE 38, 152-153 (in Russian). Mathematics, in BSE 38, 359-402 (in Russian). Mathematical induction, in BSE 38, 405-406 (in Russian). Measure, in BSE 38, 831-832 (in Russian). Multidimensional space, in BSE 39, 577-578 (in Russian). Probability theory and its applications, in Mathematics and Natural Sciences in the USSR (GONTI, Moscow), pp. 51-61 (in Russian). On the section on information in the first issue of Uspekhi Mat. Nauk, Usp. Mat. Nauk No. 4, 326-327 (in Russian). A remark on the foundations of geometry, Usp. Mat. Nauk No. 4, 347-348 (in Russian). From the board of editors (Series of articles on the theory of random processes), Usp. Mat. Nauk No. 5, 3-4 (in Russian). Uber die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Usp. Mat. Nauk No. 5, 5-41. Translated from German, Math. Ann. 104, 415-458 (1931). Ein vereinfachter Beweis der Birkhoff-Khintchineschen Ergodensatzes, Usp. Mat. Nauk No. 5, 52-56. Translated from German, Mat. Sb. 2, 367-368 (1937). (with G. M. Fichtengol'ts and I. M. Gel'fand) Some problems in the theory of functions of a real variable, Usp. Mat, Nauk No. 5, 232-234 (in Russian). On the solution of a problem in biology, Izv. Nil Mat. Mekh. Tomsk, Univ. 2:1, 7-12 (in Russian). Une generalisation de Finegalite de M. J. Hadamard entre les bornes superieures des derivees successives d'une fonction, C.R. Acad. Sci. Paris 207, 764-765. Editing the Russian translation of the book by H. Lebesgue: On Measurement of Quantities (Uchpedgiz, Moscow) (in Russian).

1939 103. (with P. S. Aleksandrov) Algebra, Chapter 1 (Uchpedgiz, Moscow) (in Russian). 104. Orientation, in BSE 43, 342-344 (in Russian). 105. On inequalities between least upper bounds of sequences of derivatives of an arbitrary function on an infinite interval, Uch. Zap. MGU Mat. 3:30, 3-16 (in Russian). 106. (with I. M. Gel'fand) On rings of continuous functions on topological spaces, Dokl. Akad. Nauk SSSR 22:1, 11-15 (in Russian). 107. Sur l'interpolation et extrapolation des suites stationnaires, C.R. Acad. Sci. Paris 208, 2043-2045. 1940 108. Surfaces, in BSE 45, 746-748 (in Russian). 109. Curves in a Hilbert space, invariant under a one-parameter group of motions, Dokl. Akad. Nauk SSSR 26, 6-9 (in Russian). 110. Wiener spirals and some other interesting curves in Hilbert space, Dokl. Akad. Nauk SSSR 26, 115-118 (in Russian). 111. (with V. L. Goncharov) On the 60th birthday of S. N. Bernshtein, Izv. Akad. Nauk SSSR Ser. Mat. 4, 249-260 (in Russian).

149 112. V. I. Glivenko, Usp. Mat. Nauk No. 8, 379-383 (in Russian). 113. Review of V. I. Romanovsky, Mathematical statistics, Usp. Mat. Nauk No. 7, 327-329 (in Russian). 114. On a new argument for Mendel's laws, Dokl. Akad. Nauk SSSR 27, 38-42 (in Russian). 1941 115. Stationary sequences in Hilbert space, Bull. MGU Mat. 2:6,1-10 (in Russian). 116. Interpolation and extrapolation of stationary random sequences, Izv. Akad. Nauk SSSR Ser. Mat. 5, 3-14 (in Russian). 117. Points of local topological character of countably multiple open mappings of compacta, Dokl. Akad. Nauk SSSR 30, 477-479 (in Russian). 118. Local structure of turbulence in an incompressible viscous fluid for very large Reynolds numbers, Dokl. Akad. Nauk SSSR 30, 299-303 (in Russian). 119. On the lognormal distribution for particle sizes under grinding, Dokl. Akad. Nauk SSSR 31, 99-101 (in Russian). 120. On degeneration of isotropic turbulence in an incompressible viscous fluid, Dokl. Akad. Nauk SSSR 31, 538-541 (in Russian). 121. Energy scattering for local isotropic turbulence, Dokl. Akad. Nauk SSSR 32:1, 19-21 (in Russian). 122. (with P. S. Aleksandrov) Properties of inequalities and the notion of approximate computations, Mat. Shk. No. 2, 1-12 (in Russian). 123. (with P. S. Aleksandrov) Irrational numbers, Mat. Shk. No. 3, 1-15 (in Russian). 124. Confidence limits for an unknown distribution function, Ann. Math. Stat. 12:4, 461-463. 1942 125. Definition of the scatter centre and measures of exactness using a finite number of observations, Izv. Akad. Nauk SSSR Ser. Mat. 6, 3-32 (in Russian). 126. The equation of turbulent motion of an incompressible fluid, Izv. Akad. Nauk SSSR Ser. Fiz. 6:1-2, 56-58 (in Russian). 1943 127. (with P. S. Aleksandrov) Nikolai Ivanovich Lobachevskii, 1793-1843 (Gostekhizdat, Moscow) (in Russian). 128. A great Russian scientific innovator: on the 150th birthday of N. I. Lobachevskii, Izvestiya (2 September 1943) (in Russian). 1945 129. The number of hits for several shots, and general principles for estimating the efficiency of shooting systems, Trudy Mat. Inst. Akad. Nauk SSSR 12, 7-25 (in Russian). 130. Artificial scatters in the case of hitting in one shot and scatter in one measurement, Trudy Mat. Inst. Akad. Nauk SSSR 12, 26-45 (in Russian).

150 1946 131. On the problem of the law of resistance in a turbulent flow in smooth tubes, Doki Akad. Nauk SSSR 52, 669-671 (in Russian). 132. On the substantiation of the method of least squares, Usp. Mat. Nauk 1:1, 57-70 (in Russian). 133. On the foundations of the theory of real numbers, Usp. Mat. Nauk 1:1, 217-219 (in Russian). 134. Newton and modern mathematical thought, in Moscow University - In Commemoration of Isaac Newton, 1643-1943 (Moscow Univ. Press), pp. 27-43 (in Russian). 135. Review of M. A. Velikanov's article "Transfer of suspended pumps by a turbulent flow", Izv. Akad. Nauk SSSR OTN 5, 781-784 (in Russian). 1947 136. Development of mathematics in the USSR, in BSE, Vol. "SSSR", 1318-1323 (in Russian). 137. Average quantities, in BSE 52, 508-509 (in Russian). 138. (with A. A. Petrov and Yu. M. Smirnov) A formula of Gauss from the theory of the least squares method, Izv. Akad. Nauk SSSR Ser. Mat. 11, 561-566 (in Russian). 139. (with N. A. Dmitriev) Branching random processes, Dokl. Akad. Nauk SSSR 56, 7-10 (in Russian). 140. (with B. A. Sevast'yanov) Computation of final probabilities for branching random processes, Dokl. Akad. Nauk SSSR 56, 783-786 (in Russian). 141. Statistical theory of oscillation with continuous spectrum, in Collection of Articles Dedicated to the 30th Anniversary of the Great October Socialist Revolution, Vol. 1 (USSR Acad. Sci. Press), pp. 242-252 (in Russian). 142. The role of Russian science in the development of probability theory, in The Role of Russian Science in the Development of World Science and Culture, Vol. 1 (Moscow Univ. Press), pp. 53-64 (Uchen. Zap. MGU, No. 91). 1948 143. Statistical theory of oscillation with continuous spectrum, in General Meeting of the USSR Acad. Sci. Dedicated to the 30th Anniversary of the Great October Socialist Revolution (USSR Acad. Sci. Press), pp. 465-472 (in Russian). 144. (with B. V. Gnedenko) Probability theory, in Mathematics in the USSR During Thirty Years, 1917-1947 (Gostekhizdat), pp. 701-727 (in Russian). 145. On two theorems regarding probabilities: comments, in P. L. Chebyshev, Complete Collection of Works, Vol. 3, Mathematical Analysis (Gostekhizdat), pp. 404-409 (in Russian). 146. Construction of complete metric Boolean algebras, Usp. Mat. Nauk 3:1, 212 (in Russian). 147. Notes regarding the Chebyshev polynomials of least deviation from a given function, Usp. Mat.Nauk 3:1, 216-221 (in Russian). 148. E. E. Slutskii, Usp. Mat Nauk 3:4, 143-151 (in Russian). 149. Editing and writing the foreword to the Russian translation of the book by H. Cramer: Mathematical Methods of Statistics (Inost. Lit., Moscow) (in Russian).

151 150. Algebres de Boole metriques completes, in VI Zjazd Mat. Polskich, pp. 22-30. 1949 151. (with B. V. Gnedenko) Limit Distributions for Sums of Independent Random Variables (Gostekhizdat) (in Russian). 152. On the problem of 'geometrical selection' of crystals, Dokl. Akad. Nauk SSSR 65, 681-684 (in Russian). 153. Solution of a problem in mathematical statistics, related to the problem of the mechanism of fibre-formation, Dokl. Akad. Nauk SSSR 65, 793-796 (in Russian). 154. (with Yu. V. Prokhorov) On sums of a random number of random terms, Usp. Mat. Nauk 4:4, 168-172 (in Russian). 155. A local limit theorem for classical Markov chains, Izv. Akad. Nauk SSSR Ser. Mat. 13, 281-300 (in Russian). 156. Basic problems of theoretical statistics, in Proc. 2nd USSR Conf. Math. Statistics, Tashkent, 27 Sept.-2 Oct., 1948, Uzbekgosizdat, Tashkent, pp. 216-220 (in Russian). 157. The true meaning of results by dispersion analysis, in Proc. 2nd USSR Conf. Math. Statistics, Tashkent, 27 Sept.-2 Oct., 1948, Uzbekgosizdat, Tashkent, pp. 240-268 (in Russian). 158. On the breakage of drops in a turbulent flow, Dokl. Akad. Nauk SSSR 66:5, 825-828 (in Russian). 159. Absolute quantities, in BSE-2 1, 32 (in Russian). 160. J. Hadamard, in BSE-2 1, 388 (in Russian). 161. Additive quantities, in BSE-2 1, 394 (in Russian). 162. Axioms, in BSE-2 1, 613-616 (in Russian). 163. Axonometry, in BSE-2 1, 617 (in Russian). 1950 164. Unbiased estimators, Izv. Akad. Nauk SSSR Ser. Mat. 14, 303-326 (in Russian). 165. On the problem of determining the coefficient of temperature-conductivity of soil, Izv. Akad. Nauk SSSR Ser. Geogr. Geofiz 2, 97-98 (in Russian). 166. Algebra in secondary school, in BSE-2 2, 61-62 (in Russian). 167. Algebraic expressions, in BSE-2 2, 64 (in Russian). 168. Algorithms, in BSE-2 2, 65 (in Russian). 169. Euclid's algorithm, in BSE-2 2, 65-67 (in Russian). 170. A. D. Aleksandrov, in BSE-2 2, 83 (in Russian). 171. R S. Aleksandrov, in BSE-2 2, 84 (in Russian). 172. Asymptote, in BSE-2 3, 238-239 (in Russian). 173. Asymptotic expressions, in BSE-2 3, 239 (in Russian). 174. N. I. Akhiezer, in BSE-2 3, 565 (in Russian). 175. S. Banach, in BSE-2 4, 183 (in Russian). N. K. Bari, in BSE-2 4, 245 (in Russian). 176. S. N. Bernshtein, in BSE-2 5, 52 (in Russian). 177. Infinitely large, in BSE-2 5, 66-67 (in Russian). 178. (with V. F. Kagan) Infinitesimal, in BSE-2 5, 67-71 (in Russian).

152 179. (with B. V. Delone) Infinitely distant elements, in BSE-2 Russian). 180. Infinity (in mathematics), in BSE-2 5, 73-74 (in Russian). 181. Biharmonic function, in BSE-2 5, 159 (in Russian). 182. Bilinear form, in BSE-2 5, 167 (in Russian). 183. Law of large numbers, in BSE-2 5, 538-540 (in Russian).

5, 71-72 (in

1951 184. On the problem of differentiability of transition probabilities in timehomogeneous Markov processes with countable number of states, Uch. Zap. MGU 148, Mat. 4, 53-59 (in Russian). 185. Generalizations of Poisson's formula to the case of selection from a finite sample, Usp. Mat. Nauk 6:3, 133-134 (in Russian). 186. I. G. Petrovskii, Usp. Mat. Nauk 6:3, 161-164 (in Russian). 187. Statistical quality control for admissible number of defective articles equal to zero, Leningrad, pp. 1-24 (in Russian). 188. (with S. A. Yanovskaya) L. E. J. Brouwer, in BSE-2 6, 62 (in Russian). 189. Variational series, in BSE-2 6, 641 (in Russian). 190. (with S. A. Yanovskaya) H. Weyl, in BSE-2 7, 106 (in Russian). 191. Quantity, in BSE-2 7, 340-341 (in Russian). 192. Probable deviation, in BSE-2 7, 507 (in Russian). 193. Probability, in BSE-2 7, 508-510 (in Russian). 194. (with T. I. Kozlov) Sampling method, in BSE-2 9, 417-418 (in Russian). 195. (with A. N. Khovanskii) Introduction and commentary to the book: N. I. Lobachevskii, Complete Collected Works, Vol. 5 (Gostekhizdat), pp. 329-332; 342-348 (in Russian). 1952 196. On the problem of pressure in the velocity profile under turbulent flow in tubes, Dokl. Akad. Nauk SSSR 84, 20-30 (in Russian). 197. Gaussian distribution, in BSE-2 10, 275 (in Russian). 198. Geodesic curvature, in BSE-2 10, 481 (in Russian). 199. D. Hilbert, in BSE-2 11, 370-371 (in Russian). 200. Histogram, in BSE-2 11, 447 (in Russian). 201. B. V. Gnedenko, in BSE-2 11, 545 (in Russian). 202. Homeomorphism, in BSE-2 12, 21 (in Russian). 203. Homotopy, in BSE-2 12, 35 (in Russian). 204. Motion (in geometry), in BSE-2 13, 447 (in Russian). 205. Two-term, in BSE-2 13, 518 (in Russian). 206. Real numbers, in BSE-2 13, 570 (in Russian). 207. Division, in BSE-2 13, 628 (in Russian). 208. Discreteness, in BSE-2 14, 425 (in Russian). 209. Variance, in BSE-2 14, 438 (in Russian). 210. Distributivity, in BSE-2 14, 479 (in Russian). 211. Distributive operator, in BSE-2 14, 479 (in Russian). 212. Differential, in BSE-2 14, 498-499 (in Russian). 213. (with B. P. Demidovich and V. V. Nemytskii) Differential equation, in BSE-2 14, 520-526 (in Russian).

153 214. Confidence probability, in BSE-2 14, 616 (in Russian). 215. Confidence bound, in BSE-2 14, 617 (in Russian). 216. (with I. G. Bashmakova and A. P. Yushkevich) Mathematical symbols, in BSE-2 17, 115-119 (in Russian). 217. Significant digits, in BSE-2 17, 135 (in Russian). 218. (with V. I. Bityutskov) Isomorphism, in BSE-2 17, 478-479 (in Russian). 219. Isotropic line, in BSE-2 17, 509 (in Russian). 220. Concrete number, in BSE-2 17, 557 (in Russian). 221. V. G. Imshenetskii, in BSE-2 17, 607 (in Russian). 222. On the Profession of a Mathematician: A Guide for Those Entering Colleges, (Sovetskaya Nauka) (in Russian). 1953 223. On the notion of algorithm, Usp. Mat. Nauk 8:4, 175-176 (in Russian). 224. Some papers of the last decade on the limit theorems of probability theory, Vestn. MGU 10, 29-38 (in Russian). 225. On dynamical systems with an integral invariant on a torus, Dokl. Akad. Nauk SSSR 93, 736-766 (in Russian). 226. Mathematical induction, in BSE-2 18, 146 (in Russian). 227. (with V. I. Glivenko) Integral, in BSE-2 18, 250-253 (in Russian). 228. Probability integral, in BSE-2 18, 253 (in Russian). 229. Interpolation, in BSE-2 18, 304-305 (in Russian). 230. Intuitionism, in BSE-2 18, 319 (in Russian). 231. Elimination of variables, in BSE-2 18, 483 (in Russian). 232. Testing, in BSE-2 18, 604 (in Russian). 233. Method of exhaustion, in BSE-2 19, 50-51 (in Russian). 234. Quadrant, in BSE-2 20, 434 (in Russian). 235. Compactum, in BSE-2 22, 282 (in Russian). 236. Constant, in BSE-2 22, 416 (in Russian). 237. Continuum, in BSE-2 22, 454-455 (in Russian). 238. Coordinates, in BSE-2 22, 524-525 (in Russian). 239. Correlation, in BSE-2 23, 55-58 (in Russian). 1954 240. On the preservation of conditionally periodic motions under small variations of the Hamilton function, Dokl. Akad. Nauk SSSR 98:4, 527-530 (in Russian). 241. The general theory of dynamical systems and classical mechanics, in Proc. Int. Math. Congress, Amsterdam, 1954, Review lectures (USSR Acad. Sci. Press, 1961), pp. 187-208 (in Russian). 242. Line (curve), in BSE-2 25, 167-170 (in Russian). 243. Law of small numbers, in BSE-2 26, 168 (in Russian). 244. A. A. Markov, in BSE-2 26, 294 (in Russian). 245. Mathematics, in BSE-2 26, 464-483 (in Russian). 246. Mathematical statistics, in BSE-2 28, 485-490 (in Russian). 247. Mathematical physics, in BSE-2 26, 490 (in Russian). 248. R. von Mises, in BSE-2 27, 414 (in Russian). 249. Multidimensional space, in BSE-2 27, 660 (in Russian). 250. (with P. S. Aleksandrov) Set theory, in BSE-2 28, 14-17 (in Russian).

154 251. (with S. V. Fomin) Elements of Theory of Functions and Functional Analysis: A Course of Lectures. No. 1: Metric and Normed Spaces (Moscow Univ. Press) (in Russian). 1955 252. Estimates of the minimal number of elements of £-nets in various function classes and their application to the problem of representing functions of several variables by superpositions of functions of a smaller number of variables, Usp. Mat. Nauk 10:1, 192-194 (in Russian). 253. Orientation, in BSE-2 3 1 , 188-189 (in Russian). 254. Foundations of geometry, in BSE-2 3 1 , 296 (in Russian). 255. (with L. A. Skornyakov) Surface, in BSE-2 33, 346-347 (in Russian). 256. Ordinal number, in BSE-2 34, 238 (in Russian). 257. Statistical quality control, in BSE-2 34, 498-499 (in Russian). 258. Estimates of the minimal number of elements of e-nets in various function classes and their application to the problem of representing functions of several variables by superpositions of functions of a smaller number of variables, Dokl. Akad. Nauk SSSR 101, 192-194 (in Russian). 1956 259. On Skorokhod convergence, Teor. Veroyainosi. Primenen. 1, 239-247 (in Russian). 260. Two uniform limit theorems for sums of independent terms, Teor. Veroyainosi. Primenen. 1, 426-436 (in Russian). 261. (with Yu. V. Prokhorov) Zufallige Funktionen und Grenzverteiilugssatze, in Bericht iiber die Tagung Wahrscheinlichkeitsrechnung und Mathematische Statistik, Berlin, pp. 113-126. 262. Some principal problems in the approximate or exact representation of functions of one or several variables, in Proc. 3rd USSR Math. Congress, Vol. 2 (Moscow Univ. Press), pp. 28-29 (in Russian). 263. On the Shannon theory of information transmission in the case of continuous signals, IEEE Trans. Inform. Theory, IT-2, 102-108. 264. On the representation of continuous functions of several variables as superpositions of continuous functions of a smaller number of variables, Dokl. Akad. Nauk SSSR 108:2, 179-182 (in Russian). 265. On some asymptotic characteristics of totally bounded metric spaces, Dokl. Akad. Nauk SSSR 108, 385-388 (in Russian). 266. (with I. M. Gel'fand and A. M. Yaglom) On a general definition of amount of information, Dokl. Akad. Nauk SSSR 111, 745-748 (in Russian). 267. Probability theory, in Mathematics, Its Content, Methods, and Significance, Vol. 2 (USSR Acad. Sci. Press), pp. 252-284 (in Russian). 268. (with S. B. Stechkin) S. M. Nikol'skii: on his 50th birthday, Usp. Mat. Nauk 11:2, 239-244 (in Russian). 269. E. E. Slutskii, in BSE-2 39, 378 (in Russian). 270. N. V. Smirnov, in BSE-2 39, 406 (in Russian).

155 1957 271. Theory of information transmission, in USSR Acad. Sci. Session on Scientific Problems in Automatization of Industry, 15-20 October 1956: Plenary meetings (USSR Acad. Sci. Press), pp. 66-69 (in Russian). 272. On the representation of continuous functions of several variables as superpositions of continuous functions of one variable and addition, Dokl. Akad. Nauk SSSR 114:5, 953-956 (in Russian). 273. On the foundation of the theory of real numbers, Mat. Prosveshchenie 2, 169-173 (in Russian).

274. 275.

276. 277. 278. 279. 280. 281. 282.

283. 284. 285. 286. 287. 288.

1958 On the Profession of a Mathematician (2nd ed.) (Moscow Univ. Press) (in Russian). (with I. M. Gel'fand and A. M. Yaglom) The amount of information and entropy for continuous distributions, in Proc. 3rd USSR Math. Congress, Vol. 3 (USSR Acad. Sci. Press), pp. 300-320 (in Russian). Sufficient statistic, in BSE-2 51, 106 (in Russian). Information, in BSE-2 51, 129-130 (in Russian). Cybernetics, in BSE-2 51, 149-151 (in Russian). A new metric invariant of transitive dynamical systems and automorphisms of Lebesgue spaces, Dokl. Akad. Nauk SSSR 119, 861-864 (in Russian). Sur les proprietes des fonctions de concentrations de M. P. Levy, Ann. Inst. Henri Poincare 16:1, 27-34. On the linear dimension of topological vector spaces, Dokl. Akad. Nauk SSSR 120, 239-241 (in Russian). (with V. A. Uspenskii) On the definition of an algorithm, Usp. Mat. Nauk 13:4, 3-28 (in Russian). 1959 On entropy in unit time as a metric invariant of automorphisms, Dokl. Akad. Nauk SSSR 124, 754-755 (in Russian). (with V. M. Tikhomirov) e-entropy and e-capacity of sets in function spaces, Usp. Mat. Nauk 14:2, 8-86 (in Russian). Transition of branching processes into diffusion processes and related problems of genetics, Teor. Veroyatnost. Primenen. 4, 233-236 (in Russian). Notes on the works of R. A. Minlos and V. V. Sazonov, Teor. Veroyatnost. Primenen. 4, 237-239 (in Russian). Probability theory, in Mathematics in the USSR During 40 Years, Vol. 1, Fiz-matgiz, Moscow, pp. 781-795 (in Russian). Foreword to the Russian translation of the book by W. R. Ashby: Introduction to Cybernetics (Inost. Lit., Moscow) (in Russian).

1960 289. (with O. V. Sarmanov) On Bernshtein's work in probability theory, Teor. Veroyatnost. Primenen. 5, 215-221 (in Russian). 290. On the classes $( n ) of Ford and Blanc-Lapierre, Teor. Veroyatnost. Primenen 5, 373 (in Russian).

156 291. Random functions of several variables, almost all realizations of which are periodic, Teor. Veroyatnost. Primenen. 5, 374 (in Russian). 292. (with B. V. Gnedenko, Yu. V. Prokhorov and 0 . V. Sarmanov) On N. V. Smirnov's work in probability theory: on his 60th birthday, Teor. Veroyatnost. Primenen. 5, 436-440 (in Russian). 293. (with Yu. A. Rozanov) On strong mixing conditions for Gaussian stationary processes, Teor. Veroyatnost. Primenen. 5, 222-227 (in Russian). 294. (with B. V. Gnedenko) A. Ya. Khinchin, Usp. Mat. Nauk 15:4, 97-110 (in Russian). 295. On the Profession of a Mathematician, 3rd ed. (Moscow Univ. Press) (in Russian). 296. (with S. V. Fomin) Elements of the Theory of Functions and Functional Analysis, No. 2: Measure, Lebesgue Integral and Hilhert Space (Moscow Univ. Press) (in Russian). 1961 297. Automata and life, Mash. Perevod Prikl. Lingvistika 9, 3-8 (in Russian). 298. Automata and life, Tekh. Molodezhi, No. 10, 16-19; No. 11, 30-33 (in Russian). 299. Note on the paper by V. K. Lezerson, Teor. Veroyatnost. Primenen. 6, 367 (in Russian). 300. (with P. S. Aleksandrov) Properties of inequalities and the notions of approximate computations, in Questions of Mathematics Teaching in School (Uchpedgiz, Moscow) (in Russian). 301.

302. 303. 304. 305.

306.

1962 (with M. Arato and Ya. G. Sinai) On an estimate of parameters of a complex stationary Gaussian Markov process, Dokl. Akad. Nauk SSSR 146, 747-750 (in Russian). (with A. M. Kondratov) The rhythm of Mayakovskii's poem, Vopros. Yazykoznaniya, No. 3, 62-74 (in Russian). (with L. S. Pontryagin and E. F. Mishchenko) On a probabilistic problem in optimal control, Dokl. Akad. Nauk SSSR 145, 993-995 (in Russian). On the work of B. V. Gnedenko in probability theory, Teor. Veroyainost. Primenen. 7, 323-329 (in Russian). A refinement of the conception of the local structure of turbulence in an incompressible viscous fluid at large Reynolds numbers, in Mecanique de la Turbulence: Colloq. Int. CNRS, Marseille, Aout-Sept 1961, Paris, pp. 447-458. A refinement of the conception of the local structure of turbulence in an incompressible viscous fluid at large Reynolds numbers, J. Fluid Mech. 13:1, 82-85.

1963 307. On approximating the distribution of a sum of independent terms of infinitely divisible distributions, Trudy Moskov. Mat. Obshch. 12, 437-451 (in Russian). 308. Discrete automata and finite algorithms, in Proc. 4th USSR Math. Congress, Vol. 1 (Leningrad Univ. Press), p. 120 (in Russian).

157 309. Various approaches to estimating the difficulty of approximately specifying and computing' functions, in Proc. Int. Congress Math. Stockholm, pp. 369-376 (in Russian). 310. On tables of random numbers, Ind. J. Stat. Ser. A 25:4, 369-376. 311. (with A. V. Prokhorov) On the down-to-earthness of modern Russian poetry: a general characteristic, Vopros. Yazykoznaniya No. 6, 84-95 (in Russian). 312. On the study of Mayakovskii's rhythm, Vopros. Yazykoznaniya No. 4, 64-71 (in Russian). 313. (with A. V. Prokhorov) Statistics and probability theory in the study of Russian poetry compositions, in Abstracts and Annotations Symp. on Complex Investigation of Artistic Creative Work (Nauka), p. 23 (in Russian). 314. How I became a mathematician, Ogonek No. 48, 12-13 (in Russian). 315. Foreword to the Russian translation of the book by C. A. Shannon: Works on Information Theory and Cybernetics (Inost. Lit., Moscow) (in Russian). 1964 316. (with A. V. Prokhorov) On the down-to-earthness of modern poetry: a statistical characteristic of this with Mayakovskii, Bagritskii, Akhmatova, Vopros. Yazykoznaniya No. 1, 75-94 (in Russian). 317. On the metre in Pushkin's "Songs for Eastern Slaves", Rus. Lit. No. 1, 98-111 (in Russian). 318. Foreword to the Russian translation of the book by W. Feller: Introduction to Probability Theory and Its Applications (2nd ed.) (Mir, Moscow) (in Russian). 1965 319. Three approaches to the definition of "amount of information", Prohlemy Peredachi Inform. 1:1, 3-11 (in Russian). 320. Geometric education in geometry courses at school, Mat. Shk. No. 2, 24-29 (in Russian). 321. (with I. M. Yaglom) On the contents of the school course of mathematics, Mat. Shk. No. 4, 53-62 (in Russian). 322. Functions, graphs, continuous functions, Mat. Shk. No. 6, 12-21 (in Russian). 323. Notes on the analysis of the rhythm of Mayakovskii's "Verses on a Soviet passport", Vopros. Yazykoznaniya No. 3, 70-75 (in Russian). 1966 324. Introduction to Analysis (Moscow Univ. Press) (in Russian). 325. P. S. Aleksandrov and the theory of 6s-operations, Usp. Mat. Nauk 21:4, 275-278 (in Russian). 326. On the school definition of identity, Mat. Shk. No. 2, 33-35 (in Russian). 327. Geometry on the sphere and geology, Nauka Zhizii, No. 2, 32 (in Russian). 1967 328. (with Ya. M. Bardzin) On the realization of sets in three-dimensional space, Probt. Kibnernetiki 19, 261-268 (in Russian). 329. New programmes and some basic questions on the advanced course in mathematics at secondary school, Mat. Shk. No. 2, 4-13 (in Russian).

158 1968 330. Some theorems on algorithmic entropy and algorithmic amount of information, Usp. Mat. Nauk 21:2, 201 (in Russian). 331. On the study of the exponential function and logarithms in the 8-year school, Mat. Shk. No. 2, 23-25 (in Russian). 332. Generalizations of the notion of power and exponential functions, Mat. Shk. No. 1, 24-32 (in Russian). 333. Introduction to probability theory and combinatorics, Mat. Shk. No. 2, 63-72 (in Russian). 334. (with S. V. Fomin) Elements of the Theory of Functions and Functional Analysis (2nd ed. revised and extended) (Fizmatgiz, Moscow) (in Russian). 335. (with A. V. Prokhorov) On the foundations of classical Russian metre, in Cooperation of Sciences and Mystery of Creative Work (Iskusstvo, Moscow), pp. 397-432 (in Russian). 336. Example of the study of metre and its rhythmic variants, in Prosody (Nauka), pp. 145-167 (in Russian). 1969 337. On the logical foundations of information theory and probability theory, Problemy Peredachi Inform. 5:3, 3-7 (in Russian). 338. S. N. Bernshtein, Usp. Mat. Nauk 24:3, 211-218 (in Russian). 339. Scientific foundation for a school course in mathematics, Mat. Shk. No. 3, 12-17; No. 5, 8-17 (in Russian). 340. New material in school mathematics, Nauka Zhizri No. 3, 62-66 (in Russian). 1970 341. V. N. Zasuhin: to the memory of a mathematician who died in World War II, Usp. Mat. Nauk 25:3, 243 (in Russian). 342. G. A. Seliverstov, Usp. Mat. Nauk 25:3, 244-245 (in Russian). 343. Generalizations of the notion of number, a non-negative rational number, Mat. Shk. No. 4, 27-32 (in Russian). 344. What is a function?, Kvant No. 1, 27-36 (in Russian). 345. What is the graph of a function?, Kvant No. 2, 3-13 (in Russian). 346. Tilings from regular polyhedra, Kvant No. 3, 24 (in Russian).

347. 348. 349. 350. 351. 352. 353.

1971 Quantity, in BSE-3 4, 456-457 (in Russian). N. Wiener, in BSE-3 5, 72 (in Russian). D. Hilbert, in BSE-3 6, 519 (in Russian). On a system of basic notions and notation for a school course of mathematics, Mat. Shk. No. 2, 17-22 (in Russian). Modern mathematics and mathematics at modern school, Mat. Shk. No. 6, 2-3 (in Russian). (with V. A. Gusev, A. B. Sosinskii and A. A. Shershevskii) A Course of Mathematics for Physico-Mathematical Schools (Moscow Univ. Press) (in Russian). Summer School at Rubskoe Lake, Prosveshchenie, Moscow (in Russian).

159 1972 354. Complexity of specification and complexity of construction of mathematical objects, Usp. Mat. Nauk 27:2, 159 (in Russian). 355. Integral, in BSE-3 10, 300-302 (in Russian). 356. Method of exhaustion, in BSE-3 10, 586 (in Russian). 357. (with S. V. Fomin) Elements of the Theory of Functions and Functional Analysis (3rd revised ed.) (Fizmatgiz, Moscow) (in Russian). 358. Scientific handbook, in L. Neiman, Delight of Discovery (Detskaya Literatura, Moscow), pp. 160-164 (in Russian). 359. There is no substitute for teachers, Koms. Pravda (19 January 1972) (in Russian). 360. A qualitative study of mathematical models of population dynamics, Probi. Kibernetiki 25:2, 101-106 (in Russian). 1973 361. Continuum, in BSE-3 13, 64 (in Russian). 362. Semilogarithmic and logarithmic sets, Kvant No. 4, 2 (in Russian). 363. On the profession of the mathematician, Kvant No. 4, 12 (in Russian). 1974 364. Basic Notions of Probability Theory (2nd ed.) (Nauka) (in Russian). 365. (with P. S. Aleksandrov and O. A. Oleinik) To the memory of I. G. Petrovskii (18 January 1901-15 January 1974), Trudy Moskov. Mat. Obshch. 31, 5-16 (in Russian). 366. I. G. Petrovskii, Usp. Mat. Nauk 29:2, 3-5 (in Russian). 367. A. A. Markov, in BSE-3 15, 379 (in Russian). 368. Mathematics, in BSE-3 15, 467-478 (in Russian). 369. (with Yu. V. Prokhorov) Mathematical statistics, in BSE-3 15, 480-484 (in Russian). 370. Multidimensional space, in BSE-3 16, 372 (in Russian). 371. Orientation, in BSE-3 18, 508-510 (in Russian). 372. (with I. T. Tropin and K. V. Chernyshev) Concern about adequate stock replacement, Vestn. Vyssh. Shk. No. 6, 26-33 (in Russian). 373. New programmes: specialized schools, in Mathematics Education Today (Prosveshchenie, Moscow), pp. 5-12 (in Russian). 374. The sieve of Eratosthenes, Kvant No. 10, 2 (in Russian). 1975 375. (with Yu. K. Belyaev) Statistical quality control, in BSE-3 20, 572-573 (in Russian). 376. Elements of combinatorics, Mat. Shk. No. 2, 16-25 (in Russian). 377. (with O. S. Ivashev-Musatov) Real numbers, infinite sequences and their limits, Mat. Shk. No. 2, 25-35 (in Russian). 1976 378. (with S. V. Fomin) Elements of the Theory of Functions and Functional Analysis (4th revised ed.) (Nauka) (in Russian). 379. (with S. I. Shvartsburd) Trigonometric functions, their graphs and derivatives, tenth grade school fashion, Mat. Shk. No. 1, 10-25 (in Russian).

160 380. (with G. A. Galperin) Moscow 38th Math. Olympiad, Mat. Shk. No. 4, 68-72 (in Russian). 381. Integral, tenth grade school fashion, Mat. Shk. No. 6, 15-17 (in Russian). 382. Transformation groups, Kvant No. 10, 2-5 (in Russian).

383. 384. 385. 386.

1977 Infinity, in Math. Encyclopedia, Vol. 1, 455-458 (in Russian). Quantity, in Math. Encyclopedia, Vol. 1, 651-653 (in Russian). Probability, in Math. Encyclopedia, Vol. 1, 667-669 (in Russian). (with V. V. Vavilov) The physics-mathematics school at MGU, Kvant No. 1, 56-57 (in Russian).

1978 387. On education on the levels of mathematics and physics of a dialecticalmaterialistic world view, Mat. Shk. No. 3, 6-9 (in Russian). 388. (with I. G. Zhurbenko) Estimates of the spectral functions of random processes, in 11th European Conf. Statistics, Abstract of Papers. 14-18 August 1978, Oslo, p. 36 (in Russian). 1979 389. (with A. V. Bulinskii) Linear sample estimates of sums, Teor. Veroyatnost. Primenen 24, 241-251 (in Russian). 390. (with V. V. Vavilov and I. T. Tropin) Fiz.-Mat. Shkola at MGU: 15 years, Kvant No. 1, 55-57 (in Russian). 1980 391. The dialectical-materialistic world view in school courses of mathematics and physics, Kvant No. 4, 15-18 (in Russian). 1981 392. (with S. V. Fomin) Elements of the Theory of Functions and Functional Analysis (5th ed.) (Nauka) (in Russian). 393. (with V. V. Vavilov and I. T. Tropin) Physico-Mathematical School at Moscow University (Moscow Univ. Press) (in Russian); (Mat. Kibemetika, No. 5). 394. (with A. F. Semenovich and P. S. Cherkasov) Geometry for 6-Sth Forms: Textbook (3rd ed.), Prosveshchenie, Moscow (in Russian) (1st ed., 1979). 395. On the notion of vector in courses of secondary school, Mat. Shk. No. 3, 7-8 (in Russian).

396. 397. 398. 399. 400.

1982 (with A. G. Dragalin) Introduction to Mathematical Logic (Moscow Univ. Press) (in Russian). (with I. G. Zhurbenko and A. V. Prokhorov) Introduction to Probability Theory (Nauka) (in Russian); (Bibliotechka "Kvani", No. 23). Mathematics, in Math. Encyclopedia Vol. 3, pp. 560-564 (in Russian). (with Yu. V. Prokhorov) Mathematical statistics, in Math. Encyclopedia Vol. 3, pp. 576-581 (in Russian). Newton and modern mathematical thought, Mat. Shk. No. 6, 58 (in Russian).

161 1983 401. (with A. M. Abramov, B. E. Veits, O. S. Ivashev-Musatov and S. I. Shvartsburd) Algebra and Elements of Calculus: Textbook for the 9th and 10th Forms (4th ed.) (Prosveshchenie, Moscow) (in Russian) (1st ed., 1980). 402. On the science textbook "Geometry 6-10", Mat. Shk. No. 2, 45 (in Russian). 403. Combinatorial foundations of information theory and computation of probabilities, Usp. Mat. Nauk 38:4, 27-36 (in Russian).

162 OBITUARY M R A N D R E I KOLMOGOROV Giant of Mathematics Mr Andrei Kolmogorov, Russian mathematician, died in Moscow on October 20. He was 84. His unique distinction was to have been the living embodiment of the theory of mathematical probability, and to have had countless pupils and disciples the world over others. Kolmogorov was born at Tambov on April 25, 1903. He was educated at Moscow University, where he subsequently taught as an instructor from 1925 until 1931, in which year he was made professor. He had held the chair of theory of probability since 1937. Much important work in probability had already been done without benefit of any foundations, but his little book, published in German, "Foundations of the Calculus of Probabilities" (1933), immediately became the definitive formulation of the subject. This was no mere codification of the findings of others. It is studded with beautiful theorems now bearing his name: the famous Strong Law of Large Numbers is just one of these. It further provided the essential basis for the study of stochastic ("random") systems dispersed in space and/or developing in time, and set the stage for the fruitful concept of a filtration of sigma-algebras representing the progressive swelling of the corpus of available information. This lies at the heart of the modern theory of random processes; it is an essential concept in the very idea of control theory, and it plays a vital role in Kolmogorov's later synthesis of information theory and ergodic theory. Another later theme was a penetrating analysis of randomness itself from the standpoint of the theory of the complexity of algorithms. This has become a key idea hinting at a link between classical and statistical mechanics. The theory of the smoothing and prediction of stationary time-theories is usually associated with the name of Norbert Wiener, another scholar of universal scope, but in fact it was developed simultaneously and independently by Wiener and by Kolmogorov during the Second World War. Those random processes called "Markovian" have a specially simple predictive structure. Kolmogorov created the basic theory for these and also constructed pathological examples which are still receiving study. He was interested in every branch of science; he and his pupils wrote about crystal growth, about the geometry of the interactions of plants, and also made significant contributions to "birth and death" processes and to genetics. One of these led to a head-on confrontation with Lysenko. Another joint work (with Piscounov and Petrovsky) treated the rate of advance of an advantageous gene in a linear environment (a topic studied independently by R. A. Fisher, for whom Kolmogorov had a high regard). This was later adapted to describe spreading of epidemics of innovations, and of rumours.

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In 1941, Kolmogorov made a contribution of the greatest importance to the theory of turbulent fluid-flow; in two short papers he proposed hypotheses putting into concise mathematical form ideas about the structure of the small-scale components of turbulent motion latent in earlier experimental work, particularly by G. I. Taylor. According to these, the small-scale structure has a universal form determined by two parameters: one was the rate of dissipation of energy in the fluid and the other (relevant only to the smallest scales in which dissipation occurs) was the fluid viscosity. These hypotheses imply many qualitative results (following from dimensional arguments) that are widely applicable — what goes on, for example, within the turbulence that occurs in the wake of a jet aircraft. Subsequent research has shown the hypotheses to be valid to a good approximation, though subtle refinement is needed because of the highly non-uniform or "spotty" nature of the energy dissipation in the fluid, as he himself showed two decades later. In the long (and unfinished) history of research on turbulence there has perhaps been no contribution more important than these two papers published in English in 1941 in the comptes rendus (record of proceedings) of the Soviet Academy and (remarkably) delivered to libraries in Britain during the war. In 1954, Kolmogorov made a seminal contribution to the fundamental problem of classical dynamics identified some fifty years earlier by the French mathematician Henri Poincare in his study of the motion of the planets round the sun. The motion of a single planet (neglecting the others) provides an example of an "integrable" problem and is well understood. The small effects associated with gravitational interaction between the planets introduces a profound qualitative change related to the fact that the governing equations are now "non-integrable". The phase trajectories of the unperturbed integrable system lie on a family of nested toroidal (ring-shaped) surfaces (or "tori") in the phase space. Kolmogorov's great achievement in 1954 was to demonstrate that a large set of these tori (the so-called non-resonant tori) survive the perturbation. This was a breakthrough in the study of general Hamiltonian systems in which the "K. A. M. tori" (named after Kolmogorov, his brilliant student V. I. Arnold, and J. Moser) play a pivotal role. Subsequent computational studies aptly confirm Kolmogorov's insights and have opened up the enormously fruitful field of "chaos in dynamical systems" which is currently attracting much attention. These studies could lead, for example, to more accurate weather forecasting. Kolmogorov enjoyed the company of young people, and encouraged them by pointing out possibilities latent in their scientific ideas. He devoted much of his time to improving the teaching of mathematics in secondary schools in the Soviet Union, and in providing special schools for the mathematically gifted. His dacha near Moscow became a meeting place for mathematicians. One of them said: "It is just like Oberwolfach (a mathematical institute in the Black Forest), except that here Kolmogorov buys all the drinks."

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For many years he attempted to capture in quantitative form some aspects of Russian poetry, especially that of Pushkin. To hear him lecture about this, with examples, was fascinating whether one understood Russian or not. It is, perhaps, a matter of regret that none of his work in this area has been translated into English. Kolmogorov was loved and admired by colleagues in all parts of the world. For two generations, each probabilistic compass needle has pointed to Moscow. The shock of losing him will be tempered only by gratitude for having known him and by the pleasure of recalling his memory with others.

Wolf Prize in Mathematics, Vol. 2 (pp. 165-217) eds. S. S. Chern and F. Hirzebruch © 2001 World Scientific Publishing Co. Integr. equ. oper. theory 30 (1998) 123 - 134 © Birkhauser Verlag, Basel, 1998

Integral Equations and Operator Theory

MARK GRIGORIEVICH KREIN: RECOLLECTIONS

ISRAEL GOHBERG

1. C U R R I C U L U M VITAE Mark G. Krein was born into a Jewish family of modest means in Kiev,on April 3, 1907. His father was a lumber merchant. As a youngster he exhibited a talent for mathematics. At the age of 14 he was already attending research seminars. He never obtained an undergraduate degree. In 1924 he ran away from home to Odessa and in 1926 he was accepted for doctoral studies by N.G. Chebotarev at Odessa University. He completed the degree requirements in 1929. M. G. Krein was an excellent and enthusiastic teacher. He attracted many students. In the thirties he created at Odessa University one of the strongest centers of functional analysis throughout the world. His interests included matrices and integral equations, geometry of Banach spaces, moment problems, spectral theory of linear operators, extension problems and applications. Many of the results of his outstanding students of this period (A.B. Artemenko, M.S. Livsic, D.P. Milman, M.A. Naimark, V.P. Potapov, M.A. Rutman, V.L. Shmuljan), as well as joint results together with his friends and colleagues (N.I. Achiezer, F.R. Gantmacher), are now characterized as classical and appear in textbooks on functional analysis. During World War II, from 1941 to 1944, M. G. Krein held the Chair of Theoretical Mechanics at the Kuibyshev (now Samara) Industrial Institute. M. G. preferred this position over the Chair in Mathematics because he thought that in technical institute of higher education it offered more interesting work and a wider set of possibilities and responsibilities. In 1944 he returned to Odessa. Soon he and his closest friend, B.Ja. Levin, were dismissed from Odessa University. This step was a direct result of the antisemitic Communist Party policy and corruption of the administration of the university (by the

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way t h e letter of dismissal was h a n d e d to M . G . on his b i r t h d a y ) . Also their s t u d e n t s h a d to leave. This was t h e end of t h e famous center of Functional Analysis at Odessa University. T h e a d m i n i s t r a t i o n of Odessa University replaced M. G. Krein by more "reliable" m a t h e m a t i c i a n s , such as N. A. Lednev, a n d N. I. Gavrilov.

T h e former became

famous for his Marxist critique of Einstein's T h e o r y of Relativity, while t h e l a t t e r claimed publicly t h a t he h a d resolved a n u m b e r of t h e o u t s t a n d i n g open problems in M a t h e m a t i c s including t h e R i e m a n n hypothesis. E a c h time elementary mistakes were found, a n d he tried t o force t h e acceptance of his a r g u m e n t s u n d e r t h e pressure of t h e C o m m u n i s t P a r t y a d m i n i s t r a t i o n . F u r t h e r c o m m e n t s , I t h i n k are superfluous. ^From 1944 to 1952 M . G . K r e i n held a p a r t - t i m e position as head of t h e Departm e n t of Functional Analysis a n d Algebra at t h e M a t h e m a t i c a l I n s t i t u t e of t h e Ukrainian A c a d e m y of Science in Kiev. He was dismissed from this post in 1952. T h e official reason given was t h a t he was not a p e r m a n e n t resident of Kiev. T h e real reason is easy to guess, it h a p p e n e d soon after the tragedy of t h e Jewish physicians. In t h e period 1944-1954 M . G . was a professor a t t h e Chair of Theoretical Mechanics a t t h e Odessa M a r i n e Engineering I n s t i t u t e . For reasons t h a t a r e stiE unclear he did not t r y to extend his contract with this institution, instead he moved to a less prestigious i n s t i t u t e - Odessa Civil Engineering I n s t i t u t e . Here he held the Chair of Theoretical Mechanics till his retirement. During t h e last period of his life he was a consultant to the I n s t i t u t e of Physical Chemistry of t h e Ukrainian A c a d e m y of Sciences. 2.

ACHIEVEMENTS

One of t h e most eminent m a t h e m a t i c i a n s of our t i m e , M a r k Grigorievich Krein is t h e a u t h o r of m o r e t h a n 270 p a p e r s a n d m o n o g r a p h s of u n s u r p a s s e d b r e a d t h and quality. His work opened up new areas of m a t h e m a t i c s a n d greatly enriched t h e more traditional on.es. He e d u c a t e d dozens of brilliant s t u d e n t s in t h e U S S R a n d inspired t h e work of m a n y m a t h e m a t i c i a n s , engineers a n d physicists all over t h e world. A list of themes where M . G . Krein's research was f u n d a m e n t a l , a n d in m a n y cases even d e t e r m i n e d t h e future of t h e field, includes: oscillating (totally positive) kernel functions =and matrices; p r o b l e m of m o m e n t s , orthogonal polynomials, a n d approximation theory; cones a n d regular convex sets in B a n a c h spaces; t h e theory of gaps between subspaces a n d operators in spaces with two n o r m s ; the extension theory of Hermitian o p e r a t o r s , continuation of H e r m i t i a n positive definite functions a n d helical arcs, entire o p e r a t o r s ; integral o p e r a t o r s , direct a n d inverse spectral problems for nonhomogeneous strings a n d Sturm-Liouville equations; t r a c e formula a n d scattering theory; m e t h o d of directing functionals; stability theories for differential equations; Wiener-Hopf,

Toeplitz

a n d singular integral operators; o p e r a t o r theory in spaces with indefinite metric, indefinite

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extension problems; nonselfadjoint o p e r a t o r s , characteristic operator-functions a n d triangular models; p e r t u r b a t i o n a n d F r e d h o l m theories; interpolation a n d factorization theories; prediction theory for s t a t i o n a r y stochastic processes; problems in elasticity theory, and ship waves a n d water resistance. A profound intrinsic unity is characteristic of Krein's work. Interlacing of general a b s t r a c t a n d geometric ideas with concrete a n d analytical results a n d applications are also characteristic of his work. Krein was a very fine p e d a g o g u e a n d lecturer. He would always share his new ideas a n d plans w i t h his s t u d e n t s a n d colleagues. He was known for his scientific generosity a n d e n t h u s i a s m , as well as his kindness a n d a t t e n t i o n to young m a t h e m a t i c i a n s . In 1982 M . G . Krein was a w a r d e d the prestigious i n t e r n a t i o n a l Wolf Prize in M a t h e m a t i c s in J e r u s a l e m . T h e citation to this prize reads in p a r t as follows: "His work is t h e culmination of t h e noble line of research begun by Chebyshev, Stieltjes, S. Bernstein a n d Markov a n d continued by F . Riesz, B a n a c h a n d Szego. Krein b r o u g h t t h e full force of m a t h e m a t i c a l analysis to bear on p r o b l e m s of function theory, o p e r a t o r theory, probability a n d m a t h e m a t i c a l physics. His contributions led to i m p o r t a n t developments in the applications of m a t h e m a t i c s to different fields ranging from theoretical mechanics to electrical engineering. His style in m a t h e m a t i c s a n d his personal leadership a n d integrity have set s t a n d a r d s of excellence." A m o n g his h o n o r a r y a w a r d s , he was elected corresponding m e m b e r of the U k r a i n i a n Academy, 1939; h o n o r a r y m e m b e r of the American A c a d e m y of Arts and Sciences, 1968; Foreign M e m b e r of t h e National Academy of Sciences of t h e United States of America,1979. He was also awarded t h e S t a t e prize of Ukraine, 1988. 3.

OUR FIRST

MEETING

I m e t M a r k Grigorievich Krein in the a u t u m n of 1950. M . G . Krein was already t h e n a world renowned m a t h e m a t i c i a n with a list of publications containing m o r e t h a n 75 very i m p o r t a n t p a p e r s a n d books. He was considere'd to be one of t h e strongest experts in functional analysis a n d its applications. His m a i n position at t h a t time was in t h e Marine Engineering I n s t i t u t e in Odessa at t h e Chair of Theoretical Mechanics. I was a t t h a t t i m e a s t u d e n t a t t h e Kishinev University, in m y fifth a n d final year.

O u r meeting was at m y initiative, a n d my a i m was to discuss with him m y first

results in o p e r a t o r theory. T h e y were a b o u t Fredholm theory a n d index. After I met him he invited m e to listen to his seminar talk a b o u t e x t r e m a l distributions of t h e m a s s of a string a n d inverse problems. T h e talk was a masterpiece a n d m a d e a great impression on m e .

He invited m e to his h o m e in t h e evening a n d we h a d detailed discussions on

m a t h e m a t i c s , m y future education a n d other subjects.

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He drew my attention to two topics, Toeplitz operators and normed commutative rings. He presented the topics in such an interesting and attractive way that I already fell in love with them, and planned to spend a lot of time studying them. During that evening M.G. told me that his first impression was that I am not Jewish, and he had planned to take me on as a graduate student. He explained to me that antisemitism is very strong and that the administration is not allowing acceptance of Jewish graduate students. He also told me that he had recently been dismissed from Odessa University, which had become very antisemitic, and relations towards him were personally hostile and extended also to his family and his friends. M.G. invited me to write to him and to visit him. After this conversation it became clear to me that my dream of becoming his graduate student would never be realized. It was difficult to imagine then that I would one day be privileged to have M.G. Krein as a teacher, coauthor and friend. Joint work with M.G. Krein was an outstanding university in the best and most friendly atmosphere. I will always remember M.G. Krein with gratitude, affection and admiration. 4. J O I N T W O R K For more than twelve years I worked intensively with M.G.. During those years we wrote twenty papers and two books. A third book was in our plans, but we never finished it. For the remaining twelve years M.G. was mostly busy with other projects and our joint work was less intensive. At other times I remember him to be busy with large joint projects with V.M. Adamjan and D.Z. Arov; and with each of the following colleagues: M.Sh. Birman, Yu.L. Daletsky, I.S. Iokhvidov, V.A. Jakubovich, I.S. Kac, H. Langer, G.Ya. Liubarsky, A.A. Nudelman, Yu.L. Shmuljan, and I.Spitkovsky. Over a long period of time I spent two or three months a year in Odessa for joint work. Most of the time we worked in Krein's apartment; a small room there got the official title of Izea's office. At the beginning of our joint work we worked late into the night. In fact we often stopped working in the early hours of the morning, and that would cost me a ruble to get the porter of the house to open the door, which was already locked for the night. Later we changed our hours of work for more convenient ones. Some of the time was during my official vacation when we would work on M.G.'s dacha on the shores of the Black Sea. On occasions we would even allow ourselves an hour off for a swim. Mark Grigorievich was extremely demanding of himself and of everyone working with him. If, by a total change in the writing of a paper or a chapter he could improve it by even a small fraction, he would not think twice. It was never a question of whether it was worthwhile or not. Papers which were almost ready would lie for years until they attained perfection. He maintained that the writing of a paper is no less scientifically important than

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proving t h e result. As a rule t h e polishing process leads to new proofs, new connections a n d new m a t h e m a t i c s . At t h e beginning of our collaboration M . G . himself wrote t h e formulas in by h a n d in t h e first copy of t h e p a p e r , he did not t r u s t m e enough t o let m e d o this. W h e n I asked if this was so, he answered t h a t he likes to do it himself because this gives h i m a n o p p o r t u n i t y t o think again over t h e proofs a n d t h e whole m a t e r i a l . T h e writing of t h e joint books took u p m o r e a n d m o r e t i m e . E a c h t i m e m y wife would ask, "Is it finished?" a n d each t i m e I would reply t h a t a little m o r e needs to be done. It is still told in Odessa, now as a joke, t h o u g h at t h e time it was t r u e , how L.A. Sakhnovich m e t m e a n d asked, "How is t h e book going?" "Well, it is 85 percent ready," I replied. " T h e n why do you look so sad? T h a t is wonderful." "Yes," I answered, " b u t if you h a d asked m e yesterday I would have said it was 95 percent ready." O u r s t u d e n t s a n d friends m a d e fun of us; I.S. Iokhvidov even wrote a little p o e m on t h e eve of t h e new year in 1963 ( t h a n k s to C. Davis for t r a n s l a t i n g it into English). A r o u n d t h e festive table, all our friends Have come to m a r k our new book's publication. T h e fresh a n d shiny volume in their h a n d s , T h e y offer Izia a n d m e congratulations. T h e long awaited h o u r is here at last. T h e sourest sceptic sees he was mistaken, A n d , smiling, comes to cheer us like t h e rest, A n d I ' m so delighted,....I awaken.

W h e n m a k i n g selections or discussing expositions or proofs we sometimes h a d differences of opinion, a n d this often led t o an a r g u m e n t . In general M . G . was a fighter a n d could m a k e himself very convincing. B u t he would also listen attentively to m y a r g u m e n t s . D u r i n g such discussions we sometimes exchanged positions; I would a d o p t his position a n d he mine. B u t we always ended up on a cheerful n o t e with a c o m m o n point, jokes often being used to illustrate a r g u m e n t s . After we wrote the first joint p a p e r I suggested t h a t his n a m e a p p e a r before m i n e on the m a n u s c r i p t . My a r g u m e n t was based on t h e fact t h a t this would be to the benefit of t h e p a p e r , which would be m o r e appreciated a n d a t t r a c t m o r e a t t e n t i o n with his n a m e first. M . G . refused, maintaining t h a t he would not write joint work with a person w h o could not be an equal p a r t n e r , a n d h e always used t h e alphabetical order of n a m e s in joint p a p e r s . Any change m a y cause speculations in the m i n d s of t h e reader. After a while I drew his a t t e n t i o n to a p a p e r by " M . G . Krein a n d M.A. Krasnoselsky". He laughed a n d r e m a r k e d t h a t t h e difference is in t h e t h i r d letter. M . G . considered joint work as t h e work of a t e a m a n d after deciding t h a t we would work together on certain topics thereafter we

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would not distinguish who did what in these topics. M . G . would c o m p a r e joint work with a t e a m of s p o r t s m e n , especially in a long t e r m collaboration. E a c h player passes t h e ball on to a n o t h e r player t o t h e best of his ability, a n d it is impossible to delineate which pass was t h e most i m p o r t a n t . T h e s e principles worked very well a n d during twenty four years of our collaboration we never h a d any m i s u n d e r s t a n d i n g . Our joint work usually s t a r t e d with plans, discussions, p r e p a r a t i o n s of ingredients for proofs a n d changes, changes, changes. In such discussions a p p e a r e d also t h e basis a n d topic for a new p a p e r . T h e n t h e joint work was realized in different ways. Sometimes one of us would write t h e first draft of t h e m a n u s c r i p t a n d t h e second would t h e n polish the m a n u s c r i p t a n d m a k e additions. This process would continue a n d t h e m a n u s c r i p t would change h a n d s m a n y times until it was considered ready for publication. Sometimes each of us would p r e p a r e a p a r t of t h e m a n u s c r i p t a n d we would p u t t h e p a r t s together at a very late stage. Sometimes we would write t h e m a n u s c r i p t together, discussing t h e presentation on t h e spot. Usually t h e final step in t h e process of p r e p a r a t i o n of t h e m a n u s c r i p t was its joint reading. We continued thinking a b o u t a p a p e r even after it was considered ready for publication. M a n y times this led to surprising results a n d to m a n y last m i n u t e changes. Often we continued our m a t h e m a t i c a l discussions d u r i n g walks in town. P a r t of t h e walk would take us along t h e m a i n street - Deribasovskaya . It was especially pleasant on a s u n n y day.

T h e street was very crowded a n d usually M . G . m e t m a n y people he

knew. Sometimes he would stop for a short conversation a b o u t t h e latest Odessa news. Next we would visit t h e m a i n bookstore. M.G. would check t h e newly arrived books in m a t h e m a t i c s , mechanics, physics a n d astro-physics. He usually m a d e an order for a book in advance a n d his copy would be waiting for him. O u r next stop would be the office supplies store. Here he would b u y p a p e r , ink, p e n s , pencils, glue a n d o t h e r accessories which we used in our work. L a t e r I found out t h a t M.G. liked to meet a young saleswoman in this store. She was really beautiful.

She h a d a biblical look, with d a r k hair a n d black eyes.

She always welcomed M . G . with a m o d e s t smile. We would also visit a second-hand store in t h e neighborhood. M . G . was looking for a typewriter with L a t i n T e t t e r s . He planned to replace his old one (in t h e usual stores one could buy only typewriters with Russian lettering). He never found one. Sometimes we would also visit a h a r d w a r e store. M.G. liked to b u y new original tools for t h e household. He greatly a d m i r e d t h e ingenuity of such tools. M . G . a n d his family were very kind to m e . This wonderful family always insisted t h a t I eat with t h e m . Once M . G . i n t r o d u c e d m e to a nice looking girl who was a friend of his d a u g h t e r . M u c h later I learned t h a t M.G. t h o u g h t we would suit each other a n d t h a t we would m a k e t h e perfect couple. At t h e time I did n o t u n d e r s t a n d his intentions a n d I p r o b a b l y disappointed him. Working with M . G . was a joy. I was never a formal s t u d e n t of his, but for m e

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he was m u c h m o r e t h a n an o r d i n a r y instructor. W h e n I s t a r t e d to work with M.G. I was warned by a friend t h a t I wo'uld get little credit for this joint work. I h a d also h e a r d m a n y tales a b o u t w h a t a difficult person M.G. Krein was, b u t during t h e twenty four years in which we worked together I found Krein to be otherwise. He set very high s t a n d a r d s for himself a n d I found his d e m a n d s on others to be fair a n d reasonable. 5.

SEMINARS AND

LECTURES

For m a n y years M.G. Krein r a n a m a t h e m a t i c a l seminar at t h e Scientists House in Odessa. This house used to be a n old palace, b u t since t h e revolution it served as a kind of club for scientists. T h e r o o m s with high ceilings were very luxurious, covered with expensive m a r b l e a n d wood paneling. Missing was only a good blackboard. T h e meetings were always in a very pleasant a t m o s p h e r e . M.G. would usually open with an interesting i n t r o d u c t i o n to t h e subject u n d e r discussion, a n d from this i n t r o d u c t i o n one could feel how rich a n d wide was his experience. During t h e talk he would ask i m p o r t a n t questions a n d s o m e t i m e s a d d r e m a r k s with a n historical perspective, or other approaches a n d proofs. As a rule he would present details, closing t h e talk with illuminating c o m m e n t s . All was done in a very friendly m a n n e r a n d with a great respect for t h e lecturer a n d audience. At t h e e n d of such seminars I felt myself enriched. M . G . would leave t h e Scientists House for his h o m e a c c o m p a n i e d by a large group of his a d m i r e r s . O u t s t a n d i n g m a t h e m a t i c i a n s p a r t i c i p a t e d in these seminars, some of t h e m m e m bers of t h e famous older generation of M . G . ' s s t u d e n t s a n d colleagues, a n d also t h e younger generation. A m o n g t h e m was V.M. A d a m j a n , D.Z. Arov, M.L. Brodsky, J u . P. Ginsburg, I.S. Iokhvidov, V . A . J a v r j a n , I.S. K a c , K . R . Kovalenko, H. Langer, F.E.Melik-Adamjan, S.M. M k h i t a r j a n , B.R. M u k m i n o v , A.A. N u d e l m a n , I. Ovcharenko, G . J a . P o p o v , Sh.N. Saakjan, L.A. Sakhnovich, P.A. S h w a r t s m a n , J u .

L. Schmulyan, V.G. Sizov, a n d I.

Spitkovsky. M . G . Krein also regularly held other interesting seminars on a smaller scale in t h e institutions where he was working. For instance he r a n a seminar in H y d r o d y n a m ics in t h e Marine Engineering I n s t i t u t e . A.A. Kostiukov, V . G . Sizov a n d Yu.L. Vorob'ev were p a r t i c i p a n t s in this seminar. In Kiev were Yu.M. Berezansky, B.I. K o r e n b l u m , M.A. Krasnoselsky, a n d S.G. Krein (M.G.'s younger b r o t h e r ) , in Kuibyshev, G.Ya. Liubarsky, A.V. S t r a u s s a n d O.V. Svirsky. I recollect M.G.'s seminars a n d lectures as one of t h e best I ever h e a r d .

His

lectures a n d talks were always well p r e p a r e d with historical r e m a r k s a n d examples. Almost every year M . G . gave courses of lectures for experts a n d P h . D . s t u d e n t s . Usually these lectures were based on his recent results.

Many of t h e m were even unpublished.

His

lectures on t h e theory of entire operators (prepared for publication by the Gorbachuks) will a p p e a r only now in print, in Birkhauser Verlag.

Lectures on t h e theory of cones

were circulating a m o n g experts for m a n y years a n d were never published. Lectures on t h e

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theory of the string exist only in rough handwritten notes. A manuscript of a book on group representations (written together with M.S. Brodsky) is not finished. As a matter of fact, almost no one from Odessa University attended.M.G. Krein's seminars and lectures. They probably did not even realize how much they were missing. 6. TRAVEL A B R O A D M.G. Krein never traveled abroad. The reason for this was that he would not have been granted an exit visa had he apphed. An exit visa for a trip abroad was considered by the authorities as a special privilege given only to the very "reliable". He received many invitations, then he would apply for an exit visa, entailing the filling in of thick official forms. He would then get the inevitable negative reply, or no reply at all. He even developed a sort of allergy to the filling in of forms, and during his last years it was very difficult to get him to undertake such a task, even on his wife's insistence. He already knew it would be of no use, so why bother. In one case was he given permission to travel abroad. That was in 1970 to attend a conference in Tihany on the Balaton Lake in Hungary. It probably worked out this time because he used a private invitation which did not have to go through the high official channels. But this time he could not use the visa because precisely at that time there was an epidemic of cholera in Odessa and no one was allowed to leave Odessa. I took part in this conference and at the request of M.G. I gave Professor B. Sz.-Nagy (the organizer of the conference) regards from M.G. and told him the reason why M.G. could not come. Professor Sz.-Nagy smiled and answered, "So it's now called cholera, is it?" In the West people were already used to the various reasons that were invented to justify the absence of M.G. Krein. But this was the only time that the reason given was the true reason. M.G. liked very much to meet and to receive foreign colleagues. However those meetings always brought repercussions. He would have to answer unpleasant questions put to him by KGB agents (in the best case). The small technical institutions that he worked for did not have the official facilities to receive foreigners and to organize such visits (usually they existed only in large universities or research institutes). For instance, in the case of the one year visit of H. Langer an unusual solution was found. Langer was well established at Odessa University. There was his residence and there he received his salary, but scientifically he was connected only with M.G., with his seminars and his School. Very often Mark Grigorievich would avoid a meeting with a colleague in order to prevent the inevitable repercussions and difficulties. In order to avoid similar difficulties and problems with the administration he did not send preprints and reprints abroad.

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Unfortunately foreign colleagues did not always understand these sad facts of real life in the USSR. 7. T H E R E S I D E N C E M.G. Krein loved Odessa. He was well acquainted with the history of the town and even the history of the main streets. He liked especially the satirical books written about Odessa. Often he read out the funny bits for his guests. All his mature life (with a relatively short exception during World War Two) the Kreins lived in Odessa in an old apartment house on 14 Artema Str., App. 6. This was the apartment in which his wife and her parents lived previously. The apartment building, which had been built at the beginning of this century, was luxurious. It had an elevator, heating, bathrooms, and every apartment had many large high ceilinged rooms, and separate quarters for servants. The owner of the building used to send a cleaner every week to polish the parquet flooring and to take care of other small things. In general he wanted to make sure that the house would be suitably maintained as such a building deserves. After the revolution normal life in such apartments became extremely difficult and in many cases impossible. First of all the houses were nationalized and the newly appointed caretakers were more involved in watching the occupants and reporting on their doings than in maintaining the building. In each apartment new families were installed. Very often each family had only one room, sharing the use of the kitchen and other facilities. Very soon the elevator and heating broke down and the water failed to reach the upper floors. Everyone knew what was going on in each of the neighboring rooms and what was being served for dinner in each room. Situations arose which were described with great humor in Russian literature of the post revolution period. All of this was the lot also of the Krein's apartment. There were times when this a-partment accommodated six families (according to the number of rooms). Sometimes running water did not reach the apartment (certainly not in the summer). Hot water they never had. Each morning there was a queue for the toilet. For many years heating was a serious problem. I remember one occasion during lunch or dinner while a hvely discussion was taking place one of the members of the family made a sign, and the topic under discussion was automatically changed. The sign meant that someone from another room w,as passing by their room on the way to the bathroom or to the telephone and their discussion could be overheard. People from other rooms would often call in on the Kreins during a visit by colleagues. Television was introduced to the Krein family relatively late. The family would watch together different programmes. They liked theater, opera, classical music, as well as other programmes. The Kreins liked the summer when they moved to the dacha on the shores of the Black Sea (in Arcadia) without the television.

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Mark Grigorievich was happily married to Raisa L'vovna Romen. She was. an expert in naval architecture and worked in the Marine Engineering Institute. They had only one child. Their daughter, Irma Markovna Krein (Kozdoba), has a Ph.D. in Philology and is an expert in cybernetics. They also had a grandson Aleosha and and a great grandson, also called Mark. 8. B A T T L I N G HOSTILITIES In general, M.G. Krein was a fair, very amiable and kind person. However, all of his life he battled against mediocrity. After the Second World War he had to contend with hostile elements which fought fiercely against him using the officially supported antisemitism which was rife in the Ukraine, and especially so in Odessa. He was accused of Jewish nationalism, presumably for having had too many Jewish students before the War. This accusation was certainly entered into his classified file and was held against him all of his fife. Presumably, it played a significant role in his two dismissals which were mentioned earlier. He was not allowed to have Jewish students and was deprived of a university base. All attempts on the part of various societies, academies and individuals in the Soviet Union to gain for him some measure of the official recognition which he so richly deserved, were unsuccessful. Remarkably enough, he was never elected as a full member of the Ukranian Academy of Sciences. Their standards must have been very high. Worse than that, there were times when his friends feared that he was in serious danger of arrest. In 1948, in an article in a local Odessa newspaper, M.G. Krein was called a "rootless cosmopolitan." He was accused of too often quoting foreign mathematicians and following too much their ideas and ignoring the achievements of Russian and Soviet mathematicians. The official establishment considered this to be a crime. This accusation was supported publicly by a professor of the University in Astronomy. The latter based his conclusions on the famous paper of M.G. Krein and M.A. Rutman (Linear operators leaving invariant a cone in Banach space, Uspekhi Mat.Nauk 3,No 1 (1948), 3-95; English translation: Amer. Math.Soc. Transl. (1), 10 (1962), 199-325). I guess he could understand only the general introduction to this paper and the fist of references; remarkable that he did not need to know any more in order to make the accusation. This was the usual way of starting many of the campaigns against Jewish scientists, writers and cultural activists which often led to arrests. M.G.was lucky, he escaped without an arrest. M.G. Krein gave a plenary talk at the Mathematical Congress held in Moscow in 1966. It was a very deep talk with a detailed analysis of the state of affairs in operator theory and its applications. Special attention was paid to the theory of nonselfadjoint operators. In this talk M.G. introduced, for convenient references, a mathematician Gokr (as a shortened form of Gohberg-Krein). My friend V.G. Boltjansky was working in the Press Committee of the Congress and mentioned this talk and Gokr in the public press.

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After the Congress he had serious problems with the mathematical authorities. He was accused of passing to the press information which had a "wrong political orientation." In the second half of the sixties the situation worsened. In particular two very important and influential committees became almost openly antisemitic. I have in mind the Committee of Experts (Chairmen V.A. Il'in, V.S. Vladimirov) which dealt with approval of degrees in mathematics in USSR, and the Publication Committee (chairman L.S. Pontrjagin) which over-saw all publications in mathematics in USSR. Approval of the degree of "Doctor of Sciences" was absolutely refused for Jews, and projects for books by Jewish authors were rejected. Laughable reasons were given to justify all of this. M.G. Krein responded to these hostile surroundings in the only way open to him, by deep research and hard work. He and many of his students were protected by virtue of his outstanding achievements. In retrospect, it seems clear that he won this very difficult struggle. Firstly, he was able to devote all his life to mathematics (teaching and research), the work he loved so much. Secondly, he was able to spend most of his life in Odessa, a town which he had always regarded with love and affection (some of his friends thought that his life would have been much easier in Moscow or Leningrad). Thirdly, he was always the leader of a strong and dedicated group of colleagues and followers who loved and respected him. (This group existed almost on a private basis, holding many of its meetings in his house, or at the Scientists Club.) Fourthly, he had a great impact on the development of mathematics and its applications throughout the world. Even though he was never allowed to travel abroad, his brilliant work knew no borders. In spite of all the difficulties that surrounded him, M.G. was cheerful and optimistic. He liked to tell stories and jokes, some of which he invented himself. At one time he went to the rector of his institute and asked if there was any danger that he could be accused of Armenian nationalism since he had four graduate students who were Armenians. The rector did not understand the joke and tried, in all seriousness, to explain that in this case there was no danger. 9.

EPILOGUE

This fight took a heavy toll on his health, and towards the end of his life he suffered from depression. This condition worsened after the tragic loss within one year of his wife, and his only grandson, Aleosha. On October 17, 1989, M.G. Krein died in Odessa (USSR). There he is buried. The passage of time brought about considerable changes. In the Institute of Mathematics of the Ukrainian Academy and the Institute of Mathematics of Odessa University some of M.G. Krein's followers and outstanding former students are now working. Preparation of an international conference on the occasion of the ninetieth anniversary of

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the birth of M.G. Krein is in progress. This conference is being organized by the two above mentioned institutes, which just half a century earlier had dismissed M.G. The conference is even partially supported by the city of Odessa. The conference will take place in Arcadia (in the suburbs of Odessa) on the shores of the Black Sea, not far from the Krein family dacha, where M.G. liked so much to work. There he spent many hours working with students and friends. And there he also spent the last days of his life. Unfortunately the dacha no longer belongs to the Krein family. It was taken away from them recently. We hope that this conference will help everyone to understand in full measure who was Mark Grigorievich Krein, and also to understand the bitter and tragic mistakes and injustices of the past and the present. 10.

REFERENCES

The following sources were used in these recollections: 1. M.G. Krein. Autobiography, 1966. Private communication (Russian). 2. I. Gohberg and M. Kac. Biography of M.G. Krein. Topics in functional analysis. Essays dedicated to M.G. Krein on the occasion of his seventieth birthday, I.Gohberg and M.Kac editors, Academic Press, 1979. 3. I. Gohberg, Mathematical Tales. Gohberg Anniversary Collection, vol. 1, edited by H. Dym, S. Goldberg, M.A. Kaashoek and P. Lancaster. Birkhauser Verlag, 0 T 40, 1989, pp. 17-56. 4. I. Gohberg. Mark Grigorievich Krein (1907-1989), Notices of the American Mathematical Society, vol. 37, no. 3, 1990, pp. 284-285.

Raymond and Beverly Sackler Faculty of Exact Sciences School of Mathematical Sciences Tel-Aviv University Ramat-Aviv, 69978 Tel-Aviv, Israel

MSC 1991: primary 01A70, secondary 01A99

Submitted: June 15, 1997

177 Ukrainian Mathematical Journal, Vol. 46, No. 3, 1994

MARK GRIGOR'EVICH KREIN (RECOLLECTIONS) B. JA. LEVIN From 1924 and until his death in October 1989, Mark Grigor'evich Krein—one of the most brilliant and famous mathematicians of our century—lived and worked in Odessa. I had a great opportunity to work with him for many years. He greatly influenced my scientific interests, directions, and the character of my investigations. For the first time, I met M. Krein in June 1934 at the Second Mathematical Congress in Leningrad. I attended his talks at this Congress devoted to loaded integral equations and the L-moment problem. In the second talk, he presented his work with N. I. Akhiezer. His talks always made a strong impression on the audience not only by their important and profound contents but also by their brilliant form. M. Krein.often said: "I want to be understandable." His concern for those who listened to him was always evident: He tried to make his presentations as clear and well thought-out as possible, to create a festive atmosphere on his talks and lectures. The same was my impression about his talks at this Congress. There or, more precisely, as we used to say, "in the couloirs of the Congress" I found the opportunity to read a manuscript of an unpublished article by M. G. Krein and M. A.Naimark, "A Method of Symmetric and Hermitian Forms in the Theory of Separation of Roots of Algebraic Equations," which was quite close to my scientific interests. This article was written for the encyclopedia. In 1936, it was published as a separate paperback edition and very soon became a bibliographic rarity. I decided to meet M. Krein and discuss with him the problems we were both interested in. We got acquainted but found no possibility for discussion. There were too many people who wanted to speak with Krein but absolutely no time for this 1 . After I had told him that I often came to Odessa, M. Krein invited me to visit him during my next visit. This happened a year later, when I came to Odessa for summer vacations from Rostov-on-Don, where I was completing my post-graduate studies. M. Krein asked me about my scientific activities. At that time, I was carried away by the construction of the theory of entire functions of completely regular growth and was very happy to have the opportunity to speak about my ideas and results with a great scientist. M. Krein listened to me very carefully, rarely interrupting with questions. Some of these questions were so important that I undertook a serious additional investigation and when they were finally solved, I suddenly understood that this investigation was indeed necessary to make the entire theory more precise and complete. That summer we met quite often, spoke about mathematics, went for walks near the sea. At that time, M. Krein studied, together with F. R. Gantmakher, oscillatory matrices and kernels and his talks about these problems were very exciting. I was deeply surprised by his observation that a simple mechanical fact made it possible to extract an important class of matrices about which it was clear that it would play an important role in algebra, analysis, and other fields of mathematics. This was the following: Consider an elastic continuum, e.g., a string in tension, and apply to it n transverse forces lying in the same plane; then the deflection curve of this continuum has at most n - 1 knots. (A knot is defined as a point where the ordinate changes its sign.) Later, I often noticed in Krein's works how the ideas of the theory of oscillatory matrices penetrate in profound problems in the theory differential and integral operators, and then understood that this method of investigation represents the general style of Krein's scientific work. For example, his well-known theory of extensions of semibounded operators appeared in connection with certain problems in the theory of bending of plates, and then found numerous applications both in Krein's works and in the works of other mathematicians. Large series of his investi1

"These congresses and conferences are so efficient in separating mathematicians" — Alexander Kurosh once joked on a conference.

Translated from Ukrainskii Matematicheskii Zhumal, Vol. 46, No. 3, pp. 305-309, March, 1994. 318

© 1995

Plenum Publishing Corporation

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gations were often stimulated by the problems in mechanics, electromechanics, the theory of communications, and the theory of shipbuilding. Mathematics is a science dealing with mathematical structures. Formally speaking, one absolutely arbitrarily chooses an independent system of axioms that describes an object and then, according to the rules of logic, the properties of the indicated object are derived from these axioms. This is just a description of a mathematical structure. In principle, one can construct a great number of these structures but their scientific importance will be. as a rule, quite low. It is probable that many structures of this sort give about the same amount of knowledge about the world and are of the same level of importance in cognition of the truth as chess-playing. It seems to be quite a difficult problem to select in this collection of structures those that are indeed useful and important. To solve this problem, the scientist must have high scientific culture, profound knowledge, and great intuition. (I do not even mention such necessary requirements as desire and ability to be objective.) M. Krein possessed in a remarkable degree the ability to understand and evaluate the importance of various structures. His strong mathematical intellect allowed him to reveal the finest and deepest relations between different objects and this often determined his interests and directed his investigations. Let us present several examples. At the beginning of the 1930s. M. G. Krein and N. I. Akhiezer performed an extensive and very fine investigation of the trigonometric L-moment problem. More than 30 years later, this great abstract investigation found, in the works of N. N. Krasovskii, important applications to the theory of optimal control. At first sight, the abstract of the Krein- Mil'man theorem on extreme points published in 1940, became one of the most important in many branches of contemporary mathematical analysis. This list of examples can easily be continued but I present only one example. In 1943, M. G. Krein posed and studied the problem of extension of helix lines in Hilbert spaces 2 and soon applied the results obtained in the study of important problems of the theory of predictions. The high practical value of this work was mentioned by N. Wiener in his book "I Am a Mathematician." This ability to discover general fundamental ideas in particular problems and construct, on this basis, theories that are very abstract on the one hand and, on the other hand, deeply penetrating in various fields of mathematical science was a distinctive feature of Krein's creative work. "Certainly",— M. Krein said one day — "science preserves only simple rough results and all fine points are forgotten little by little but how can one determine what is rough and what is fine? For example, could you say when it became clear that L2-metric is simpler than uniform?" In his works, M. Krein originated many new trends in science which were than developed and applied in the works of other mathematicians. At the same time, he paid great attention to the development of old classical fields. Being a qualified expert in the works of the Petersburg school of mathematicians and related researches, he devoted many of his works to the development of this important and extensive field of science. Let us now recall the characteristics of this part of his creative activities, given in 1983 when he was awarded the International Wolff Prize in mathematics: "His work represents a culmination of the noble series of investigations initiated by P. L. Chebyshev, T. Stieltjes, and S. N. Bernstein and then continued by F. Riesz, S. Banach, and H. Szego...". A major part of these results can be found in his books and surveys. His last work published after his death also belongs to this series. He established the priority of A. M. Lyapunov in the theory of stability regions for the Hill equation. Later, he generalized the entire Lyapunov theory for a large class of systems, namely, for the so-called canonical systems of differential equations of an arbitrary order with periodic coefficients. Note that in this case, a decisive role in the selection and study of these systems was also played by the mechanical considerations. M.Krein always gave extremely high praise to the results of A.M.Lyapunov; once, he even said that Lyapunov's works in the theory of stability of the shape of rotating liquid are "of genius." He did not make this remark, 2

By the way, it is worth noting that in the 1920s Krein was a post-graduate student of N. G. Chebotarev and often said that he was indebted to Nikolai Grihor'evich for the idea of studying general problems of extension with preservation of a certain property. Thus, as an example of a problem of this sort, one can mention the problem (posed by Chebotarev) of extension (i.e., of adding higher powers) of a polynomial whose roots are all real with preservation of this property. M. Krein also regarded as problems of this type the following problems posed and studied by him: On the extension of a Hermitian positive function defined on an inten-al to a Hermitian positive function on the enure real axis and on the extension of a helix line in a Hilbert space or in a Lobachevskii space to a complete helix line in the corresponding space.

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made during one of our long walks around the town, more precise. All these walks were, as a rule, finished in a bookstore on the famous Deribasovskaya street. M. Krein loved these walks around Odessa very much, he loved his sunny town with fresh air and a wonderful mixture of sea and steppe smells. (As it was in the 1930s.) He liked the irregular but very expressive and vivid "Odessa" language and rendered with great humor the picturesque Odessa phrases he had heard in the streets. He had a great skill of Ukrainian, enjoyed this language and often gave lectures in it. In the 1930s, M. Krein could never imagine himself without Odessa and the Odessa University. The first edition of his book with F. R. Gantmakher "Oscillatory Matrices and Kernels" published in 1940 was dedicated to the 75 th anniversary of the Odessa University. For M. Krein, it was always interesting to share his ideas and results with his mathematician friends, colleagues, post-graduate students, and students. He was an excellent narrator able to create an impression that new ideas and thoughts came to his mind just in a precise finished form. This feature distinguished him from the majority of people whom I know. Usually, a new thought first is not absolutely clearly connected with complicated processes in the subconscious and observed as an obscure "distant prospect of a free novel," only far later the picture acquires clear contours, and it becomes possible to distinguish principal facts and exact arguments. For Krein, this period was always a "thing in itself," when listening to him, you could never get rid of the impression that these new results appeared in his mind just as they were delivered to us—in a precise form and separated into lemmas and theorems. This was always interesting. We felt his enthusiasm, a specific atmosphere of creation, of finding the truth. This feature of Krein was very attractive. He was always surrounded by young mathematicians. A picture of slowly moving groups of young people speaking with enthusiasm on a mathematical language hardly understandable for normal people was typical of the Paster street in the 1930s. These were Krein's students and post-graduate students accompanying their professor from the university after a lecture or a seminar. Walking near the sea in the summer of 1935, we spoke not only about mathematics. We were of the same age, less than 30, and both were going in for sport. Once, Krein told me that, at the age of 17-18, he tried to get a job in the circus as an acrobat but failed for some reasons. This was a great event for mathematics, since he was born as a mathematician. In 1935,1 finished my post-graduate studies in the Rostov University and moved to Odessa and, in September, began my career as a lecturer at the Physico-Mathematical Department of the Odessa University. By that time, I had already got certain experience as a lecturer in high schools. To understand the situation in the Odessa University at that time, one should make a small excursion into the near past. At the end of the 19 th and the beginning of the 20th century, Odessa or. more precisely, Novorossiiskii University played an important role in the scientific life of the country. The Department of Natural Sciences of the Physico-Mathematical Faculty was famous for the prominent scientists working there. It suffices to mention M. I.Sechenov, I.I. Mechnikov, a famous physicist N. A. Uraov, and the lectures of a prominent zoologist A. O. Kovalevskii. All these scientists caused a great influence on the spiritual life of the university and created good scientific traditions. Of great importance for the development and direction of the mathematical investigations at the faculty, were the activities of Prof. I. V. Sleszczynski who was a highly educated mathematician and one of the founders of the "Mathesis" Publishing House, which played an important role in the dissemination of mathematical knowledge in Russia. This Publishing House published the translations of the famous works of Bolzano, G. Cantor, Dedekind, and other classics of contemporary mathematics. P. S. Aleksandrov was sure that the ideas of the theory of sets came to Russia through Odessa. S. O. Shatunovskii and V. F. Kagan who worked at the faculty took part in the activities of the "Mathesis" Publishing House. Mathematics was taught at a very good level. Samuil Osipovich Shatunovskii was a brilliant lecturer and a talented mathematician who often left time behind in his lectures and works. As early as at the end of the 19th century, i.e., before the appearance of Gilbert's book "Foundations of Geometry," Shatunovskii gave, in his course of lectures, "Introduction to Analysis," an axiomatic definition of the notion "magnitude" and verified the independence of these axioms by using models. Later, this course was published. He also gave a definition of the limit of a function given on a semiordered set. (The Moore-Shatunovskii limit.) Shatunovskii often criticized in his lectures the unlimited use of the law of excluded middle and not only criticized but also gave in his master's thesis (in 1917) a constructive, i.e., without the law of excluded middle, presenta-

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tion of the Galois theory. N. G. Chebotarev specially emphasized in his book "Galois Theory" that the presentation is based on Shamnovskii's ideas. The lectures of a prominent geometrician Veniamin Fedorovich Kagan were also of great scientific importance. These lecturers, their colleagues and disciples guaranteed a permanently high level of the faculty. But 1920 was marked with great trouble. As a result of the so-called "administrative rapture" of bureaucratic 'reformers' all the universities in the Ukraine were closed by a special order of the Ukrainian People's Commissariat of Education. Hard times lasted until 1933, when the "disfavor" of the universities ceased and they were restored. (In Russia, the same "reform" was undertaken later, was milder, and did not touch the Moscow and Leningrad Universities.) For these 13 years, many professors moved to other places. University or, as they are sometimes called now, "fundamental" sciences were kept down. Traditions were lost. It was necessary to begin from the very beginning: to invite and teach new experts, form new educational plans and programs of lecture courses, take care of the quality of lectures, and so on. This work aimed at the restoration and modernization of the faculty was headed (not formally but in fact) by M. Krein, who worked much charging his colleagues, disciples, and students with his enthusiasm. Parallel with compulsory courses, many special courses were delivered to the students including Functional Analysis, Hilbert Spaces and the Theory of Operators, the Method of Fixed Points and Its Applications in Analysis, and Harmonic Analysis on Groups. There were also classical courses: Equations of Mathematical Physics, Abstract Theory of Groups, Galois Theory, Theory of Algebraic Functions, Almost Periodic Functions, Theory of Entire Functions, etc. The students also had the opportunity to attend special courses in mechanics, also patronized personally by M. Krein. An incredible role in raising the scientific level of the faculty was played by Krein's scientific seminar. It deserves to be described in more detail. The subjects of the seminar were not restricted at all. Sometimes, the talks were suggested by the members of the seminar, sometimes they were offered by Krein. Many talks were delivered by him personally. M. Krein listened to all the talks very carefully and understood the authors even in the situations when everybody caught nothing; as a rule, his brief summaries after the talk made the entire picture much clearer. New problems were also posed in these brief summaries and it often happened that their solutions were also later presented at the seminar. Krein created there an atmosphere of free scientific discussion. It looked like a group of young people gathered to discuss something interesting for everybody. In 1936, the first issue of the journal "Uspekhi Matematicheskikh Nauk" appeared; it was mainly devoted to the problems of functional analysis and played a significant role in the popularization of this science. The survey article of L. A. Lyustemik published in it was, in fact, the first manual in functional analysis. The attention of mathematicians was also attracted by the work of V. V. Nemyckii dealing with the method of fixed points in analysis. Krein's scientific interests, and then the interests of the other members of the seminar, moved toward the theory of Banach spaces. This very soon resulted in the appearance of a series of works devoted to cones in Banach spaces, positive functionals, operators preserving an invariant cone, to the theorems on regularly closed sets, and many other functional-analytic papers. Among the members of the seminar one should mention M. A. Naimark, V. L. Shmul'yan, D. P. Mil'man, M.A. Rutman, V.P. Potapov, S.A.Orlov, A. P. Artemenko, M. S. Livshits, I.M. Glazman, and others. The works and names of these scientists then became widely known. It was, in fact, a new mathematical center in the south of the USSR. This fact was recognized by the mathematical community and mentioned in many surveys. In 1939, Krein was elected a corresponding member of the Ukrainian Academy of Sciences and soon, after the annexation of the Western Ukraine, he was sent, together with M. A. Lavrentyev, to the Lvov University to establish scientific relations with mathematicians living in Lvov. There, at the seminar of S. Banach, he presented his results and results of his colleagues in the theory of Banach spaces. Stefan Banach highly appreciated his talk and said: "Your talk is a progress." We were witnesses of the rapid progress of the Physico-Mathematical Faculty of the Odessa University both in the scientific and pedagogic senses, proud of its successes and considered ourselves as cocreators of this process. Beginning with 1936, M. Krein was a collaborator of the Institute of Mathematics at the Kharkov State University; in 1940, he occupied a position (not leaving his work in Odessa) of the Head of the Department of Functional Analysis at the Institute of Mathematics of the Ukrainian Academy of Sciences in Kiev. His regular visits to Khar-

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kov and then to Kiev, talks delivered there, discussions with mathematicians and common works greatly contributed to the development of functional analysis in these scientific centers. During the Second World War, many employees of the university including colleagues and followers of M. G. Kxein went to the regular army, the others were evacuated to various places in the East of the country. Krein was evacuated to Kuibyshev (now Samara). There, he worked as a professor in the Kuibyshev Pedagogical Institute, taught post-graduate students, and was one of the organizers of the Aviation Institute. In these three years, M. Krein published his first works devoted to the theory of entire operators, to representations of functions by the FourierStieltjes integrals, and to the solution of Kolmogorov's problem of prediction. In 1944, M. Krein returned to Odessa after it was liberated from Nazi occupation. One after one, his disciples and colleagues also returned to Odessa. But some of them did not return. V. L. Shmul'yan, always merry and tireless, perished near Warsaw; even in the regular army, he found possibilities to continue extensive studies in mathematics and published several articles during the war. A. G. Okun' was lost near Korsun-Shevchenkovsky, V. N. Shmushkovich also died, extraordinary gifted A. P. Artemenko was missing. But the others returned. The university also returned to Odessa. But they got into unexpected trouble. Neither M. Krein, nor his intimate colleagues were allowed to return to the university. An attempt undertaken by Mark Grigor'evich to convince the Rector that this was not reasonable failed. M. Krein occupied a position at the Chair of Mechanics of the Odessa Institute of Navy Engineers (OENE); his colleagues had to work in different high schools of Odessa. But what had happened? What was the reason to keep a recognized mathematical school separate from the mathematical youth? Why it was necessary to weaken the Physico-Mathematical Faculty of the Odessa University once again? No reasonable answer was received. The situation was so unnatural and weird that the university authorities were forced to give explanations and, in particular, they alluded to the orders "from above." Certainly, such orders indeed existed at that time, but I think, that the university authorities were too active in realizing these general rules. Scientific community of Odessa and many Soviet mathematicians tried to protest against this unnatural situation by the sole method possible at that time, namely, by writing letters of protest to higher authorities, which, of course, sent inquiries and the university authorities, to defend themselves, replied with various accusations against M. G. Krein. This caused the appearance and propagation (among the Communist Party activists and philistines) of a fairy tale that Krein, in their frame of references, is a "Zionist." Of course, this version had nothing common with reality. As the flow of accusations increased, Krein's secret "dossier" lying somewhere in the KGB increased as well. Since that time, M. Krein lived under the permanent pressure of the authorities combined with an unfriendly attitude to him by some mathematicians. (The accusation against Krein became evident every time when he tried to go abroad, or when the problem of awarding him a prize was discussed, etc.). In 1952. he was dismissed from the Kiev Institute of Mathematics with a simple motivation that he did not live in Kiev permanently. This weight pressed M. Krein uninterruptedly for the second half of his life, beginning in 1948. Only once he complained to me: "I read sometimes in books that somebody was slandered but never could imagine what it means in reality." What a powerful spirit he possessed and how strong was his love of mathematics to continue in that situation, as he did, extensive studies, undertook all new researches, making one discovery after another. This permanent and successful work supported his high life tone for many years. In this period of his life, Krein carried out thorough investigations of the regions of stability of canonical systems of differential equations with periodic coefficients and the Wiener-Hopf equation, classical investigations in the theory of operators, wrote a large series of works in the spectral theory of a string, the works in the theory of operators in spaces with indefinite metric, works devoted to Hankel operators, etc. Even a superficial analysis of Krein's scientific heritage would cover many pages. In the last period of his life, he wrote six books and more than 150 articles. He delivered large lecture courses in the mathematical schools. His investigations caused more and more profound influence on various branches of mathematics and, correspondingly, his results were more and more highly appreciated throughout the world. In 1968, he was elected an Honored Member of the American Academy of Science and Arts; in 1987, he became a foreign member of the National Academy of the USA, and, in 1987, an Honored Member of the Moscow Mathematical Society. In a well-known book by P. D. Lax and R. S. Phillips "Scattering Theory for Automorphic Functions," the introduction is completed by the words: "This book is dedicated to Mark Krein, one of the mathematical giants of our

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century, as a recognition of his extremely versatile and deep contribution to mathematics. As any analyst, we are indebted to him very much." This wide appreciation of his great role in the world of science was his spiritual victory over his opponents. M. Kxein was always friendly and enjoyed personal contacts with different people being interested in their problems. At the same time, he was an interesting narrator himself and often told about his meetings with interesting people and about interesting events. Scientific contacts were necessary for him. Probably this was one of the main reasons for which he always organized seminars that attracted many participants. In addition to the seminar in the institute where he worked, Krein organized a city seminar in the House of Scientists, his seminar in hydromechanics organized at the OINE was of great importance for shipbuilding engineers. Among the participants of this seminar, one should mention A. A. Kostyukov, V. G. Sizov, Yu. L. Vorob'ev. and others. As before, gifted young people were attracted by Mark Grigor'evich, many students and post-graduate students of the university and pedagogical institute attended his seminars. Young scientists came to him for consultations from other cities and republics. Some universities sent their graduates to Krein for post-graduate studies. He always worked much. Krein's life was especially intense in the summer on his dacha in Arcadia (near Odessa). A major part of the day he worked at the table on a large veranda. This was also a place for receiving visitors. Once Boris Nikolaevich Delone delivered a lecture entitled "What is necessary for anybody who wants to become a good scientist ?" Prof. Delone told about many important things in his traditional witty manner but finished his lecture quite unexpectedly. He said: "However, everything that I have already told you is not so important, the most important problem is to marry successfully." Krein's wife, Raisa Lvovna Romen was an attractive educated lady and good witty company. All her life she was dedicated to her husband, created good conditions for his scientific work; in difficult situations, she tried to protect him against all troubles; in hard times, she never allowed him to give up. She always deserves to be mentioned when anybody speaks about Mark Krein. Raisa Lvovna died in May, 1989. Mark Krein died five months later.

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ABSTRACT to papers by M.G.Krein submitted by Moscow Mathematical Society and Department of Mathematics and Mechanics of Moscow State University for the Lenin Prize on 1963.

In accordance to the presentation of the mentioned departments for the Lenin Prize on 1963 my works in 10 fields of mathematical analysis are submitted. A short description of these works in each field separately is given below.

1. The theory of oscillatory matrices, oscillatory integral and differential operators On this subject the following papers were published: 1.1 Sur quelques applications des noyaux de Kellog aux problemes d'oscillation, Soobshch. Nauchn.-Issled. Inst. Mat. Mekh. Khar'kov. Gos. Univ. i Khar'kov. Mat. o-va (4) 11 (1935), 3-19. 1.2 Sur les matrices oscillatoires, C.R. Acad. Sci. Paris, 201 (1935), 577-579 (with F.R. Gantmakher). 1.3 On oscillation differential operators, Dokl. Akad. Nauk SSSR 4 (1936), 379-382. 1.4 Sur les vibrations propres des tiges dont l'une des extremites est encastree et I'autre libre, Zap. Khar'kov. Mat. O.-va (4) 12 (1936), 3-11. 1.5 On a special class of determinants in connection with Kellog's integral kernels, Mat. Sb., 42 (1935), 501-508 (with F.R. Gantmakher). 1.6 Sur quelques proprietes de noyaux de Kellog, Zap. Khar'kov. Mat. O.-va (4) 13 (1936), 15-28. 1.7 On some properties of the Kellog's resolvent kernel, Zap. Khar'kov. Mat. O.-va (4) 14 (1937), 61-73. 1.8 Sur les matrices complement non-negatives et oscillatoires, Comp. Math., 4 (1937), 445-476 (with F.R. Gantmakher). 1.9 On totally nonnegative Green functions of ordinary differential operators, Dokl. Akad. Nauk SSSR 24 (1939), 220-223 (with G.M. Finkel'shtein). 1.10 On nonsymmetric oscillating Green functions of ordinary differential operators, Dokl. Akad. Nauk SSSR 25 (1939), 643-646. 1.11 Oscillation theorems for ordinary linear differential operators of arbitrary order,

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Dokl. Akad. Nauk SSSR 25 (1939), 717-720. 1.12 Oscillation Matrices, Kernels a n d Small Oscillations of Mechanical Systems, 2nd edition, Moscow, Leningrad (1950) (with F.R. G a n t m a k h e r ) . 1.13 Oszillationsmatrizen, Oszillationskerne u n d kleine Schwingungen Mechanischer Systeme, Akademie Verlag, Berlin, (1960) (with F . R . G a n t m a k h e r ) . T h e m a t h e m a t i c a l theory of oscillatory properties of small vibrations of mechanical systems is presented in t h e books 1.12 and 1.13. In t h e papers 1.9, 1.10, 1.11 the a u t h o r showed t h a t in this way one can obtain oscillatory theorems for b o u n d a r y value problems, selfadjoint and non-selfadjoint, moreover of any order (even or o d d ) .

2. G e o m e t r y o f B a n a c h s p a c e s a n d its a p p l i c a t i o n s in a n a l y s i s On this subject the following papers were published: 2.1 On several questions concerning the geometry of convex sets belonging to a linear n o r m e d a n d complete space, Dokl. Akad. Nauk SSSR 14 (1937), 5-7. 2.2 On positive additive functionals in linear n o r m e d spaces, Zap. Khar'kov. M a t . O.-va (4) 14 (1937), 227-237. 2.3 O n positive functionals in linear n o r m e d spaces, In the book "Some questions in t h e Theory of M o m e n t s " , Gos. Nauchn. Tekhn. Izd-vo Ukr., Kharkov (1938), (with F . R . G a n t m a k h e r ) ; English Transl.: Transl. M a t h . M o n o g r a p h s , Vol. 2, A m e r . M a t h . S o c , Providence, R.I. (1962). 2.4 On the linear operators t h a t leave invariant a conic set, Dokl. Akad. Nauk SSSR 23 (1939), 749-752. 2.5 Basic properties of n o r m a l conic sets in B a n a c h space, Dokl. Akad. Nauk SSSR 28 (1940), 13-17. 2.6 On the minimal decomposition of a linear functional into positive c o m p o n e n t s , Dokl. Akad. Nauk SSSR 28 (1940), 18-22. 2.7 O n an intrinsic characteristic of t h e space of all continuous functions denned on a bicompact Hausdorff space, Dokl. Akad. Nauk SSSR 27 (1940), 427-430 (with S.G. Krein). 2.8 On regularly convex sets in t h e space conjugate to a B a n a c h space, Ann. M a t h . 41 (1940), 556-583 (with V.L. S h m u l ' a n ) .

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2.9 On extreme points of regularly convex sets, Studia Math. 9 (1940), 133-138 (with D.P. Mil'man). 2.10 Linear operators leaving invariant a cone in Banach space, Usp. Mat. Nauk 3, no. 1 (1948), 3-95 (with M.A. Rutman); English transl.: Amer. Math. Soc. Transl. (1), 10 (1962), 199-235. Offprints of the papers 2.3, 2.9, 2.9 (4 copies) and 2.4, 2.5, 2.6 (4 copies) were submitted to the Lenin committee. The notion of the "cone" in Banach space (as the term itself) were introduced in the papers 2.2, 2.3. In the papers 2.2 and 2.3 and 2.5-2.7 the geometry of these cones, and in the paper 2.4 and in the joint paper 2.10 the theory of operators leaving a cone invariant with its applications were developed. These papers were continued and used by soviet and foreign mathematicians. At the last time the theory of cones got a series of essential applications in nonlinear functional analysis in papers by the group of Voronezh mathematicians headed by M.A.Krasnosel'skii. In the papers 2.1, 2.8, 2.9 convex sets in Banach spaces are studied. In 2.8 using using results of the paper 2.9 the theorem on extreme points of a regularly convex set which is known in the mathematical literature as the "Krein-Mil'man" theorem was established. Thanks to the well known paper by I.M.Gel'fand and D.A.Raikov this theory got important applications in the theory of representations of unitary groups. Thanks to papers by N.N.Bogol'ubov and N.N.Krylov it is used in the theory of dynamical systems. This theorem is presented in various papers and monographs.

3. Theory of Hermitian operators with finite deficiency numbers and method of directing functionals On this subject the following papers were published: 3.1 On the continuation problem for Hermitian-positive continuous functions, Dokl. Akad. Nauk SSSR 26 (1940), 17-22. 3.2 On Hermitian operators with deficiency indices equal to one, I, Dokl. Akad. Nauk SSSR 43 (1944), 323-326. 3.3 On Hermitian operators with deficiency indices equal to one, II, Dokl. Akad. Nauk SSSR 44 (1944), 131-134. 3.4 On a remarkable class of Hermitian operators, Dokl. Akad. Nauk SSSR 44 (1944), 191-195. 3.5 On a generalized moment problem, Dokl. Akad. Nauk SSSR 44 (1944), 239-243.

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3.6 On the logarithm of an infinitely decomposable Hermitian-positive function, Dokl. Akad. Nauk SSSR 45 (1944), 99-102. 3.7 On t h e continuation problem for helical arcs in Hilbert space, Dokl. Akad. N a u k SSSR 45 (1944), 139-142. 3.8 On a generalization of investigations of G.Szego, V.I.Smirnov and A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 46 (1945), 91-94. 3.9 On an extrapolation problem of A.N.Kolmogorov, Dokl. Akad. N a u k SSSR 46 (1945), 306-309. 3.10 On t h e resolvents of a Hermitian operator with deficiency index ( m , r a ) , Dokl. Akad. Nauk SSSR 52 (1946), 657-660. 3.11 On a general m e t h o d of decomposing positive-definite kernels into elementary p r o d u c t s , Dokl. Akad. Nauk SSSR 53 (1946), 3-6. 3.12 On Hermitian operators with directing functionals, Sb. Trudov Inst. M a t . Akad. Nauk Ukr. SSR 10 (1948), 83-106. 3.13 F u n d a m e n t a l aspects of t h e representation theory of Hermitian matrices with deficiency index

(m,m),

Ukr. M a t . Zh., 1, no.2 (1949), 3-66; English t r a n s l : A m e r . M a t h . Soc. Tranl. (2) 97 (1970), 75-143. 3.14 Infinite J - m a t r i c e s a n d t h e m a t r i x m o m e n t problem, Dokl. Akad. Nauk SSSR 69 (1949), 125-128. 3.15 On a one-dimensional singular b o u n d a r y value problem of even order on t h e interval (0,oo), Dokl. Akad. N a u k SSSR 74 (1950), 9-12. 3.16 Analogue of t h e Chebyshev-Markov inequalities in a one-dimensional b o u n d a r y value problem, Dokl. Akad. Nauk SSSR 89 (1953), 5-8. 3.17 On the theory of entire functions of exponential t y p e , Izv. Akad. Nauk SSSR Ser. M a t . , 11 (1947), 309-326. Offprints of the p a p e r s 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 2.10, 2.11, 2.12, 2.13, 2.14, 2.15, 2.17 (4 copies) were s u b m i t t e d to t h e Lenin committee. In t h e papers of this set the a u t h o r developed t h e theory of representations of of Hermitian operators (an a b s t r a c t analogue of Fourier transforms) which allowed to consider a new class of "entire" Hermitian operators. In t h e theory of entire operators it was a success to unite a n d to elaborate t h e solution of such problems as t h e problem of continuation of helical lines in Hilbert spaces, t h e problem of filtration a n d extrapolation of stationary casual processes, t h e problem of description of all spectral functions of t h e

187 non-determined Sturm-Liouville problem, - it was a success also to carry over the famous Chebyshev-Markov inequalities on boundary value problems. These researches are supplemented by elaborating of the method of directing functionals thanks to which the gap which was existed between abstract spectral theory and spectral theory of concrete differential operators was removed. The method of directing functionals allowed to obtain by the unite method for spectral partitions assertions of the Plancherel theorem type and of the Bochner-Khintchine type. Strictly analytic papers of the set 5 on inverse problems are in close connection with these papers. All this papers were a channel for penetration of different methods of the analytic functions theory to the operator theory and new facts from the functions theory were required; one of them was obtained in the paper 3.17. This work found applications in the theory of non-selfadjoint operators (see 10.4). It was a starting point in new researches of young soviet mathematicians I.V.Ostrovsky, V.I.Matsaev and A.A.Gol'dberg.

4. Theory of semibounded Hermitian operators On this subject the following papers were published: 4.1 On self-adjoint extensions of bounded and semibounded Hermitian operators, Dokl. Akad. Nauk SSSR 48 (1945), 323-326. 4.2 The theory of self-adjoint extensions of semibounded Hermitian operators and its applications. I, Mat. Sb., 20 (62) (1947), 431-495. 4.3 The theory of self-adjoint extensions of semibounded Hermitian operators and its applications. II, Mat. Sb., 21 (63) (1947), 365-404. 4.4 The basic theorems concerning the extension of Hermitian operators and some of their applications to the theory of orthogonal polynomials and the moment problem, Usp. Mat. Nauk 2, no.3 (1947), 60-106 (with M.A.Krasnosel'skii). Offprints all of these papers were presented to the Lenin committee, 4 copies of 4.2. In the papers 4.1, 4.2 the complete description of all positive selfadjoint extensions of a positive selfadjoint operators is given, the complete solution of the Newman-Friedrichs on existence criterion for positive selfadjoint extension is given, it was proved that the "Friedrichs extension" is the "most hard", the "most soft" extension is found. In the papers 4.3 and 4.4 an illustration of applications of these results to ordinary boundary value problems and to the moment problem is given.

188

5. I n v e r s e p r o b l e m s for t h e s p e c t r a l t h e o r y o f differential o p e r a t o r s On this subject the following papers were published: 5-1 Solution on t h e inverse Sturm-Liouville problem, Dokl. Akad. Nauk SSSR 76 (1951), 21-24. 5.2 Determination of t h e density of a nonhomogeneous symmetric string from its frequency spectrum, Dokl. Akad. Nauk SSSR 76 (1951), 345-348. 5.3 On inverse problem for a nonhomogeneous string, Dokl. Akad. N a u k SSSR 82 (1952), 669-672. 5.4 On a generalization of investigations of Stieltjes, Dokl. Akad. Nauk SSSR 87 (1952), 881-884. 5.5 On some new problems of t h e oscillation theory for S t u r m i a n systems, Prikl. M a t . Mekh. 14 (1952), 555-568. 5.6 On t h e transfer function of a one-dimensional second-order b o u n d a r y value problem, Dokl. Akad. Nauk SSSR 88 (1953), 405-408. 5.7 On some cases of effective determination of t h e density of a nonhomogeneous string from its spectral function, Dokl. Akad. N a u k SSSR 93 (1953), 617-620. 5.8 On inverse problems of t h e theory of filters a n d A-zones, Dokl. Akad. N a u k SSSR 93 (1953), 767-770. 5.9 A fundamental approximation problem in t h e theory of extrapolation a n d filtration stochastic processes, Dokl. Akad. N a u k SSSR 94 (1954), 13-16; English transl.: Selected Transl. in M a t h . Statistics and Probability, Inst, of M a t h . Statistics and Amer. M a t h . Soc. 4 (163), 127-131. 5.10 On a m e t h o d of effective resolution of an inverse b o u n d a r y value problem, Dokl. Akad. Nauk SSSR 94 (1954), 987-990. 5.11 On integral equations t h a t generate second-order differential equations, Dokl. Akad. Nauk SSSR 97 (1954), 21-24. 5.12 On the determination of the potential of a particle from its ,9-function, Dokl. Akad. N a u k SSSR 105 (1955), 433-436. 5.13 Continual analogues of propositions on polynomials orthogonal on t h e unit circle, Dokl. Akad. N a u k SSSR 105 (1955), 637-640. 5.14 On t h e theory of accelerants a n d 5-matrices of canonical differential systems, Dokl. Akad. Nauk SSSR 111 (1956), 1167-1170.

189

5.15 On a continual analogue of a formula of Christoffel from t h e theory of orthogonal polynomials, Dokl. Akad. N a u k SSSR 113 (1957), 970-973. Offprints of these papers are presented to t h e Lenin committee in 4 copies. In the p a p e r 5.1 of t h e enclosed list for t h e first time was shown a chain of analytic operations which allows to solve inverse problem for t h e S t u r m - Liouville equation in t h e Borg formulation defined more precisely by V.A. Marchenko. In t h e same p a p e r for t h e first time a transfer function for solution of inverse problem was introduced. Mechanical sense of this function a n d simultaneously a series of its basic properties was t h e n clarified in 5.3 and 5.6. In 5.2, 5.3 conditions of uniqueness a n d solvability of inverse problems for nonhomogeneous string are established. In 5.5 results of t h e p a p e r s 5.2, 5.3 are illustrated by example of mechanical systems with a finite n u m b e r degrees of freedom. In 5.8 an inverse problem for t h e Hill equations is solved, i.e. t h e problem of reconstruction of a periodic coefficient in equation by its A-zones of stability (on t h e physical language by zones of letting pass of corresponding

filter).

After papers 5.1, 5.2 a p p e a r e d p a p e r s by I.M.Gel'fand a n d B.M.Levitan in which they starting from t h e same transfer function obtained simple integral equations which allow t o recover one-dimensional radial Schroedinger equation from its spectral function. Uniqueness of solution for this problem was established earlier by G.Borg a n d V.A.Marchenko. In 5.6, 5.11 a gap in necessary a n d sufficient conditions of solvability of t h e problem for recovering of a potential by spectral function which took place in p a p e r s by I.M.Gel'fand a n d B.M.Levitan is removed. Simultaneously was obtained a simplification of their integral equation which allowed to see a series of new cases when the inverse problem is solved effectively. Different cases of effective solving of inverse problems of classical a n d q u a n t u m mechanics are shown in 5.7, 5.10, 5.15. In 5.15 it is shown how to reduce t h e solving of inverse problem for radial s t a t e equation of a particle with an a r b i t r a r y value of the q u a n t u m n u m b e r to the corresponding problem for its s t a t e . In 5.12 t h e p r o b l e m of d e t e r m i n a t i o n of t h e potential of a particle by its limit p h a s e is solved. In 5.15 this result is generated on t h e case of a system of differential equations (the case of tensor forces in inverse problems of q u a n t u m mechanics). In 5.13 it is found out t h a t a n a t u r a l generalization a n d in certain cases also a simplifica-

190

tion of previous researches is obtained by replacing of equation of the radial Schroedinger equation t y p e by a certain system of equations of t h e first order (of radial Dirac equations type). In 5.4 it is shown t h a t t h e successive generalization of well-known researches by Stieltjes in chain fractions leads to complete theory of spectral functions of regular and singular structure. In 5.9 it is established t h a t the problem of extrapolation a n d filtration for real casual process by its d a t a for finite time interval is reduced to solving of a problem of the foundation of the density of nonhomogeneous string by its spectral function.

6. T h e o r y o f linear H a m i l t o n i a n s y s t e m s o f differential e q u a t i o n s w i t h p e r i o d i c coefficients O n this subject t h e following papers were published: 6.1 Generalization of a certain investigations of A.M.Lyapunov concerning differential equations with periodic coefficients, Dokl. Akad. Nauk SSSR 73 (1950), 445-448. 6.2 On t h e application of an algebraic proposition in t h e theory of m o n o d r o m y matrices, Usp. M a t . N a u k 6, n o . l (1951), 171-177. 6.3 T h e basic propositions of the theory of A-zones of stability of a canonical system of linear differential equations with periodic coefficients, Memorial volume dedicated to A.A.Andronov, Moscow (1955), 413-498; English transl.: Topics in Differential a n d Integral E q u a t i o n s and O p e r a t o r Theory, O p e r a t o r Theory: Advances a n d Applicatins, Vol. 7, Birkhauser, Basel (1983), 1-70. 6.4 On tests for stable boundedness of solutions of periodic canonical systems, Prikl. M a t . Mekh., 19 (1955), 641-680; English transl.: A m e r . M a t h . Soc. Transl. (2) 120 (1983), 71-110. 6.5 Hamiltonian systems of linear differential equations with periodic coefficients, P r o c . I n t e r n a t . S y m p . Linear Vibrations, Kiev, (1961) (with V.A.Yakubovich). 6.6 Analytic properties of multipliers of periodic canonical differential systems of positive type, Izv. Akad. Nauk SSSR Ser. Mat., 26 (1962), 549-572 (with G.Ya.Lyubarskii). English transl.: Amer. M a t h . Soc. Transl. (2) 89 (1970), 1-20. T h e papers 6.3, 6.4, 6.5 (4 copies) are s u b m i t t e d to t h e Lenin committee. T h e m a i n p a p e r is 6.3 in which there are established t h e classification of multiplicators, their rules of motion on t h e unit circle, existence of central zone of stability for systems

191 of positive type and a basic rule for determination of critical frequencies in phenomenon of parametric resonance. After the papers 6.1 and 6.2 for the first time became possible the extension of classical results by A.M.Lyapunov obtained at 90-th of the past century for a single equation on systems. Some of results of the author were rediscovered later by foreign physicists in connection with theory of sinchrotrons with strong focusing (Courant and Snider, Seiden, Luders and others).

7. Generalization of L.S.Pontrjagin's duality principle On this subject the following papers were published: 7.1 On positive functionals on almost-periodic functions, Dokl. Akad. Nauk SSSR 30 (1941), 9-12. 7.2 The duality principle for a bicompact group and square block-algebra, Dokl. Akad. Nauk SSSR 69 (1949), 726-728. 7.3 Hermitian-positive kernels on homogeneous spaces, II, Ukr. Mat. Zh., 2, no.l (1950), 10-59; English transl.: Amer. Math. Soc. Transl. (2) 34 (1963), 109-164. Offprints of the papers 7.2, 7.3 are submitted to the Lenin committee (4 copies). In the paper 7.1 was obtained a partial generalization of the classical L.S.Pontrjagin's duality principle, obtained by him for locally-compacted commutative groups on, the case of compact noncommutative groups. This result as it turned later was obtained somewhat earlier by Japanese mathematician Tanaka. In 7 years in 7.2 the author obtained complete generalization of the principle containing the precise description of duality object which formed the most difficult part in the establishing of the new principle. The detailed exposition with certain further development is given in 7.3.

8. Spectral theory of operators in spaces with indefinite metric On this subject the following papers were published: 8.1 Lorentz transformation in a real Hilbert space, Chapter 5 of the paper 2.10. 8.2 Helical lines in an infinite-dimensional Lobachevsky space and Lorentz transformations, Usp. Mat. Nauk 3, no.3 (1948), 158-160. 8.3 On an application of the fixed point principle in the theory of Hnear transformations of spaces with an indefinite metric,

192

Usp. Mat. Nauk 5, no.2 (1950), 180-190; English transl.: Amer. M a t h . Soc. Transl. (2) 1 (1955), 27-35. 8.4 Spectral theory of operators in spaces with an indefinite metric. 1,11, Tr. Mosk. Mat. O.-va 5 (1956), 367-432; 8 (1959), 413-496 (with I.S.Iokhvidov); English transl.: Amer. M a t h . Soc. Transl. (2) 13 (1960), 105-175; (2) 34 (1963), 283-373. 8.5 On an integral representation of a continuous Hermitian-indefinite function with a finite n u m b e r of negative squares, Dokl. Akad. Nauk SSSR 125 (1959), 31-34. Offprints of t h e papers 8.4, 8.5 are s u b m i t t e d to t h e Lenin committee (4 copies). T h e first fundamental result on the theory of self-adjoint operators in spaces with an indefinite metric was obtained by L.S.Pontrjagin. To the a u t h o r belongs t h e elaboration of a new m e t h o d (8.1, 8.3) which allowed to obtain easier a n d at t h e same time to generalize t h e result of L.S.Pontrjagin. T h e essentially new application of t h e theory of spaces with an indefinite metric was the creation by t h e a u t h o r of foundations of the theory of helical lines in Lobachevsky spaces of infinite n u m b e r of dimensions and establishing of t h e t h e o r e m on integral representations of a Hermitian-indefinite function (a generalization of the Bochner-Khintchine t h e o r e m ) . Except of the last result of 8.5, all the others are presented in t h e joint s u m m a r y paper 8.4.

9. T h e o r y of singular integral e q u a t i o n s and s y s t e m s of such equations On this subject the following papers were published: 9.1 On a new m e t h o d of solving linear integral equations of first and second kind, Dokl. Akad. Nauk SSSR 100 (1955), 413-416. 9.2 Integral equations on t h e half-line with kernel depending on the difference of the arguments, Usp. M a t . N a u k 13, no.5 (1958), 3-120; English transl.: Amer. M a t h . Soc. Transl. (2) 22 (1962), 163-288. 9.3 On some basic propositions of the theory of systems of integral equations on the half-line with kernels depending on t h e difference of t h e a r g u m e n t s , P r o c . 3rd All-Union M a t h . Conf., Vol. 2 (1956), 37-38 (with I.C.Gohberg). 9.4 Systems of integral equations on t h e half-line with kernel depending on t h e difference of a r g u m e n t s , Usp. Mat. Nauk 13, no.2 (1958), 3-72 (with I.C.Gohberg); English transl.: Amer.

193

Math. Soc. Transl. (2) 14 (1960), 217-287. 9.5 On the stability of the systems of partial indices of the Hilbert problem with several unknown functions, Dokl. Akad. Nauk SSSR 119 (1958), 854-857 (with I.C.Gohberg). Offprints of the papers 9.1, 9.2, 9.4, 9.5 are submitted to the Lenin committee (4 copies). In connection with inverse problems the author elaborated (the paper 9.1) a new method for solution of Hnear integral equations of first and second kind in a finite interval. This method became to be used in contact problems of the elasticity theory (the papers by N.H.Arutjunjan, P.E.Prokopovich, G.Ya.Popov). In the paper 9.2 for the first time is established the structure of simple or generalized resolvent of Wiener-Hopf integral equation moreover in the unique assumption of belonging of the kernel function to the class Li[ — oo,oo). It allowed to give to all the theory the harmony and to formulate a series of final theorems. In joint papers 9.3, 9.4 based on the paper 9.2 for the first time the theory of systems of Wiener-Hopf integral equations was elaborated. Let note separately criterion of stability of the systems of partial indices of integral system which gave a reason for establishing of the corresponding result in the RiemannHilbert problem for several unknown functions.

10. Method of infinite determinants of a perturbation On this subject the following papers were published: 10.1 On a trace formula in perturbation theory, Mat. Sb., 33 (75) (1953), 597-626. 10.2 On perturbation determinants and a trace formula for unitary and self-adjoint operators, Dokl. Akad. Nauk SSSR 144 (1962), 268-271; English transl.: Soviet Math.. Dokl. 3 (1962), 707-710. 10.3 On the theory of wave operators and scattering operators, Dokl. Akad. Nauk SSSR 119 (1962), 475-478 (with M.Sh.Birman); English transl.: Soviet Math. Dokl. 3 (1962), 740-744. 10.4 On the theory of linear non-selfadjoint operators, Dokl. Akad. Nauk SSSR 130 (1960), 253-256; English transl.: Soviet Math.. Dokl. 1 (1960), 38-40. Offprints of all these papers are submitted to the Lenin committee (4 copies).

194 In 10.1 by this method is given a strict basis for a trace formula in perturbation theory derived for the first time (not perfectly strictly) for finite-dimensional perturbations by I.M.Livshits (in connection with problems of quantum statistics of solid solutions). In 10.2 further generalizations of this formula are established. Using of papers 10.1, 10.2 allowed to establish in the joint paper 10.3 a first theorem on structure of scattering matrix giving the Heisenberg scattering operator in elementary spaces. In the paper 10.4 the method of perturbation determinants is used to establish the completeness of system of the root subspaces for non-selfadjoint operator with an imaginary kernel component.

/M.G.Krein/ 10/XI.1962

195

SOLUTION OF T H E I N V E R S E STURM-LIOUVILLE P R O B L E M

M. G. KREIN

1. Let q(x) (0 < x < I, I < oo) be a real-valued function that is integrable in its interval of definition (0,1), and let Lq = —d2/dx2 + q(x) be the differential operator corresponding to q. We denote by Sq(a, (3) (—oo < a, (3 < oo) the spectrum, arranged as an increasing sequence of numbers, of the boundary value problem (1)

Lqy = Ay;

cos a y(0) + sin a y'(0) = 0, cos/3 y(l) + sin (3 y'(l) = 0.

As is known, the number I is completely determined by a given spectrum Sq(a, /3) = {-\j}i°i s m c e (2)

I2 = 7T2 lim ( n 2 / A „ ) . n—Voo

Borg [1] showed that if for two functions q\(x) and q[2(x) and some numbers a\, a.2, and (3 ( t a n a i ^ tan«2) we have Sqi(ai,f3) = Sq2(ai,(3) (i = 1,2), then almost everywhere qi(x) = q2(x) (0 < x < I).1 Recently, Marchenko [3], using other techniques, obtained a more complete result; namely, he showed that the function q(x), as well as the numbers t a n a i , tana2, and tan/3 ( t a n a i 7^ tan02) are uniquely determined by the corresponding spectra Sq(ai, (3) (i = 1,2). Moreover, changing slightly the setting of the question, Marchenko generalized this uniqueness theorem to a singular case where I is not necessarily finite and the function q(x) (0 < x < I) is absolutely integrable, but possibly in subintervals (0, a) with a < I only, and not in the entire interval (0,1). We call l-sequence any sequence {An} that satisfies condition (2) with a given l>0. As is known, the numbers of two distinct spectra Sq(oti,(3) (i = 1,2) strictly alternate. The above-mentioned important studies by Borg and Marchenko left the following basic problem unsolved: P r o b l e m A. Given two interlacing l-sequences of numbers, S = {A„} and S' =

(3)

Ai < A'i < A2 < A'2 < • • • ,

Doklady Akademii Nauk SSSR, vol. 76 (1951), pp. 21-24. ^ o r g assumed that one of the numbers a or a' equals zero, but it is easy to get rid of this restriction [2].

196

it is required to determine whether there exist a function q (0 < x < I) and numbers a, a', and (3 such that Sq(a,(3) = S and Sq(a',(3) = S', and if they do, to find them. In the sequel, we present a chain of analytic procedures that give a solution to this problem. The solution is based on the following idea: I. Just as to each Jacobian matrix J corresponds a certain power moment problem for every solution of which (mass distribution function) the matrix J itself is completely determined, so to a second order differential operator L with boundary condition on one end corresponds a "generalized" moment problem such that for every distribution function of it, is completely determined, together with the boundary condition, the operator itself (if the latter is given in a "canonical" form of some kind). For operators of "sufficiently regular type" this generalized moment problem is the problem of continuation of Hermitian-positive functions developed by the author [4-7]. This idea is applied here to the solution of Problem A. The Borg-Marchenko uniqueness theorem will be obtained incidentally. 2. We denote by Pa (0 < a < oo) the set of real-valued continuous even functions F(t) (\t\ < a) to which corresponds a positive definite kernel F(s — t) (0 < s,t < a). The central mass M of a function F(t) € Pa is the greatest number p for which F(t) — p G Pa. There are many ways to find M. In the most difficult case where a < oo, we can obtain M, for example, using the formula M

~

1 =

X>;(/

Xj(.s)ds)

,

where Xj(s) (i = 1,2,...) is a complete orthonormal system of fundamental functions of the integral equation X(s)

= p f F(sJo

t)X(t)dt,

and {MJ}I° is the corresponding sequence of characteristic numbers [4]. If F(t) £ Pa, then, considering F in a shorter interval (—x, x), we obviously have F(t) G Px (0 < x < a). To the function F as an element of Px (0 < x < a) will correspond a central mass M(x) (0 < x < a). The function M(x) (0 < x < a) so defined will be called the central function for F(t) € Pa- It turns out that this function is always lower semicontinuous and nonincreasing as x grows; moreover, M(x) < F(0) (0<x +0. 3. When solving Problem A, it can obviously be assumed without loss of generality that all the numbers in (3) are positive. To every two infinitely increasing interlacing sequences 5 = {Aj}J° and {\'j}f (0 < Ai < \[) corresponds an absolutely convergent expansion OO

3= 1

-i

\

OO

l\l

'

J

where 7 > 0, and rrij > 0 (j = 1,2,...).

3= 1

J

197 The existence of the expansion (4) for the product on the left-hand side can be proved by elementary means, but it also results directly from the following general proposition: In order that a function $(z) defined for all z £ (0, oo) be a regular analytic function that assumes only nonnegative values for negative z and maps the upper half-plane into its part, it is necessary and sufficient that $(z) have, for each z £ (0, oo), the representation [°° do-(X)

da(X)
where 7 > 0 and cr(\) (0 < A < 00) is a nondecreasing function. We now use (4) to construct the function 00

F(t) = S2 -T1 COS y/Xjt 3= 1

(-00 < t < OO)

J

that obviously belongs to the class Poo. Its central function will be denoted by M{x;S,S') (0
M (2a:) - M(2l) = const (^

- ^^]

forO<x
where 21. Since, in view of (5), (6)


4—^)

(0<x
we arrive at the following rule of recovering the operator Lq from its two spectra S and S'. We construct the function M(x) = M(x;S,S') (0 < x < I), find the function (p(x) from formula (6), and using the

198 Now, constructing the solution x(x> -M that satisfies the conditions x(^>^) = sin/3, X'(l; A) = — cos/3, we obtain

<*>

'*•""'—m u=^->-

Thus, for the solvability of Problem A, the following conditions are necessary: 1°. The function M(x) = M(x; S, S') (0 < x < 21) is absolutely continuous, together with its first two derivatives. 2°. The derivative dM/dx (0 < x < 21) is different from zero. If these conditions hold, we can use formula (6) to construct the function
G. Borg, Acta Math. 78 (1946), 1-2. L. A. Chudov, Mat. Sbornik 25 (67) (1949), no. 3. V. A. Marchenko, Dokl. Akad. Nauk SSSR 72 (1950), no. 3. M. G. Krein, Dokl. Akad. Nauk SSSR 26 (1940), no. 1. , Dokl. Akad. Nauk SSSR 4 4 (1944), no. 5. , Dokl. Akad. Nauk SSSR 4 5 (1944), no. 3. , Ukrain. Mat. Zh. 1949, no. 2.

I N S T I T U T E O F M A T H E M A T I C S , A C A D E M Y OF SCIENCES O F T H E U K R A I N .

SSR

199

ON A FUNDAMENTAL APPROXIMATION PROBLEM IN THE THEORY OF EXTRAPOLATION AND FILTRATION OF STATIONARY RANDOM PROCESSES M. G. KREIN Let o{\) (- M < A < m) be a nondecreasing function which satisfies the condition —oo

Denote by Ac the Hilbert space of functions F{X) (- oo < A < oo) a-measurable and a-integrable, together with their squares, with the usual definition of the scalar product (F, G) {F, G G Aoc) as the integral over the whole axis of F(A)GOOrfoU). Let / = (a, b) (- °o < a < b < °°) be some interval of the real axis - o° < t < «>. Denote by A/ the closed linear manifold in A^ of the family of functions of argument A: exP (it2A) - exp (itlA)

=

,2

e x p (-A5)

^

(

Note that if a (A) is a function of bounded variation,

j

b J2

g ^_

then A/ can be defined

more simply, viz. as the closed linear manifold in A,*, of the family of functions exp (itX) {t G / ) . Two questions can be posed: I.

Under what criteria A/ = Aoo?

II.

/ / A / ^ A^, then how to express analytically

the orthogonal projection

P[F of any element F G A,*, on A/? For the case of / = (- oo, a) the first question was already answered by die author in 1944 [1] and (when a (A) is of bounded variation) he also obtained PjF for F(A) = exp(itA) {t > a) *.

For special cases of functions a and F the question

of the effective determination of P[F for / = (- oo, o) was studied by N. Wiener [3] who explained its applications in technology (see also [4]). The analytically formulated problems I and II can be reformulated as problems in the theory of prediction (extrapolation) and filtration of stationary random processes from their observation in the time interval /. Naturally problems I and II are of greatest interest when / is a finite interval. However their investigation is particularly difficult in this case.

Unaware of the applications which problems I and II have, the

author had already obtained in 1940 [5] and then in 1943—1944 [6] a series of essential results directly connected with them. Certain results have been obtained 'These results were obtained by the author in answer to the question raised by A. N. Kolmogorov [1]; K. Karhunen L^J repeated them in 1950. Dokl. Akad. Nauk SSSR 94 (1954), 13-16, Selected translation in Math., Statistics and Probab., Inst, of Math. Statistics and AMS 4 (1963), 127-131. 127

200 128

M. G. KREIN

in recent times which give the complete solution of t h e s e problems. It i s derived below for an odd function a{\) b e c a u s e the investigation of the solution in the most general c a s e would require u s to undertake much preliminary work.

Besides

in problems of the theory of real stationary random p r o c e s s e s only this c a s e occurs. In considering a finite interval /, it is possible without loss of generality to assume that / = (— a, a) and then it will be convenient to write A a and Pa respectively for A/ and Pi. 1. Solution

of problem I. Thus l e t cr{— A) = - a(\)

Assume r(A) = 2a(yJ\-

(0 < X < C«D; a ( 0 ) = 0).

0) (0 < A < <*,; r(0) = 0). By virtue of (1) DO

f dr{\)/{l + .X)
and sufficient

that

r — —

t(x) = / y/M'{s)ds
the

function M (x) will have a derivative almost everywhere equal to zero but this property no longer holds in any larger interval containing the interval {xa_ Q, xa ( Q)Let us find die solution of the integral equation {x; X) = 1 - A fix 0

- s) 6 is; X)dM{s) (0 < x < I).

Denote by L(2) (0, f ) , where 0 < £ < I, the set of all functions f{x) (0 < x <
which are M-measurable and which satisfy the condition

/l/(x)|2^f(.x) <„,. 0 Theorem 2. Entire functions F{X) of form *a-0 F(X) = / f(x) ix; X2)dMix) ifG 0 exhaust

all even functions

Entire functions

of

Aa.

G(A) of form

iJV

(o, *„_<,))

201 APPROXIMATION PROBLEM IN RANDOM PROCESSES G(A) = T

J

£*) '{x; W ix

A „

exhaust

*a+0 / f{x) 0

F{\) of form. {%• \2)dM(x)

all even entire functions

FiX) satisfying

/|F(A)|2,fo(A)
lim

'

( / G ' L ( ^ ( 0 , *a+0)) the '
conditions (£=A+ip).

(2)

G(A) of form i *a+0

G(A) = i exhaust

d*

of Ka.

Theorem 3. Entire functions

exhaust

|g(»)|2rfx
0

all odd functions

F(A) =

i 7

129

7

*a+0

g(x) <£' U ; X 2 ) ^ ( J" | g U)|2rf a ; <

0 all odd entire functions

GiX) satisfying

0 condition

(2).

F o r c/a = dX/rr we w i l l h a v e : Z < °o,
cos

X*> <£'(*; A 2 )/A =

- sin Xx, and Theorems 2, 3 transform into the well known Paley— Wiener [9] theorem. 3.

Solution of problem II. Let F G Aoo. Form the functions

fix) = l.i.m. / F{X) $ ix; X2)doiX),

gix) = l.i.m. / FiX) 0 ' ix;

/v->oo -yv

X2)-j^;

/v-*oo -yv

the limit in the first e x i s t s in view of convergence in L ( 2 ) ( o , I) and in the second in view of convergence in L

(0, I), i.e., convergence in mean square in the in-

terval (0, I). The solution of problem II for I = i— a, a) is given by the formula X

PaFW

=

a-Q / fix) 0

. *a-0 $ ix; X2)dMix) + Y f gix) '{x; X2)dx. A 0

Let us also derive the formula for the square of the distance of F from A a :

f \FiX) - PaFiX)\2do{X) -°°

= ;

\fix)\2dMix)

r

+

*a-0

\g{x)\2dx.

*a_0

The family of projective operators Pa (a > 0, Prj = /) form some spectral decomposition of identity; here the operator Pa is a weakly continuous function, from the left, of the argument o: P a _ o = Pa {a < 0). The point a > 0 is a point of weak continuity on the right:

Pa+o = Pa if and only if Mixa

+

Q + 0) = M r a _ 0 - 0), i-e.,

when the points x a _o and xa+Q are the points of continuity of the function

Mix)

and between them the function Mix) is of constant value. 4.

Let I* (< Z) be the greatest of the points of growth of the function Mix).

the function Mix) lor x > + 0 and < /* has two absolutely continuous derivatives

If

202 130

M G. KREIN

where M {x) > 0 for x > + 0 then taking ; T = fjM'(x)dx,

4 $ U ; A) = V « ' U ) <M*;A)

x

t = fy/M'{s)ds, (0

<x
we get $ U ; A) as the normalized solution ( (0; A) = [M (+ 0)]^) of the differential system ~ -

q{t)u + Au = 0,

u'(0) + U A - A) u (0) = 0 (0 < t < T),

where q(t) =
(3)

and

/ <MA) (> 0). —oo

In this c a s e for any F G Aoo we will have* i a

a

PaF(X)

= ffU)${t;k2)dt 0

°°

+ ±fg(t)yHt;>a)dt+m A 0

f F(\)do{\), -~>

(4)

where V{t, X) = *Ct; 0)-£-[<&(«; A)/<3(«; 0)] / W = l.i.m. fK\)$U;\*)M\),

g(0=l.i.m.

/

(0 < t < I ) ,

(5)

F(A) * (t; A2) - ^ l i ,

(6)

where the limits are understood in the s e n s e of convergence in mean square in the interval (0, 7). B e s i d e s this we have f | F ( A ) - PaF{\)\2do{\) —°°

T = f\f{t)\2dt

T + f \ g{t)\*dt

a

(0
T).

a

Conversely if for some system (3) (with m > 0, — °° < A < °o) the spectrum is positive and $ ( j ; A) is the solution of the system (3), normalized, for example, by the condition (0; A) = 1 and r(A) (0 < A < oo; r(0) = 0) i s , in the corresponding manner, die normalized orthogonal s p e c t r a l function (any, if it is not uniquely determined) then for the function a (A) = sign \-r(A 2 ) (— oo < A < oo) A a will be different from Aoo only for a < T, and for a < Tproblem

II is solved by the formulas (4),

(5) and (6). Thus the effective solution of problems I and II depends on the effective construction of the functions M(x), and <£>{x; A) (and in some c a s e s only on one function $ U ; A)) corresponding to the s e n s e of the given function CT(A) indicated above. The several c a s e s where it is possible to make such effective construction have been mentioned in the preceding note [8]. In a following note we will show that this is p o s s i b l e in all c a s e s where the function a is analytic and da/dX is a *For m = 0 the last term on the right side of (4) is assumed to be zero quite tndependendy of whether the integral multiplying m has sense or not.

203 APPROXIMATION PROBLEM IN RANDOM PROCESSES

131

rational function and a l s o in many other c a s e s (for example when da/d\

can be ob-

tained by dividing unity by a trigonometric polynomial).

BIBLIOGRAPHY [l] M. G. Krein, On a problem of extrapolation Nauk SSSR 46 (1945), 306-309.

(Russian)

[2] K. Karhunen, Uber die Struktur station'drer (1950), 141-160. [3J N. Wiener, Extrapolation,

of A. /Y. Kolmogoroff, Dokl. Akad.

interpolation

zufdlliger

Funktionen,

and smoothing

Ark. Mat. 1

of stationary

time

series,

Wiley, New York, 1949. [4] A. Jaglom, Introduction

to the theory of stationary

Mat. Nauk 7 (1952), no. 5 (51), 3-168.

random functions,

Uspehi

(Russian)

[5] M. G. Krein, Dokl. Akad. Nauk SSSR 24 (1940), no. 1. [6]

, ibid. 44 (1944), no. 5; 45 (1944), no. 4; no. 5.

[7J

, On a generalization 884.

[8]

of investigations

of Stieltjes,

ibid. 87 (1952), 881-

(Russian) , ibid. 93 (1953), no. 4.

[9] N. I. Ahiezer, Lectures (Russian)

on the theory of approximation,

Kharkov, 1940.

Translated by: D. V. Thampuran

204 O N T H E D E T E R M I N A T I O N OF T H E P O T E N T I A L OF A PARTICLE F R O M ITS

S-FUNCTION

M. G. K R E I N

Results of the paper [1] allow one to indicate a simple criterion that a given function S(k) (—00 < k < 00) be the S'-function (S'-matrix) of s-state of a particle in a centrally symmetric field with a certain potential V(r), and also to obtain a simple rule of recovering the potential from the function S(k). 1. Let H(t) (0 < t < 2T) be a real-valued measurable locally integrable function that satisfies the condition (Y) For each r (0 < r < T) the equation (1)

q{t;r)+

f

H(\t-s\)q(s;r)ds

= l

{~r
has a unique bounded (and therefore continuous) solution

q(t;r).

For any complex k we set (2)

x(r;k)

=k

q(s; r) cos ks ds Jo Directly from the results of [1] follows

(0 < r < T).

Theorem 1. The function x(r', k) (0 < r < T) is the solution of the differential system (3)

Tr(9{rr2%)+k29{rr2x

= Q]

x(0;*) = o, x'(Q;k) = k,

where g(r) = q(r; r) (0 < r < T). We set ip(r; k) = x{r\ k)/g{r) (0 < r < T); then ip is the solution of the system (4)

{al~ig~){di

+ i

g-)i,

+ k2 P

'

= 0''

^(0; A;) = 0,

^'(0;k)

= k.

The function g(r) (0 < r < T) is always absolutely continuous and positive. If H(t) (0 < t < 2T) is absolutely continuous, then the first derivative g'(r) is also absolutely continuous, in which case system (4) can be rewritten in the form (5) (6)

V" - V(r)V> + fc2V = °; V

V'(0;A;) = 0,

( r ) = s ( r ) ^ ^

ip'(0;k)=k,

(0
We will associate system (5) to a function H(t) (0 < t < 2T) in all cases, thus allowing V(r) (0 < r < T) to be a generalized function of the type D\ (derivative of locally integrable function). Doklady Akademii Nauk SSSR, vol. 105 (1955), no. 3, pp. 433-436.

205

2. Later on, we will be interested in the case where T = oo. In this case, system (5) has a completely determined spectrum, which is nonnegative. Integrating (3) by parts we obtain for tp the following expression: V(r; k) = Im leikr

(7)

(l - J * V2r(s)e-iks

ds) j ,

where T2 r (s) denotes a solution of the integral equation /•2r

T2r(t)+

(8)

Jo

H(\t-s\)T2r(s)ds

= H{t)

(0<*<2r).

It is natural to assume that under certain conditions the asymptotic formula for ip(r; k) as r —• oo will include a function T(t) (0 < t < oo) that is a solution of the integral equation /•OO

V(t) +

(9)

H(\t-s\)T{s)ds

= H{t)

(0
Jo

Consider the case where H(t) € £i(0, oo). As is easily seen, in this case the condition (Y) is equivalent to the condition of nonnegativity of the function /•oo

(10)

p(k) = 1 + 2

H(t) cosktdt (-co < k < oo). Jo It turns out that if p{k) > 0 (0 < k < oo), then equation (9) has one and only one (real-valued) solution Y £ £i(0, oo), which can be obtained as follows. By virtue of the general Wiener-Levy theorem (see [2], Chapter VI, subsection 101), there is a real-valued function 7 G la(0, 00) such that (11)

\ogp(k) = 2

/*oo

~/(t)cosktdt

(-00 < k < 00).

Jo Then T(t) can be found from the relation (12)

1-

T(t)e-ikt

[

dt = exp (-

f

It follows from (11) and (12) that p(k) =

ikt ikt )e-— dt r(t)<

/

(Imfc < 0).

2

/f•O OO O

(13)

e-iktj(t)dt\

( — 00 < k < OO),

Jo

so that we can set /•OO

(14)

1- /

T(t)e-ikt

dt = p'll2{k)eiS^

(-00 < k < 00).

If we normalize the continuous function d(k) by the condition 6(0) = 0, then (15)

S(~k) = -6{k),

6(±oo) =

lim 5(k) = 0. k—>±oo

The second equality follows from the fact that £(—00) — J(+00) = 2S(—00) is the number of zeros of the function (14) inside the lower half-plane Im/e < 0, which is obviously equal to 0.

206 Theorem 2. If the conditions (16)

H{t) £ Za(0,oo),

p(k) > 0

(0 < fc < oo)

are satisfied, then the following asymptotic formula holds: (17)

\im{i(;(r;k)-p(k)-1/2sm(kr

+ 5(k))} = 0

(-co < k < oo).

r—^oo

3. Consider now the 5-function S(fc) = exp(2i<S(fc)). Obviously, 1) |5(fc)| = 5(0) = 1; 2) S(-jfe) = 5(fc) ( - o o < fc < oo);

3) argS(k) —>• arg 5(0) as A; —» oo. Since, according to (12) and (14), (18)

S(k) = exp ( 2i /

sinfct7(£)d£ 1

(—oo < k < oo),

we have for S(k) also the representation 4) S(k) = 1 + / ^ e ifct s(i) di, where s(t) € Li(0, oo). Theorem 3. / / 5(&) (—oo < k < oo) is a function with properties l)-4), then there is always an ordinary or generalized function V(r) (of the type D\) such that the spectrum of system (5) will be nonnegative and, for S(k) = arg S(k) and the corresponding definition of p{k) — p(—k) (> 0), the solution ip of system (5) will satisfy the asymptotic relation (17). Indeed, as is easily seen, the properties l)-4) imply the existence of a real-valued function 7 G £i(0,00) for which (18) holds, and then (10) and (11) yield a function H(t) G Li(0, 00) from which the desired potential V(r) will be found using formula (6). As is easy to see, a rational function S(k) (5(0) = 1) satisfies conditions l)-4) if and only if a) all zeros and poles of S(k) are nonreal, b) for every zero of S(k) of certain multiplicity there is a pole of the same multiplicity, symmetric to the former with respect to the real axis, c) for a zero (a pole) of S(k) that does not lie on the imaginary axis, there is a zero (a pole) of the same multiplicity, symmetric to the former with respect to the imaginary axis, and d) the function S(k) has equal numbers of zeros and poles (counted with multiplicities) inside the upper (and hence the lower) half-plane. We note that, for a rational function S(k) with given zeros and poles, the function p(k) is also rational and can be found directly. After that, H(i) (0 < t < 00) is obtained as a sum of products of exponential functions by polynomials, and the solution of equation (1), and thereby the function V(r) will be expressed in explicit form [3, 4]. 4. Now let Si(k) (—00 < k < 00) be a function that satisfies conditions 1), 2), and 4), but not 3). Then there is an integer m (> 0 or < 0) such that 3') arg S(k) -> arg 5(0) - 2mn as k -» 00. Suppose m > 0. We choose arbitrary positive numbers K\ < K2 < • • • < Km and make up a function m

/h —'

\

2

207 It is easily seen that the function S(k) satisfies all conditions l)-4); therefore, a system (5) corresponds to it. Using a formula by Jost and Kohn [5], V(r) can be transformed into a new potential Vi(r) for which the system (5) has exactly m negative eigenvalues Ej = —K2 (j = 1,2,..., m) with their respective eigenfunctions i>i{r; IKJ) {j = 1,2,..., m) satisfying ipi(r; iKjf dr = p- l

(j = 1, 2 , . . . , m),

Jo

where pj (j = 1,2,..., m) are given positive numbers. It turns out that if V(r) is replaced with Vi(r), the function S\(k) given before corresponds to system (5). For instance, if m = 1 («i = K, p\ = p), then the formula of transition from V(r) to V\{r) has the form Vl{r) =

dP V{r)-2—,

where P(r) = pip(r;iK,)2/(l+p

ip(s;iK)2 ds)

We note also that in this transformation of the potential V(r), the function xjj changes according to the formula •>Pi(r;k) = ip(r;k) + (k2 + / t 2 ) - 1 P ( r ) ^ ( r ; ZK) — Mr; k)/tp(r;iK)]. dr It seems that this result has not been noticed by the authors. By the way, it remains true also for complex «, as well as after the special initial conditions in (5) changed to the general initial conditions ^(0; k) = CQ, ij)'(Q; k) = c\. 5. It remains to observe how general is the class of potentials V(r) that are obtained by the above-described procedures. One of the results in the paper [6] allows us to assert that this class contains every function that satisfies the following two conditions: I. rV{r) e Xi(0,oo). II. The solution (p of the system 0) is the number of negative eigenvalues of system (5). REFERENCES 1. 2. 3. 4. 5. 6.

M. G. Krein, Dokl. Akad. Nauk SSSR 9 7 (1954), no. 1. N. I. Akhieser, Lectures on approximation theory, 1947. L. A. Zadeh and J. R. Ragazzini, J. Appl. Phys. 2 1 (1950), no. 1. M. G. Krein, Dokl. Akad. Nauk SSSR 9 4 (1954), no. 1. R. Jost and W. Kohn, Kg. Danske Videnskab Selsab, Mat.-fys. Medd. 2 7 (1953), no. 9. , Phys. Rev. 8 7 (1952), no. 6. ODESSA HYDROTECHNICAL INSTITUTE

208 CONTINUAL ANALOGUES

OF

P R O P O S I T I O N S O N POLYNOMIALS ORTHOGONAL O N T H E U N I T CIRCLE

M. G.

1. Let (i > 0, and let H(t) = H(-t) in every interval (—r,r) (r
KREIN

( - T < t < T; T < oo) be a function integrable

If for each continuous function 0 (0
da{\) o c l + A2

^

i»-"<" 4 =/,'^-^ T + ( " - H ' +

(2)

1 + A2

, do-(X) ;' A2

( - T < t < T), where 7 is a reo^ constant. This proposition, in a slightly different formulation, was established by the author back in 1944 [1, Theorem 6]. For the sake of simplicity, in what follows, only the case where \x = 1, T = 00 will be considered. In this case the spectral function a in (2) is determined uniquely. If, in addition, it is assumed that the equality in (1) is possible only for ip = 0, then condition (1) (for fi = 1, T = 00) means that, for any positive r, the Hermitian kernel H(t — s) (0
(0 < s, t < r).

According to the general formula (4) in [2], we have (3)

dTr(t, s)/dr = -V(r, s)Tr(t,r)

(0 < r < 00; 0 < s, t < r).

It is evident also that (4)

rr(t,s)=rr(r-s,r-t). Doklady Akademii Nauk SSSR, vol. 105 (1955), no. 4, pp. 637-640.

209 2. We set (0 < r < oo) (5) (6)

P(r; A) = eiXr (1 - / Tr(s, 0)e~iXs ds P*(r;A) = l -

rr(0,s)eiAsds.

/ Jo

With the help of (3) and (4) we easily obtain {A(r) = Tr(0,r)) dP(r; X)/dr = iXP(r; A) - A(r)P.(r; A),

(7)

dP*(r;X)/dr =

-A(r)P(r;X).

From this, without any difficulty, we get the relation | P * ( r ; A ) | 2 - | P ( r ; A ) | 2 = 2ImA f

(8)

\P(r;X)\2 ds

(0 < r < oo),

Jo

which obviously implies that for any r > 0 all the zeros of the entire function P(r; A) lie inside the upper half-plane Im A > 0. It turns out that for any finitely supported (terminating) function f(r) G £2(0, oo) we have /•OO

/

/•OO

\f(r)\2dr=

\F(X)\2da(X),

JO

J -00

where

/•oo

F(X)=

f(r)P(r;X)dr. Jo Thus, the correspondence / ( r ) —> F(X) generates a unitary mapping Up of the entire space £2(0) 00) onto a part of L£ • From an old result of the author [3] follows Theorem 1. The mapping Up is a unitary mapping of the entire 1/2(0, 00) onto the entire L^ if and orih) tf the integral

/ J — 00

loga'(X)dx + A2

T x

diverges (equals —00). By means of other techniques, the following theorem can be established. Theorem 2. The following assertions are equivalent: I. Integral (9) has a finite value. II. The integral JQ \P(r; A)|2 dr has a finite value for at least one A (Im A > 0). III. The function P*(r; A) (0 < r < 00) is bounded for at least one X (Im A > 0). IV (V). On every bounded closed set of points A of the open half-plane Im A > 0 integral (10) is uniformly convergent (there exists a uniformly convergent limit) (10)

n ( A ) = lim P*(r;A)). r—>oo

210 We note that the following relations also hold under conditions I-V: POO

n(A)n(/x) = (A - p) /

(11>

P(r; X)P(r;(i)dr

(Im A > 0, I m p > 0);

^HMk:^w-+<^4

where a € (—oo, oo) and (3 > 0. Formula (11) is interesting in that it shows that the function 11(A) does not depend on the functions CT,J(A) and as(X) in the decomposiotion a = aa + ad + <JS of a into its absolutely continuous part aa, jump function <Jd, and singular part as. According to (11), at every point A at which the derivative cr'a(X) exists, we have 27rcra(A) =lim|II(A + i e ) r 2

as e | 0.

We note also that conditions I-V are satisfied if A{r) £ 1/2(0, oo) or A(r) € Li(0, oo), and in this case we also have />oo

n(A) = l - / Jo

(i2)

A(s)P(s; A) ds

(ImA>0),

nW-«,(-jf 4 . ) ^ * = P,(l;r)+o(

f

\A(r)\dr\

(ImA > 0, I -» oo).

Therefore, for A{r) € Li(0,oo) the function 11(A) (ImA > 0) is continuous and upper and lower bounded: (13)

^(A)^1 <expf /

\A(r)\dr)

(ImA>0).

Moreover, in this case the function cr(X) is continuously differentiable and 2na'(X) = |II(A)|- 2 (-oo < A < o o ) . The propositions and relations given in subsections 2 and 3 are continual analogues of the results obtained for orthogonal polynomials by Szego [4], Kolmogorov, Geronimus [5, 6], and others. In particular, compare Theorem 2 with Theorem 21.1 in [5], and formulas (12) and (13) with Geronimus' formulas (26.7) and (26.6) in [5]. Also the original work by Kalafati [7] has led us to studying the functions P(r; A). 3. We set S(r; A) = e-iXrP{2r;

A) = $(r; A) + M(r; A),

£(-r; X) = 5(r; A) = $(r; A) - M(r\ A). Using the first equation (7) we easily get d$/dr = - A * - a(r)§ + b(r)V, ^

d$/dr=

A$ + 6(r)$ + a(r)tf,

$(0;A) = 1, tf(0;A)=0,

where a(r) — 2ReA(2r) and b(r) = 21mA(2r) are locally integrable functions.

211 Investigation of differential system (14) allows us to assert that the correspondence oo

/

f(r)£(r;X)dr, -oo

defined on compactly supported functions f(r) £ Li{—00,00), generates a unitary operator Us that always maps the entire L,2(—00,00) onto the entire L2 . The functions £(r; A), $(r; A), and $(r; A) are similar to the functions exp(iAr), cosAr, and sinAr, respectively. Naturally, this property of the generalized Fourier transform has a consequence that if !F(X) = Usf and .T^A) (—00 < A < 00) is a compactly supported function in 4 , then f(r) is the integral of Jr(X)£(x; A) da(X) from —00 to 00. Here we have started with the kernel H(t — s) (or the spectral function u(A) generating it) and arrived at system (14). We could proceed vice versa: take arbitrary locally integrable measurable functions a(r) and b(r) and prove directly the existence of a spectral function cr(A) for which the closure of transformation (15) (as an operator) will give a unitary operator that maps the entire Z/2(—00,00) onto the entire L 2 . Formula (2) and the relaton a(r) + ib(r) = 2r2 r (0, 2r) solve the problem of recovering system (14) from its spectral function a(X). We note that if the function 27rcr(A) is an integral of a rational function -R(A) = 1 + 0(1/A) as |A| —> 00, then system (14) (that is, functions a(r) and b{r)) is obtained by rational operations, provided the poles and zeros of the function R(X) are determined. Here we have obtained a generalization and at the same time, a simplification of the conclusions of the paper [2]. To understand this better, consider also the case where the function H(t) is real-valued, that is, <J(A) is an odd function. In this case b(r) = 0. Then any of the functions $,\& can be easily eliminated from equations (14). Assume for simplicity that H(t), and therefore the function a{r) = 2r2 r (0, 2r) are absolutely continuous; then it is not difficult to find that ^ ( r ; A) and $(r; A) are definite solutions of second order differential systems $ " - V ( r ) * + XH = 0, *(0;A) = 0, ^'(0;A) = A; $ " - V i ( r ) $ + A 2 $ = 0,

$(0;A) = 1,

2

$'(0; A) + a(0)$(0; A) = 0,

2

where V(r) = a (r) + a'(r), V^r) = a (r) - a'(r) (0 < r < 00). Thus, the function £(r; A) is constructed from the solutions $, ty of different equations. As is easily seen, \I>(r; A) = - A _ 1 $ ( r ; 0 ) [ $ ( r ; A)/$(r; 0)]'. These facts were already established by the author before, [8]. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

M. G. Krein, Dokl. Akad. Nauk SSSR 4 5 (1944), no. 3. , Dokl. Akad. Nauk SSSR 9 7 (1954), no. 1. , Dokl. Akad. Nauk SSSR 46 (1948), no. 8. G. Szego, Orthogonal Polynomials, 1939. Ya. L. Geronimus, Zapiski Kharkov. Mat. Obshch. Ser. 4, 1948, 19. , The theory of orthogonal polynomials, 1950. P. D. Kalafati, Dokl. Akad. Nauk SSSR 105 (1955), no. 4. M. G. Krein, Dokl. Akad. Nauk SSSR 9 4 (1954), no. 1. ODESSA HYDROTECHNICAL INSTITUTE

ANALYTIC PROBLEMS AND RESULTS IN THE THEORY OF LINEAR OPERATORS IN HILBERT SPACE (Abstract)

M. G. KREIN In the last 25 years the influence of the theory of linear operators in Hilbert space has been greatly extended, partly because of its various applications to many wide-ranging branches of mathematics (the theory of differential equations, group representation, probability, harmonic analysis and so forth), and partly because of the exceptional role that this theory has played in contempotary quantum p h y s i c s . In the same period the theory has been enriched by the creation of basic new s u b d i v i s i o n s . Among them let us mention the following. I. Theory of canonical r e p r e s e n t a t i o n s of Hermitian operators Here we particularly have in mind the new results and problems that have arisen in dealing with a question which had seemed to be very simple and to have received a definitive answer, namely the extension of Hermitian operators. The theory of generalized resolvents and of resolvent matrices has been worked out. A truly remarkable c l a s s of operators, namely the entire Hermitian operatots, has been discovered and studied. Representations of Hermitian operators by means of canonical differential operators have been obtained. Connected with these results are the extensive studies on the so-called inverse problems of the spectral theory of differential operators. In many of these studies the final goal was reached by the methods and ideas of the abstract theory of operators. The theory of entire Hermitian operators and their canonical representations has provided an answer to several difficult questions in the theoty of interpolation and extrapolation of stationary stochastic p r o c e s s e s . The theory of canonical representation of Hermitian operators throws new light on the theoty of generalized eigenelements of selfadjoint operators with its numerous results in a n a l y s i s . An important role in the c l a s s i f i c a t i o n of entire Hermitian operators has been played by the entire operators corresponding to the indefinite case in the problem of moments and in the problem of the extension of positive Hermitian functions. Amer. Math. Soc. Transl. (2) 70 (1968), pp. 68-72. 68

LINEAR OPERATORS IN HILBERT SPACE

69

In the theory of entire H e r m i t i a n o p e r a t o r s , and e s p e c i a l l y in the g e n e r a l theory of canonical representations of Hermitian and selfadjoint operators, certain fundamental questions have remained unanswered up to the present time. It turned out very unexpectedly that these questions can be reformulated as problems in the theory of nonselfadjoint operators. II. Theory of perturbations of operators Even in the theory of perturbations of operators of the simplest c l a s s , namely the selfadjoint operators, mathematicians and physicists have discovered much that is new and sometimes paradoxical from the point of view of linear algebra (usually these paradoxes arise from the possibility of the existence of an absolutely continuous spectrum for a selfadjoint operator, a phenomenon that has no analog in linear algebra). For scattering problems in quantum m e c h a n i c s , the theory of quantized fields, and quantum s t a t i s t i c s , p h y s i c i s t s have developed a very extensive formalism in perturbation theory. By now this formalism, at least in many c a s e s , has been put on a rigorous b a s i s by mathematicians and has been greatly extended. A general and rigorous theory has been constructed for wave operators, and for the operators and suboperators of scattering in the quantum problem of one-channel scattering; and a number of exact results have been obtained in the many-channel scattering problem, in various questions of second quantization, etc.

New mathematical interrelations have been discovered

between scattering theory and the theory of stationary stochastic p r o c e s s e s , and between the theory of dilatations and the characteristic functions of contraction operators. Studies have been made to generalize these results to the case of nonselfadjoint perturbations of selfadjoint operators. In the present lecture no d i s c u s s i o n will be given of perturbations that can be treated in the broader framework of the general theory of operators in Banach s p a c e s

( a l t h o u g h here also many significant advances have been

made ). III. Theory of nonselfadjoint operators. For completely continuous operators and for unbounded operators with discrete spectrum, methods have been worked out to establish theorems on the completeness of the system of eigenvectors and a s s o c i a t e d vectors, to obtain generalized s p e c t r a l e x p a n s i o n s , and to sum these expansions. The analogous questions have been worked out for s h e a v e s of polynomial operators in certain c l a s s e s . The theory of section III here has been developed in close contact with the theory of section II. Satisfactorily complete results are

70

M. G. KREIN

obtainable only for nonselfadjoint operators that are 'close in some sense to selfadjoint, unitary or more generally normal operators. Powerful criteria have been established for the e x i s t e n c e , for a given linear operator, of maximal invariant s u b s p a c c s corresponding to various parts of its spectrum. I'rom amongst these investigations it will be desirable to single out the following theory. IV. The theory of operators in s p a c e s with an indefinite metric The geometry of t h e s e spaces-, in particular of the so-called / - s p a c e s , has been highly developed. Various theorems have been established on the existence of invariant maximal definite subspaccs for operators of various c l a s s e s . It is customary to emphasize the role of set-theoretic and algebraic ideas in the creation of functional a n a l y s i s . But in the theory of operators at the present time increasing use is made of ideas, methods and apparatus from the theory of analytic functions of one or several complex variables. Many important objects in the theory of operators have by definition a " h y b r i d " character; the analytic

functions with operator value's (generalized resolvent,

resolvent matrix, characteristic operator-function, scattering suboperator and s o forth). A discussion of the role of t h e s e objects and of the interrelations among them will be an important chapter in the present lecture. Spectral expansions for /-selfadjoint and /-unitary operators have been obtained under certain r e s t r i c t i o n s . Many results have also been obtained in the theory of representations of groups and algebras in s p a c e s with an indefinite metric. The role of this theory in various questions of mathematical physics has also been clarified (the theory of stability, of damped oscillations, of canonical transformations, of the Hamiltonians of quantized fields and so forth). Since we will confine our attention here to the above sections I—IV, we must emphasize that one of the most important achievements in the theory of operators in Hilbert space has been the creation in the last few years of the theory of Banach algebras of operators (acting in a Banach s p a c e ) together with its numerous applications in the theory of representations (of groups and algebras) and in quantum p h y s i c s . This subject will be dealt with in two s p e c i a l lectures and therefore will not be discussed here. As mentioned above, in the theory of perturbations of selfadjoint operators we encounter facts that have no analog in linear algebra. On the other hand, it was noticed only recently that for certain well-known constructions in linear algebra no continual analogs have been found. Nevertheless

71

LINEAR OPERATORS IN HILBERT SPACE

development of the theory of such analogs is altogether necessary and opens up many new points of view. W,e have in mind the following constructions. A. Reduction to Jacobian form of a real symmetric matrix with simple spectrum by means of orthogonal transformations. D. Reduction to triangular form of an arbitrary matrix by a unitary transformation. C. Reduction of a Hermitian form to a sum of squares by the method of Lagrange (or the more general problem of representing a matrix as the product of triangular matrices). The development of continual analogs for the construction in A leads to the theory of canonical representations of Hermitian operators and to the theory of entire Hermitian operators. The development of continual analogs for the construction in B leads to the theory of an abstract

triangular

integral.

Closely connected with this

integral is an entire calculus that plays an important role in the theory both of nonselfadjoint and of selfadjoint operators (the latter fact was particularly surprising). The investigations of an abstract triangular integral allow us in their turn to work out the continual analog of construction C, namely the theory of factorization of an operator with r e s p e c t to a given chain of projectors. The connecring link in the development of continual analogs for the - constructions A, B, C is the basic concept of the characteristic

operator-function

of a non-

selfadjoint operator. This function c h a r a c t e r i z e s the difference between an operator and its adjoint operator. Its v a l u e s are contraction operators, generally with respect to an indefinite metric. This concept is still being developed. An important role in the theory of operators is played by the question of all the possible multiplicative properties of the characteristic operaror-function. A short time ago a continual multiplicative representation of the characteristic operator-function was obtainable only on the basis of certain preliminary investigations that were difficult and of a purely analytic

nature,

from the theory of analytic matrix-functions. At the present time, on the other hand, many of the results of the theory of matrix-functions are obtained as corollaries of various general s t a t e m e n t s in the theory of operators. Tliis competition between the two fields is still continuing. An important role in this direction is played by the construction of a theory of the abstract triangular integral and of the factorization of an operator with respect to a given chain of projectors.

72

M. G. KREIN The characteristic operator-function plays an exceptional role in the

construction of continual analogs to the theory of elementary d i v i s o r s . The concept of a Jordan cell in linear algebra corresponds to the concept of a unicellular

operator. For operators belonging to certain important c l a s s e s

criteria for their unicellularity have been set up. For operators of certain c l a s s e s we have also been able to prove the possibility of their expansion into a "skew s u m " of unicellular operators. Several criteria have been set up for similarity of operators. Various criteria have also been found for the similarity of a contraction (dissipative operators) to a unitary (selfadjoint) operator. Some of t h e s e criteria are quite remarkable. Translated by: S. H. Gould

217

N O T E S B Y I. G O H B E R G I. Gohberg, who compiled this section of the book, makes the following comments. 1. The first paper is taken from Integral Equations and Operator Theory, Vol. 30, 1998. Four consecutive issues of IEOT, Vol. 30, No. 4, volume 31 issues 1 to 3, are dedicated to the memory of M.G. Krein on the occasion of the 90th anniversary of his birth. 2. The paper by B. Ya. Levin is a translation into English of the paper published in the Ukrainian Mathematical Journal, Vol. 46, 1994, No. 3. The first three issues of this volume are dedicated to the memory of M.G. Krein. They contain different papers about M.G. Krein and about his mathematical work. 3. The paper "M.G. Krein, Abstract" is published for the first time. It was written originally by M.G. Krein in Russian and it contained a description of a number of papers of M.G. Krein proposed by Moscow University and Moscow Mathematical Society for the Lenin Prize in 1963. The prize was not awarded to M.G. Krein. In his later years M.G. became allergic to writing this type of descriptions. He knew that it was futile and no one could convince him to do so. This paper which is published here in translation, is from the private archive of I. Gohberg. It is translated into English by Yu. Eidelman. 4. Paper number 8 in the list is an abstract of the plenary talk presented by M.G. Krein in 1966 to the International Congress of Mathematicians in Moscow. A more complete version of this talk appears later in the American Mathematical Society Translations, (2) 90, 1970, pp. 181-209. 5. Papers 4 to 7 are translations of original papers of M.G. Krein. 6. A complete list of publications of M.G. Krein can be found in Operator Theory: Advances and Applications, Vol. 118, Operator Theory and Related Topics, Birkhauser Verlag, 2000. 7. The books Operator Theory: Advances and Applications, Vols. 117 and 118, Birkhauser Verlag, 2000, are Proceedings of the Mark Krein International Conference on Operator Theory and Applications, Odessa, Ukraine, August 18-22, 1997. These books contain a wealth of material on the life and work of M.G. Krein.

Wolf Prize in Mathematics, Vol. 2 (pp. 219-262) eds. S. S. Chern and F. Hirzebruch © 2001 World Scientific Publishing Co.

Curriculum Vitae Born Budapest, (Hungary), 1 May 1926 Education 1947 New York University, AB 1949 New York University, Ph.D.

Positions Held 1945-46 Los Alamos Scientific Laboratory, Manhattan Project 1950 Staff Member, Los Alamos Scientific Laboratory 1951 Assistant Professor, New York University 1958 Fulbright Lecturer in Germany 1958- Professor, New York University 1972-80 Director, Courant Institute of Mathematical Sciences, New York University

Honors and Awards 1966, 1973 Lester R. Ford 1969 von Neumann Lecturer, S.I.A.M. 1972 Hermann Weyl Lecturer 1973 Hedrick Lecturer 1974 Chauvenet Prize, Mathematical Association of America 1975 Norbert Wiener Prize, American Mathematical Society and Society of Industrial and Applied Mathematics 1982 Member, National Academy of Sciences of the U.S.A. Member, American Academy of Arts and Sciences Honorary Life Member, New York Academy of Sciences 1983 Foreign Associate, French Academy of Sciences National Academy of Sciences, Award in Applied Mathematics and Numerical Sciences 1986 National Medal of Science 1987 Wolf Prize 1989 Member, Soviet Academy of Sciences 1992 Steele Prize 1993 Member, Hungarian Academy of Sciences Member, Academia Sinica, Beijing 1995 Distinguished Teaching Award, New York University Member, Moscow Mathematical Society

220 Honorary Doctoral Degrees 1975 Kent State University 1979 University of Paris 1988 Technical University of Aachen 1990 Heriot-Watt University 1992 Tel Aviv University 1993 University of Maryland, Baltimore Brown University Beijing University 2000 Texas A&M University

Professional Societies 1966-67 Board of Governors, Mathematical Association of America 1986-87 New York Academy of Sciences 1969-71 Member, Society of Industrial and Applied Mathematics Vice President, American Mathematical Society 1977-80 President, American Mathematical Society

Government Service 1977 President's Committee on the National Medal of Science 1980-86 National Science Board DOE Related: Theory Division, Advisory Committee, LANL Senior Fellow, Los Alamos Scientific Laboratory Review Committee, Oak Ridge National Laboratory

221

Short Mathematical Essays

In these pages I describe some of my mathematical investigations that did not fit into Peter Sarnak's (overly generous) article. 1. Numerical Solution of Linear Partial Differential Equations I became interested in these problems during my visits to the Los Alamos Laboratory. Under the leadership of von Neumann, the numerical solution of the equations of continuum mechanics became a cornerstone for weapons design, and for much more. So did the planning, construction and acquisition of electronic computers of ever increasing speed and memory, as well as programming aids, such as flow charts, subroutines, etc. Truly, Los Alamos was the birthplace of computational fluid dynamics. Here is an abstract formulation of a typical numerical problem: Let ut

Gu,

u{o) = u{o)

be a hyperbolic system of linear equations, G a first order operator. The initial value problem is properly posed for initial values uW in L 2 . Let «("+!) = C f c u < n \

u (o) given,

be a difference approximation, where u^ is an approximation to the G solution of the differential equation at time tn and h = tn+\ — tn. The difference equation has to be consistent with the differential equation, which means that, in the sense of the energy norm, 'Cfc-I - G lim = 0 /i-s-0

for all u in a dense subset of the domain of G. Solving the difference equations recursively gives

uW = cjy°). Equivalence theorem. Suppose the difference equation is consistent with the differential equation. Then tends to the solution u(t) of the differential equation as nh tends to t, for all L2 initial data, iff the scheme is stable in the sense that ||C£|| < const, for nh < 1. The proof, see [15],* is simple; the result is very general and has been a useful guide for those who work with difference equations. The necessity of stability is merely an instance of the principle of uniform boundedness. I included it in [15] because I heard someone claim in a lecture that the trouble with unstable schemes is that they magnify roundoff errors beyond all bounds; 'Number within brackets refers to that in the List of Publications (pp. 240-251).

222 if the arithmetic operations could be carried out exactly, unstable schemes would converge! The verification of stability is not an easy task. Friedrichs has made the important observation that if Ch is a difference operator with positive coefficients: (Chu)k — 22 BjUk-j , j

where uk denotes the value of the approximate solution at the fcth lattice point, k an integer vector, and the coefficients Bj are non-negative symmetric matrices, depending Lipschitz continuously on x, and EBj = I, then ||C fc || < l + 0 ( / i ) . This implies stability. The stability of schemes that are not positive is more difficult; when the coefficients Bj of Ch are constant, then expansion into Fourier series yields

v(0 = B(Qu(Q, where « ( 0 = S«*e*fc€,

« ( 0 = Sufceifc« ,

vk = (Chu)k,

and

£(£) = E B , e ^ .

Here stability means that the powers of B(£), called the amplification matrix, are bounded. This is the case when the eigenvalues of £>(£) do not exceed 1 (apart from some interesting complications when there are multiple eigenvalues). For schemes with variable coefficients, von Neumann suggested the power boundedness of the amplification matrix B(x, £) at each point a; as a criterion of stability. I have shown in [26] that von Neumann's criterion (with some provisos) is indeed sufficient for stability. Burt Wendroff and I devised and analyzed a difference scheme that is of second order accuracy see [23]. This scheme has been used successfully for a wide variety of initial value problems. Perhaps of greater interest than the linear case is the numerical solution of hyperbolic systems of conservation laws, discussed in the next section. 2. Hyperbolic Systems of Conservation Laws A hyperbolic system of conservation laws is of the form ut + fx + gv + hz = 0, where u is a vector valued function of x, y, z and t, and / , g, and h are vector valued functions of u. This system is hyperbolic if Uu + V 9u + C.K have real eigenvalues and a full set of eigenvectors for all u and all real choices of £, r], £; here the subscript u denotes the gradient with respect to u. The equations of compressible flow can be written as a hyperbolic system of conservation laws, the conservation of mass, moment and energy; so are the equations

223 of magneto-hydrodynamics. It is an ineluctable fact that solutions of such equations develop discontinuities sooner or later, no matter how smooth the prescribed initial values are. These discontinuities are called shocks, and they caused great discomfort to such distinguished 19th century mathematical physicists as Stokes and Airy. The matter was clarified by Riemann, who pointed out that discontinuous solutions make physical sense if we interpret them by integrating the conservation laws over arbitrary smooth domains G and demand that [ udV + [ F-nds = 0, (1) dl Ja JdG where the flux vector F is (/, g, h), and n the outward normal. Today we say that the differential equations should be satisfied in the weak sense, or in the sense of distributions. For the sake of simplicity we shall restrict the discussion to one space variable,

The integral form of the conservation law requires that the rate at which each component of u flows across a discontinuity be the same on both sides. This leads to the Rankine-Hugoniot jump conditions s

=

ti si IL-JL

(2)

for all components, i = 1 , . . . ,n, where s is the speed with which the discontinuity propagates, and ur, ui the states on the right and left side of the discontinuity. The differential equation ut + uux = 0 (3) can be written in many ways as a conservation law, for instance as

Uf+

G" 2 ) = 0 ,

(4)

and also as

These two conservation laws have the same smooth solutions, but a discontinuous solution of one is not a solution of the other, for the jump condition for the two are different. The class of all discontinuous solutions is too large, as may be seen from the following example. The function ' 0 u(x, t) = < x/t 1

for x < 0 for 0 < x < t for t < x

is a continuous solution of Eq. (3) with initial values „ , ( 0 for x < 0 , u(x,0)=:4 K ' 1 1 for 0 < x .

224 On the other hand, the function 0 1

v(x,t)

for x < t/2 . for t/2 < x ,

satisfies the conservation law (4), vt + ( \v2 I = 0 , because its discontinuity propagates with speed s = 1/2, satisfying the RankineHugoniot jump condition (2), with ue = 0, ur = 1, fa = 0, fr = 1/2. The functions u and i> have the same initial values, and they satisfy the same conservation law. We reject v in favor of the continuous solution u. The general principle, a kind of an entropy condition, goes as follows: Denote by \j(u) the eigenvalues of the matrix / „ , arranged as Ai < A2 < • • • < A„ a discontinuity traveling with speed s is admissible of type k, 1 < k < n, if \k-i{ui)

< s < Xk(ue)

and

Afc(ur) < s < Xk+i(ur).

(5)

Such a discontinuity is called a k-shock. These fc-shocks exist unless the fcth characteristic speed A*; is linearly degenerate in the following sense: grad Afc7-fc = 0 . Discontinuities of this type are called contact discontinuities, and they satisfy Afc(w^) =s = A fc (u r ). Yet another important type of waves are centered rarefaction waves, of the form u = u{x/t), corresponding to each k for which the fcth characteristic speed is nondegenerate. Given U£, there is a one-parameter family of states ur(s), s < \k(ue), to which it can be connected by a fc-shock. There is another one-parameter family of states ur(s) to which it can be connected by a centered fc-rarefaction wave, s > Afc(u^). The two families fit together smoothly, twice differentiably. Using these n oneparameter families we can solve the so-called Riemann initial value problem: f uQ { uu

for x < 0 , for x > 0 .

We can connect UQ to n one-parameter families of waves; the resulting n-parameter family includes u°, if u° is close enough to UQ. The resulting solution consists of the two states uo and u°, separated by n waves, shocks, rarefactions and contacts, and n — 1 intermediate states. The Riemann problem has been used by Godunov for numerical approximation, and by Glimm as a building block for proving the existence of solutions to the initial value problem, combined with a deep estimate and an ingenious random choice of parameters. The initial value problem for ut + (^w2) = 0 has been solved by Hopf and Cole as the limit of solutions of the viscous Burger's equation

i4 e) +u< e M, e) =«4x ) .

u(e)(*,0) =«„(*)

(6)

225

as e tends to zero. It is given by the explicit formula u(x,t) = —— ,

(7)

where y = y(x, t) is that value of y which minimizes Uo(y) + ^

^

;

(8)

here Uo(y) is the integral of the initial function uo{y): dllo/dy = uo(y). I have extended this formula to all single conservation laws "t + f(u)x = 0 , where / is a convex function of u, as follows:

u(x,y)=b(^y where b is the inverse function of a(u) = g£, and y = y(x,t) minimizes

U0(y)+tg(^-y

(7)' the value y that

(8)'

where g(u) is the integral of the function b(u): dg/du = b(u). The Hopf-Cole formula has been extended by Joseph in an interesting way to solutions of mixed initial-boundary value problems. When the flux function / is not convex, interesting new considerations enter. The entropy condition A(u+) < s < A(w_) no longer picks out only admissible solutions. Oleinik has derived the additional conditions necessary. Much of the theory outlined above is contained in the two papers [7] and [18]. In another paper [58], I have denned a concept of entropy as follows: Let ut + fx =ut + Aux = 0,

A = fu(u),

(9)

be a hyperbolic system of conservation laws. A convex function U(u) is an entropy for this system if all smooth solutions of (9) satisfy Ut + Fx = 0 ,

(9)'

where F(u) is some function of u. Clearly this is the case iff U and F satisfy the linear first order equation Uufu = Fu . (9)' For n = 1 this is an underdetermined system, for n = 2 a determined system, and for n > 2 overdetermined. Yet for the equations of compressible flow it has a solution. The significance of this concept can be seen when discontinuous solutions are regarded as limits of solutions of the viscous equation

226 as e > 0 tends to zero. Multiply this equation by Uu; we get Ut + Fx=

eUuuxx.

The right-hand side can be written as eUx x

£UXUuuV'X

•

Since U is assumed convex, the second term is negative, so we conclude that Ut + Fx< sUxx , where U and F are functions of u^E\ As e —> 0, u^ tends strongly to u, so U(u^) tends strongly to U(u). Since the limit in the distribution sense of a non-negative distribution is non-negative, we conclude that Ut+Fx<0. (10) For physical systems this expresses the principle of increase of entropy. For 2 x 2 systems one can construct a one-parameter family of entropies in the form of an asymptotic series. The numerical calculation of solutions of systems of conservation laws is a formidable task, calling for pinpointing shocks as they arise, and tracking them thereafter, through collisions with other shocks. Von Neumann had the brilliant idea, as early as 1944, to ignore discontinuities during the calculations, and capture them as very rapid transitions in the numerical approximation. The results that von Neumann obtained in 1944 showed some curious oscillations that von Neumann interpreted as the conversion of kinetic energy into internal energy. I have a different explanation, which will be presented in Sec. 3. In 1949 von Neumann and Richtmyer showed how to eliminate oscillations by introducing an artificial viscosity. My idea, a few years later, was to approximate differential equations in conservation form by discrete equations in conservation form, as follows in one space dimension: At

„,"+!_„»

ff(n)

Uk

/^x^k+l/2

,(n)

C11)

4-1/2)-

Here u™, /J 1 denote approximations to u and / at x = jAx, t = nAt. The numerical flux /fc+i/2 is a function of Uk and Uk+i, or more generally of Uk-i, Wfc, Ufe+i, Uk+2, etc. The numerical flux is required to be consistent with the physical flux, in the sense that when all variables in fk+1/2 a r e equal, the numerical flux equals the physical flux: fk+\/2{u,u,u,u) = f(u). The numerical flux may very well include a viscous term, also in conservation form. In [9] I proposed as numerical flux f

_ f(uk)

/fc+i/2 -

+ /(ufc+l)

g

Ufc+1 ~ "fc A t

2

A^'

called the Lax-Friedrichs flux. The resulting difference scheme is stable if the Courant-Friedrichs-Levy condition Ax/At > |A max | is satisfied, where A max denotes the largest eigenvalue of the matrix A = fu. Even more, it is a positive scheme in the sense explained in Sec. 1.

227 The Lax-Wendroff scheme mimics the Taylor polynomial u ("+!)

=

u

+ At ut + \{At)2uu

+ 0(Ai) 3 .

Using the formulas ut = -fx

= -Aux,

utt = -fxt

= -ftx

= {-Aut)x

= {A2ux)x

= (Afx)x

,

the Lax-Wendroff flux is . /(«*) + / ( « * + i ) Jfc+i/2 = 2

At J{uk+i) Y ( k+1'2' Ax

f(uk) '

It achieves the same order of approximation of u( n+1 > as the second order Taylor polynomial. It is stable if the CFL condition is satisfied. The two-space dimensional analogue of the LW scheme is stable under a condition somewhat more stringent than CFL. Recently, [137] and [141], Xu-Dong Liu and the author have constructed a new type of difference scheme for hyperbolic conservation laws, with numerical flux of the form /fc+1/2 = /fc+Cl/2 + sk+i/2

/fc+1/2 •

(12)

Here / j S ; 2 is some numerical flux of an accurate scheme, such as LW, and f^+i/^ is a numerical flux with a lot of viscosity, such as LF or upwind. The function Sfc+1/2 is a switch, that is nearly zero in a region of smooth flow, and nearly the identity in a region of rapid variation. The idea, going back to Harten and Zwas, is that in a region of rapid variation, the use of a highly accurate scheme is not called for, since the accuracy of such schemes is based on the accuracy of Taylor polynomials. The second idea is that with the right form of a switch the resulting difference scheme, expressing u£ + as a nonlinear function of Uk-j, j ranging over a finite set of indices, can be written as a linear combination of the u^-j with positive coefficients. These schemes work astonishingly well, giving accurate and detailed resolution of complicated two-dimensional flows such as the diffraction of a shock by a wedge, resulting in double Mach reflection, or the obstruction of a shock propagating in a channel that narrows suddenly, see Figs. 1 and 2. Numerical experimentation is the life-blood of the investigation of numerical methods. My first numerical studies of the LF scheme were done in 1952, before the advent of compilers, using von Neumann's MANIAC at the Los Alamos Laboratory, built under the tutelage of Nick Metropolis. It had a huge memory of 1,024 words and used punched cards, like the voters in Palm Beach County. The initial value problem for hyperbolic systems of conservation laws differs from its linear analogue in this important respect: whereas time is reversible for linear equations, it is not reversible for systems of nonlinear conservation laws; the entropy condition (10) points in one time direction. Even more is true: the mapping of initial data u(x, 0) onto data at u(x, t) at time t > 0 is a compact mapping. This means that a lot of information is lost with the passage of time. I derived this result for single conservation laws in one space variable, with convex flux, Luc Tartar in full generality. DiPerna proved compactness for systems of two conservation laws. It would be desirable to quantify this compactness, in terms of entropy or capacity functions, see [84].

228 Density. T = 0.2

0.5

/)))

J 0.5

PWuM

^ 1

1.5 2 2.5 Stepsize dx=dy=l/l20. dt/dx=0.02

3

3.5

Pressure. T = 0.1

0.5

1

1.5 2 2.5 Stepsize d.x=dy= 1/120, dt/dx=0.02

3.5

Fig. 1.

Density. T = 4 1

^

0.8'

"tl>

0.6 • 0.4-

•

0.2 • 0

M

BpsS?\^^^

^^\M>- r~~- ^ •"*J<

I / f r f 'W^^^^^^Sm^^^^^-^i^.^^'-d^

0

0.5

1

/

k (\ $rj***r?&=

%^

'.<

1.5

I

Stepsize dx=dy=1/80, dt/dx=0.15 Pressure, T = 4

1 1.5 2 Stepsize dx=dy=1/80, dt/dx=0.15

Fig. 2.

C- C \\ 1

fcs^ltll 4<&$fe&$h*i&!^hi'-:

'&i

229 The compactness of the set of solutions for positive times shows that it is easier to compute the solutions of systems of nonlinear conservation laws than of linear ones, for there is less to compute. 3. Dispersive Oscillations Take the simplest nonlinear hyperbolic equation ut + uux = 0 u(x, 0) = uo{x) prescribed.

(1)

As stated in the last section this initial value problem develops singularities in finite time, no matter how smooth the initial function, unless UQ{X) is monotonically increasing. Hopf and Cole have shown that the solution u^ of the modified equation u[e) + ii( e ) u W = eu<>>,

«<E>(x, 0) = u0(x),

(2)

tends uniformly as e —>• 0 to the solution of (1) as long as the solution is smooth. For t beyond the smooth region, u^ tends to a discontinuous solution of the conservation law

Ut+ u2

(l )

=

(3)

°'

and the limit satisfies the entropy condition described in the previous section. In [89] and [103] I have studied, with Leverrnore, the behavior of solutions u^ of u<e)We>i4e)+«41=0, u^(x,0)=u0(x). (4) Whereas the added term in Eq. (2) represents viscosity and dissipates energy, the added term in (4) is merely dispersive. What happens when e tends to 0? As long as the corresponding solution of (1) is smooth, v,(£' tends uniformly to that solution. For t beyond the smooth region, u^ becomes oscillatory in certain regions of space-time, on a wavelength proportional to e1'2. These oscillatory solutions converge weakly but not strongly to some limit u; so do their squares: w — lim u^2 = u2 . Taking the limit as e —» 0 in the sense of distribution of the dispersive equation (4) we obtain
=0.

(5)

Since u(E> tends weakly, not strongly, to u, it follows that u2 > u2 in those regions. In other words, u is not a solution in the distribution sense of the conservation law (3). One could express the relation between the weak limit of u and u2 as i 2 = u2 + w2 and rewrite Eq. (5) as '1

ut+[^u2\

+(^

2

)

=0-

(5)'

230 The extra term (^w2)x in Eq. (5)' is reminiscent of the appearance of so-called Reynolds stress in turbulence theory. Whereas Reynolds stress is derived from a stochastic model, as a difference of ensemble averages, the additional term in (5)' arises deterministically, as a difference of weak limits. It turns out that there is a 3 x 3 system of hyperbolic equations in terms of whose solutions one can express u and u 2 , at least for a limited time. Afterwards one has to go to a 5 x 5 system, and so on. The derivation of these remarkable properties of solutions of Eq. (4) is based on the complete integrability of the dispersive equation (4), which yields semi-explicit formulas for its solution. More about this in the next section. Jonathan Goodman and I have in [120] investigated another dispersive approximation of Eq. (1). Instead of adding a dispersive term as in (4), we discretized (1), replacing ux by a centered difference quotient: J2uk + Uk

2A

u

'

fc(°) = u o ( ^ A ) .

(6)

We denote the solution of this system of equations as « A , where UA is defined as constant between lattice point: uA(x,t)

= uk(t)

for ( f c - l / 2 ) A < x < (fc + l / 2 ) A .

Equation (6) is well-posed for positive initial values. Numerical results suggest strongly that the behavior of WA as A —> 0 is entirely analogous to the behavior of u(£) as e -> 0. In the smooth regime UA tends uniformly to the solution of (1). For t beyond the smooth regime, MA becomes oscillatory in space-time on a wavelength proportional to A. These oscillatory solutions converge weakly but not strongly to some limit u. For the model (6) one can give an intuitive argument why oscillations develop. Equation (6) can be written in conservation form: jtuk

+ (/fc+i/2 - A - i / 2 ) / A = 0,

(7)

where /fc+l/2 = -UfcWfc+1 •

(8)

Dividing (6) by uk leads to yet another conservation form: d_ logwfc + (sfc+i/2 - 5 / t - i / 2 ) / A = 0 , ~dt

(7)'

where uk + uk+i

fffc+i/2 =

^

,

'

v

^'

We claim that as A tends to zero, beyond the smooth regime the solutions UA of (7) cannot remain in a compact set of function space. For then there would be a strongly convergent subsequence it^. tending to some limit u, and / A 3 would tend to f(u) as Aj tends to 0. Equation (7) would, tend in the sense of distributions to

ut + fx =

ut+(\u2)x=Q.

231 The same argument applied to Eq. (7)' shows that the strong limit u of the

subsequence is a solution of the conservation law (log u)t+ux

=0.

But these conservation laws are incompatible; their discontinuous solutions satisfy different jump conditions, as explained in Sec. 2. So we conclude that the family UA contains no strongly convergent subsequence. This lack of compactness is the source of the oscillatory behavior of the u&; the approximate solutions are desperately trying to satisfy two or more incompatible conservation laws! It is a curious fact that the discrete system (6) can be written in infinitely many ways as a conservation law. In fact the transformation uk = vk turns (6) into the Kac-Van Moerbeke system, known to be integrable. This creates the possibility of a rigorous proof of the behavior of UA as A —> 0. A similar analysis can be made of dispersive approximations of some systems of conservation laws, such as the equations of one-dimensional gas dynamic: U

t+Px

= 0,

conservation of momentum,

(9)

and Vt — ux = 0,

conservation of mass,

(9)'

where u is velocity, p pressure and V specific volume. The equation of state is simplified to specifying p as a smooth, decreasing function of V. In his 1944 report von Neumann proposed and tried out the following numerical method for solving the initial value problem for the system (9), (9)'. Discretize the differential equations by setting uk(t) as an approximation of u(kA,t), and Pk+i/2(t) a nd V fc+1 / 2 (i) as approximations of p((k + 1/2)A, t) and V((k + 1/2)A, t). Replacing the space derivatives with centered differences turns (9) into |„fc +

ft^-ft-i/*

= 0

(10)

and (9)' into d Jtvk+1/2

uk+1 - uk _

n

£—-o.

(io)

As initial data von Neumann took gas flowing with negative velocity in a tube 0 < x < 1, closed at x = 0, open at x = 1. The subsequent flow contains a shock generated instantaneously at the closed end x = 0, and a rarefaction wave centered at the open end x = 1; eventually they interact. The numerical solution based on the approximations (10) and (10)' agrees, roughly, with the flow described above, with this difference: The velocity field oscillates in a neighborhood of the shock. Von Neumann's explanation of these oscillations was based on the irreversible nature of shock formation: it converts mechanical energy into heat energy, represented statistically by an oscillation in the velocity distribution around its mean. Furthermore, von Neumann believed that as A —> 0 the solutions of (10), (10)' tend weakly, but not strongly, to a solution of the continuum equations (9), (9)'. That, of course, cannot be, because if VA tends weakly but not strongly to V, a nonlinear function of VA like P(VA) may converge weakly, but not to p(V).

232

My explanation of the oscillation is as before: too many conservation laws: For if you multiply Eq. (10) by uk, and subtract from it Eq. (10)' multiplied by Pk+i/2: you get after a cancellation, d

uk-uk

d

-

Pk+l/2Uk+l

Pk+1/2-Vk+1/2

-Pk-l/2,Uk

+

= 0

which is a conservation law of the form j t (K\

+ Pfc+i/a) + (hfc+i/2 - hfc-1/2) / A = 0 ,

(10)"

where P(V) is defined by dP/dV = p(V), and hk+i/2 = Pk+i/2uk+i- If U&,VA were a strongly convergent subsequence, their limit u, V would satisfy the two conservation laws (9), and the additional conservation law \u2 + p)

+(up)x

= 0.

(11)

This third conservation law is incompatible with the conservation laws (9) and (9)', for a discontinuity that satisfies the jump conditions derived from (9) and (9)' will in general fail to satisfy the jump condition for (11). In [129], Tom Hou and I have explored these issues. In particular, as von Neumann has already observed, one can introduce an Eulerian coordinate X so that Xt = u, Xx = V , Eq. (9)' shows that these are compatible. Settings X in (9) gives Xu +p(Xx)x

= 0,

a quasilinear wave equation. One can proceed similarly for the difference equations; here the Euler coordinate Xk satisfies d v -Xk

= uk ,

Xk+\ — Xk T/ ^ = T4+1/2 •

Equation (10)' shows that these are compatible. Setting Xk in (10) gives

£MK^)->(*¥^))A=°-

(12)

Equations (12) can be interpreted as governing the motion of a chain of unit masses located at positions Xk, connected to their nearest neighbors by identical nonlinear springs. There is one spring law, the celebrated Toda chain, p(V) — e~v, which is completely integrable. For this case the limit A -> 0 has been studied by Percy Deift and Ken McLaughlin. It is natural to ponder if too many conservation laws, incompatible for nonsmooth solutions of the incompressible Euler equations, may be responsible for initiating fluid turbulence. When I mention this to turbulence experts, the reaction ranges all the way from polite scepticism to impolite scepticism.

233

4. Completely Integrable System When mathematicians declare that they have solved a problem, very often they mean that they have proved that a solution exists. It is otherwise for completely integrable ordinary and partial differential equation; for these one can describe all solutions in a (quasi) explicit fashion, and extract information about properties of solutions - though not without much hard work. The concept of completely integrable Hamiltonian systems in 2N variables is due to Liouville; the modern examples are all infinite dimensional. Completely integrable systems have played a significant role in the development of science. It was the complete integrability of the equations governing the twobody problem of gravitation that enabled Newton to derive Kepler's three laws of planetary motion. In the early days of quantum theory, integrable systems served as quantisable models. And in 1944, Onsager was able to exploit the complete integrability of the Ising model to demonstrate the existence of a critical temperature where phase change occurs. Other classical completely integrable system are geodesic flow on ellipsoids discovered by Jacobi, the motion under gravity of a particle confined to a sphere, investigated by Carl Neumann, and Sonya Kowalewski's celebrated top. The modern era began with Kruskal and Zabusky's astonishing discovery of the role of solitons for solutions of the Korteweg-deVries equation ut + uux + uxxx = 0 ,

(1)

and the even more astonishing explanation, given by Gardner, Greene, Kruskal and Miura, that the KdV equation is completely integrable, an idea developed further by Faddeev, Zakharov and Takhtajan. My first contribution to the subject was a scheme for the contraction of nonlinear evolution equations with infinitely many conserved quantities. These are of the form L t = [P,L],

(2)

where P and L are operators, [P,L] their commutator. When P is antisymmetric, U t = P U generates a one-parameter family of unitary operators U(£). It follows that the solution L(t) of (2) has the property that U*LU is independent of t. It follows that the eigenvalues of L are conserved quantities. Setting 6

ax 3

u some function of x and t, and P = 4d + ud + \ux, the right side of Eq. (2) becomes multiplication by \\uux + uxxx\; so Eq. (2) is the KdV equation, except for a rescaled time. In case the operator L acts on functions on the whole line as a self-adjoint operator on L 2 (R), and u tends to zero as |a:| —>• oo, L has a finite number of eigenvalues; in the periodic case L has infinitely many eigenvalues. All are invariant under the KdV flow (1). There is a sequence of differential operators P „ , of order 2n + 1, each antisymmetric, and such that its commutator with L is of order zero. For each, (2) yields an equation of the form ut+Kn(u) = 0, (3) K„ a quasilinear operator of order 2n + 1. They constitute the KdV hierarchy.

234 The operators P and L in Eq. (2) are called a Lax pair; they occur in surprisingly many contexts. In [72] and [73] I constructed finite dimensional invariant manifolds for the space periodic KdV equations, characterized as the functions minimizing the functional Fn(u) among all functions for which Fn-i(u),... ,F0(u) have prescribed values. Here Fn is the sequence of functionals, constructed by Gardner, Kruskal and Miura, which are invariant under the KdV flow, each of the form dnu)dx ,

Fn{u) = / pn(u, du,...,

p a polynomial, quadratic in its last argument. The solutions u of this variational problem satisfy an ordinary differential equation of order 2n. As already mentioned in Sec. 3, [103] is a joint study with Levermore of the family of equations ut + uux + euxxx = 0 ,

u(x,0) = u0(x).

(4)

Each is integrable on the whole real axis, and their solution can be expressed in terms of the eigenvalues of the operator L = ed2 + f, as well as the so-called norming constants of the eigenfunctions. The passage to the limit e —»• 0 can be evaluated, thanks to a sequence of miracles. 5. Mathematics Light I am delighted by and look for nontechnical mathematics that yields a surprising result, or opens a new window on a classical topic. Here are some examples: 5.1. The existence of Green's function for the Laplace equation Let G be a domain in the plane with a smooth boundary B. Denote by C the space of continuous functions on B, and by H the boundary values of harmonic functions in G. In [5] I have shown that H is all of C by arguing as follows: Fix some point q in G, and for any f in H define the linear functional iq as the value at q of the harmonic function h in G whose boundary value is / : £q(f) = h(q) . By the maximum principle, |^ q | ma x = 1- So by the Hahn-Banach theorem lq can be extended to all of C as a bounded linear functional. Let p be any point not on the boundary B. Define kp in C as kp(z) = log \z — p\,

z on B .

kp is a harmonic function of p in each component of the complement of B. Therefore 9(p,q) = tq(kp) is a harmonic function of p. For p not in G, kp belongs to H; therefore g(p, q) = %(
235 B; therefore so does g(p, q) - g(p,q). Since p lies outside G, g(p, q) = log \q - p\; therefore as p approaches the boundary point z, g(p,q) tends to \og\q — p\. Thus log \q — p\— g(q,p) is Green's function. 5.2. The factorization

of matrix

valued

functions

The solution of Wiener-Hopf integral equations and the inversion of Toeplitz matrices rely on factoring continuous complex valued functions on, say, the unit circle, whose change of argument is zero, as a product of the boundary values of an analytic and an antianalytic function that are invertible in the unit disk. For scalar functions this is easily accomplished by taking the logarithm of the function and decomposing it additively. This does not work in the matrix valued case. In [76] I described the following procedure. Let F be the matrix valued function on the boundary of the unit disk, such that the change of argument of det F is zero. Let F = AB be the desired factorization, A analytic, B antianalytic. Differentiate with respect to z and ~z: FZ = AZB,

FT = ABZ,

(1)

FZZ=AZBZ.

(1)'

Using (1) to express Az and Bj in (1)' gives &F = FZF~1FZ.

(2)

Here we have used the identity that d2/dzdz- is the Laplace operator. Since det F = det A det B can be obtained by a scalar factorization, we can express F _ 1 in (2) as a polynomial in F divided by d e t F . Equation (2) is a system of n 2 equations whose main term is the Laplace operator, coupled by lower order terms; the value of F is prescribed on the boundary of the unit disk. This Dirichlet problem may or may not have a solution, just as the factoring problem may or may not have a solution, see Gohlerg and Krein on partial indices. If it does have a solution, A and B can be determined by solving the ordinary differential equations AZ = FZF'XA, 5.3. The approximation

B? = BF~lF-z.

of measure preserving

transformations

Numerical experimentation with the iterates of a measure preserving transformation often reveals interesting properties. Because computers use finite digit arithmetic, say k decimals, one could think of the numerical experiment as iterating an approximation of the given transformation that maps squares (cubes, etc.) of edge length 10 -A: into squares etc. of the same size. Such an approximation is itself measurepreserving iff it is a permutation of the squares. Impose on the approximation the requirement that the approximate image of every square have a nonempty intersection with its exact image. In [59] I deduced from the combinatorial Marriage Theorem that a permutation consistent with this requirement exists. Steve Alpern has deduced the same conclusion from the Birkhoff-Konig theorem about doubly stochastic matrices, which says that the extreme points of this set of matrices are the permutation matrices.

236

5.4. Polya's

area

filling

curve

In 1914 Polya constructed a continuous curve that goes through every point of a given right triangle T, required to be not isosceles. In this case the altitude from the right angle divides the triangle into two unequal triangles, each similar to the original one. The construction of the function P(t), 0 < t < 1 goes as follows: Expand t in binary fraction, t =

hb2....

Each binary digit bi is a zero or a one. If &i = 1, choose the larger of the two triangles into which the altitude divides T; if b\ = 0, choose the smaller one. Then repeat the process with whichever triangle was chosen, using the second digit of t to make the decision. Continue this process add infinitum; the result is a nested sequence of triangles T = T\, T%, • • •, Tn,... all similar and whose size tends to zero. Their intersection consists of a single point, defined to be P(i). Some numbers t have two distinct binary expansions, such as .1000... = .011111. It is an amusing fact that using either expansion of such t yields the same point

Pit). The continuity of the function P is now easy to see. For if t\ and t 2 are near each other, then either their first n binary digits are equal, or there is a tm between them, ti < tm < ti, such that t\ and tm have n digits in common in one representation of tm and tm and t2 have n digits in common in the other binary representation of tm. In either case, P(t\) and P(*2) are near each other. That the image of P(t) fills the triangle T is clear; for every point Q of T is the intersection of a nested sequence of triangles of the kind constructed by repeatedly drawing altitudes. Polya's example is by far the simplest of the many Peano type curves; when I taught real variables, I always presented it to my class. One time I assigned as homework: prove that Polya's function is not differentiable. Since nobody handed in the homework, I started doing it myself, and was surprised to find the following: (a) When the smallest angle of T is greater than 30°, P(t) is nowhere differentiable. (b) When the smallest angle of T lies between 30° and 15°, P(t) is differentiable on a set of measure zero. (c) When the smallest angle of T is less than 15°, P(t) is differentiable almost everywhere. The proof is given in [67]. 5.5. Linear

spaces of nonsingular

matrices

Frank Adams had proved that the largest number of real n x n matrices all whose linear combinations are nonsingular equals the Radon-Hurwitz function R(n) defined as follows: let b be the largest power of 2 that divides n; write b as 4d + c, 0 < c < 4. Then R(n) = 2c + 8d.

237 Ralph Phillips and I had been investigating scattering theory for symmetric hyperbolic systems of the form ut

£ ^ , = 0 ,

(3)

where the Aj are real; symmetric n x n matrices. In order to have a scattering theory, all signals have to propagate to infinity. The signal speeds in the direction (£ii • • • i£fc) a r e the eigenvalues of J2i €jAj\ so we required all these to be nonzero. This led naturally to the question, what is the largest number k of real, symmetric n x n matrices, all of whose linear combinations are nonsingular. We submitted a paper on this question to the Proceedings of the AMS; we received a referee's report with so many excellent suggestions for improving our arguments, that we wrote to the editor offering to withdraw our paper in favor of one by the referee. The end result was a joint paper with the referee; thus did we become coauthors with the topologist Frank Adams. The paper is [36]; the maximum value of k is 1 for n odd, and R(n/2) + 1 for n even. 5.6. Multiple eigenvalues Another incursion into topology also came from the study of first order hyperbolic systems of form (3), in three variables: ut + Aux + Buy + Cuz = 0, A, B, C real symmetric n x n matrices. The signal speeds in the direction £, 77, £ are the eigenvalues of £,A + rjB + C,C. The presence of multiple signal speeds has a significant effect on the propagation of singularities, such as conical refraction. Therefore, it is important to know in any given case if there are multiple signal speeds present or not. I proved in [99] that for n = 2(mod 4) there always exist values of £, r], £, £2 + T]2 + £2 = 1, such that £A + r]B + C,C has a multiple eigenvalue. The proof is based on the fundamental group of the space TV of all real, symmetric n x n matrices with distinct eigenvalues Ai < A2 < • • • < A„. Let N(6) be a continuous, closed curve in N, N(2TT) = N(0). The eigenvalues Xj(9) are continuous functions of 6, and the normalized eigenvectors rj{6), \\rj\\ = 1, can be chosen to be continuous. We have then rj(27r) = TJTJ(O) ,

Tj = 1

or

-1.

Clearly the Tj are homotopy invariants of the closed curve. If a closed curve is contractible to a point, all Tj must be 1. Let TV be an odd closed curve, satisfying N(9 + TT) =

-N(0).

Here \j{6 + it) = -A n _ J + i(fl), and rj(6 + n) — (ijrn-j+i(6), rj{2ir) = fijrn-j+i(n) = /j,jfj,n_j+1rj(0), it follows that Tj = The ordered basis ri(0),...,rn(Q)

/J,J = 1 or —1. Since fj,j/j.n_j+1.

238 has the same orientation as ri(7r),...,r n (7r), which is the same as Atir n (0),...,/i n ri(0). So we conclude that when n = 2(mod 4), J}™ fij = —1. Since HjUn-j+i = TJ, it follows that J]™' Tj = — 1 - I n particular, some n = — 1; so it follows that an odd curve is not contractible to a point. N(0) = A cos 6 + B sin 6 is an odd curve. It follows that not all £A + rjB + (C, £ 2 + V2 + C2 = 1) belong to; Af, for then the curve N(0) above could be contracted to a point on the sphere £2 + rj2 + £2 = 1. Friedland, Robbin and Sylvester have shown that a multiple eigenvalue must occur when n ^ 0, 1, — l(mod 8). 5.7. Huygens'

principle

for the spherical

wave

equation

In [82], Phillips and I investigated the spherical wave equation governing the propagation of waves on the unit sphere. This has the form utt - Lw = 0,

(5)

where L = Ag — 1, As the spherical Laplacian on the 3-dimensional unit sphere. As is related to the Euclidean Laplacian A4 in R 4 by

A4 = dl + -dr + ~ .

(6)

r r2 Denote by hn(u>) a spherical harmonic of order n. Then rnhn(u>) is harmonic in R 4 , so it follows from (6) that (n 2 + 2n)hn + Ashn

= 0,

and therefore Lhn — — ( n + l)2hn . Every solution with finite energy of the spherical wave equation (5) can be expanded in terms of the eigenfunctions of L: u{w,t) = J2(*nei(n+1)t

+

bne-^n+1^)hn{uj).

Since hn(w) is even or odd, depending on the parity of n, we deduce that u(—w,t + ir) = —u(Lj,t).

(7)

Now take a solution u of (5) whose initial data U(LJ, 0), ut(u, 0) are supported in a ball of radius e around some point U>Q of S3. It follows from (7) that the initial data of u at time -K are supported in a ball of radius e around the antipodal point — WQ. Take any t between 0 and 7r. Since signals for the wave equation (5) propagate with speed < 1, it follows that the support of u(u>, t) is confined to the ball of radius t + £ around UJO, and therefore, u(u>,t) = 0 at all points LJ whose distance from UJQ exceeds t + e.

239 Applying the same argument to the wave equation (5), starting at time n and going backward by IT — t, we conclude that u(w,t) is zero at all points w whose distance from — u>o exceeds IT — t + e. It follows that the support of u(u>,t) is confined to the annulus consisting of points UJ whose distance from U>Q lies between t — e and t + e. Letting e —>• 0 we conclude that a signal starting at CJQ spreads in time t to the sphere of radius t centered at, U)Q and nowhere else. Replacing the unit sphere with a sphere of radius R and letting R —>• oo yields Huygens' principle in Euclidean space. Curiously, as shown in [82], the Euclidean wave equation can be obtained from the spherical wave equation by a change of variables!

240 List of Publications

1944 [1] Proof of a conjecture of P. Erdos on the derivative of a polynomial, Bull. Amer. Math. Soc. 50, 509-513. 1948 [2] The quotient of exponential polynomials, Duke Math. J. 15, 967-970. 1950 [3] Partial Differential Equations, Lecture Notes, NYU, IMS (1950-51). 1951 [4] A remark on the method of orthogonal projections, Coram. Pure Appl. Math. 4, 457-464. 1952 [5] On the existence of Green's function, Proc. Amer. Math. Soc. 3, 526-531. 1953 [6] Nonlinear hyperbolic equations, Comm. Pure Appl. Math. 6, 231-258. 1954 [7] Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. Pure Appl. Math. 7, 159-194. [8] Symmetrizable linear transformations, Comm. Pure Appl. Math. 7, 633-648. [9] (with A. Milgram) Parabolic equations, Ann. Math. Studies 33 (Princeton) 167-190. [10] The initial value problem for nonlinear hyperbolic equations, Ann. Math. Studies 33 (Princeton) 211-229.

241 1955 [11] Reciprocal extremal problems in function theory, Comm. Pure Appl. Math. 8, 437-454. [12] On Cauchy's problem for hyperbolic equations and the differentiability of solutions of elliptic equations, Comm. Pure Appl. Math. 8, 615-633. [13] (with R. Courant) Cauchy's problem for hyperbolic differential equations, Ann. Mat. Pura Appl. 40, 161-166. 1956 [14] A stability theorem for solutions of abstract differential equations, and its application to the study of local behavior of solutions of elliptic equations, Comm. Pure Appl. Math. 9, 747-766. [15] (with R. D. Richtmyer) Survey of the stability of linear finite difference equations, Comm. Pure Appl. Math. 9, 267-293. [16] (with R. Courant) The propagation of discontinuities in wave motion, Proc. Nat. Acad. Sci. 42, 872-876. 1957 [17] A Phragmen-Lindelof theorem in harmonic analysis and its application to some questions in the theory of elliptic equations, Comm. Pure Appl. Math. 10, 361-389. [18] Hyperbolic systems of conservation laws, II, Comm. Pure Appl. Math. 10, 537-566. [19] Remarks on the preceding paper, Comm. Pure Appl. Math. 10, 617-622. [20] Asymptotic solutions of oscillatory initial value problems, Duke Math. J. 24, 627-646. 1958 [21] Differential equations, difference equations and matrix theory, Comm. Pure Appl. Math. 11, 175-194. 1959 [22] Translation invariant spaces, Acta Math. 101, 163-178. 1960 [23] (with B. Wendroff) Systems of conservation laws, Comm. Pure Appl. Math. 13, 217-237.

242 [24] (with R. S. Phillips) Local boundary conditions for dissipative symmetric linear differential operators, Coram. Pure Appl. Math. 13, 427-455. 1961 [25] Translation invariant spaces, Proc. Int. Symp. on Linear Spaces, Israeli Acad. of Sciences and Humanities, Jerusalem (1960) (Pergamon), 299-307. [26] On the stability of difference approximations to solutions of hyperbolic equations with variable coefficients, Comm. Pure Appl. Math. 14, 497-520. 1962 [27] (with R. S. Phillips) The wave equation in exterior domains, Bull. Math. Soc. 68, 47-79.

Amer.

[28] A procedure for obtaining upper bounds for the eigenvalues of a Hermitian symmetric operator, Studies in Mathematical Analysis and Related Topics (Stanford Univ. Press), 199-201. 1963 [29] On the regularity of spectral densities, Teoriia Veroiatnosteii i ee Prim. 8, 337-340. [30] An inequality for functions of exponential type, Comm. Pure Appl. Math. 16, 241-246. [31] (with C. Morawetz and R. Phillips) Exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle, Comm. Pure Appl. Math. 16, 477-486. [32] Survey of stability of difference schemes for solving initial value problems for hyperbolic equations, Symp. Appl. Math. 15, 251-258. 1964 [33] Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys. 5, 611-613. [34] (with B. Wendroff) Difference schemes for hyperbolic equations with high order of accuracy, Comm. Pure Appl. Math. 17, 381-398. [35] (with R. S. Phillips) Scattering theory, Amer. Math. Soc. Bull. 70, 130-142. 1965 [36] (with J. F. Adams and R. S. Phillips) On matrices whose real linear combinations are nonsingular, Proc. AMS 16, 318-322; correction, ibid. 17 (1966) 945-947.

243 [37] Numerical solution of partial differential equations, Amer. Math. Monthly 72 II, 74-84. [38] (with K. O. Priedrichs) Boundary value problems for first order operators, Comm. Pure Appl. Math. 18, 355-388. 1966 [39] (with R. S. Phillips) Analytic properties of the Schrodinger scattering matrix, in Perturbation Theory and its Application in Quantum Mechanics, P r o c , Madison, 1965 (John Wiley & Sons), 243-253. [40] Scattering theory; remarks on the energy theory, and on scattering theory for a geometrical optics model, in Proc. Conf. on Dispersion Theory, Cambridge, MA, 1966, M 11, pp. 36-39, 40-42. [41] (with L. Nirenberg) On stability for difference schemes; a sharp form of Garding's inequality, Comm. Pure Appl. Math. 19, 473-492. [42] (with J. P. Auffray) Aspects mathematiques de la mecanique de phase, Acad. Sci. Paris, Compt. Rend. (B) 263, 1355-1357. 1967 [43] (with J. Glimm) Decay of solutions of systems of hyperbolic conservation laws, Bull. AMS 73, 105. [44] Hyperbolic difference equations: a review of the Courant-Friedrichs-Lewy paper in the light of recent developments, IBM J. Res. Develop. 11(2), 235-238. [45] (with R. S. Phillips) Scattering Theory (Academic Press). [46] (with R. S. Phillips) The acoustic equation with an indefinite energy form and the Schrodinger equation, J. Func. Anal. 1, 37-83. [47] (with R. S. Phillips) Scattering theory for transport phenomena, in Functional Analysis, Proc. of Conf. at Univ. of Calif., Irvine, 1966, ed. Gelbaum (Thompson Book Co.), 119-130. [48] (with K. O. Friedrichs) On symmetrizable differential operators, Symp. on Pure Math., Singular Integrals, AMS 10, 128-137. 1968 [49] Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21, 467-490. 1969 [50] Nonlinear partial differential equations and computing, SIAM Rev. 11, 7-19.

244 (with R. S. Phillips) Decaying modes for the wave equation in the exterior of an obstacle, Comm. Pure Appl. Math. 22, 737-787. Toeplitz operators, in Lecture Series in Differential Equations 2, ed. A. K. Aziz (Van Nostrand), 257-282. 1970 (with J. Glimm) Decay of solutions of systems of nonlinear hyperbolic conservation laws, Memoirs of the AMS 101. (with R. S. Phillips) The Paley-Wiener theorem for the Radon transform, Comm. Pure Appl. Math. 23, 409-424; Errata ibid 24 (1971) 279. 1971 Nonlinear partial differential equations of evolution, Proc. Int. Conf. of Mathematicians, Nice, September 1970 (Gauthier-Villars), 831-840. (with R. S. Phillips) Scattering theory, Rocky Mountain J. Math. 1, 173-223. (with H. Brezis, W. Rosenkrantz and B. Singer) On a degenerate ellipticparabolic equation occurring in the theory of probability, Comm. Pure Appl. Math. 24, 410-415; appendix by P. D. Lax. Shock waves and entropy, Contributions to Functional Analysis, ed. E. H. Zarantonello (Academic Press), 603-634. Approximation of measure preserving transformations, Comm. Pure Appl. Math. 24, 133-135. (with K. O. Friedrichs) Systems of conservation equations with a convex extension, Proc. Natl. Acad. Sci. 68, 1686-1688. (with R. S. Phillips) A logarithmic bound on the location of the poles of the scattering matrix, Arch. Rational Mech. Anal. 40, 268-280. 1972 (with R. S. Phillips) On the scattering frequencies of the Laplace operator for exterior domains, Comm. Pure Appl. Math. 25, 85-101. The formation and decay of shock waves, Amer. Math. Monthly 79, 227-241. (with R. S. Phillips) Scattering theory for the acoustic equation in an even number of space dimensions, Ind. U. Math. J. 22, 101-134. [65] Exponential modes of the linear Boltzmann equation, in The Boltzmann Equation, ed. F. A. Grunbaum, NYU, CIMS, 111-123.

245 1973 [66] Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Conf. Board of the Mathematical Sciences, Regional Conf. Series in Appl. Math. {SIAM), 11. The differentiability of Polya's function, Adv. Math. 10, 456-465. (with R. S. Phillips) Scattering theory for dissipative hyperbolic systems, J. Func. Anal. 14, 172-235. 1974 Invariant functionals of nonlinear equations of evolution, in Nonlinear Waves, eds. S. Leibovich and R. Seebass (Cornell Univ. Press), 291-310. Applied mathematics and computing, Symp. on Appl. Math. (AMS) 20, 57-66. Periodic solutions of the KdV equations, in Nonlinear Wave Motion, ed. A. C. Newell, Lectures in Appl. Math. 15, 85-96. 1975 Periodic solutions of the KdV equation, Comm. Pure Appl. Math. 28, 141-188. Almost periodic behavior of nonlinear waves, Adv. Math. 16, 368-379. 1976 Almost periodic solutions of the KdV equation, SIAM Rev. 18, 351-375. (with A. Harten, J. M. Hyman and B. Keyfitz) On finite-difference approximations and entropy conditions for shocks, Comm. Pure Appl. Math. 29, 297-322. On the factorization of matrix-valued functions, Comm. Pure Appl. Math. 29, 683-688. (with R. S. Phillips) Scattering Theory for Automorphic Functions, Ann. Math. Studies 87 (Princeton Univ. Press and Univ. of Tokyo Press). (with S. Burstein and A. Lax) Calculus with Applications and Computing, Undergrad Texts in Math. 1 (Springer). 1977 [79] The bomb, Sputnik, computers and European mathematicians, the Bicentennial Tribute to American Mathematics, San Antonio, 1976 (Math. Assoc, of America), 129-135.

246 [80] (with R. S. Phillips) The scattering of sound waves by an obstacle, Comm. Pure Appl. Math. 30, 195-233. 1978 [81] (with M. S. Mock) The computation of discontinuous solutions of linear hyperbolic equations, Comm. Pure Appl. Math. 31, 423-430. [82] (with R. S. Phillips) An example of Huygens' principle, Comm. Pure Appl. Math. 3 1 , 415-421. [83] (with R. S. Phillips) The time delay operator and a related trace formula, Topics in Functional Analysis, Adv. Math. Suppl. Studies, 3, eds. I. T. Gohberg and M. Kac (Academic Press), 197-215. [84] Accuracy and resolution in the computation of solutions of linear and nonlinear equations, Recent Adv. in Numer. Anal., Proc. of Symp., Madison (Academic Press), 107-117. [85] (with A. Lax) On sums of squares, Linear Algebra Its Appl. 20, 71-75. [86] (with R. S. Phillips) Scattering theory for domains with nonsmooth boundaries, Arch. Rational Mech. Anal. 68, 93-98. [87] Chemical kinetics, in Lectures on Combustion Theory, eds. S. Burstein, P. D. Lax and G. A. Sod, NYU, COO-3077-153, 122-136. 1979 [88] (with R. S. Phillips) Translation representations for the solution of the nonEuclidean wave equation, Comm. Pure Appl. Math. 32, 617-667. [89] (with C D . Levermore) The zero dispersion limit for the Korteweg-de Vries equation, Proc. Natl. Acad. Sci. 76, 3602-3606. [90] Recent methods for computing discontinuous solutions — a review, in Computing Methods in Applied Sciences and Engineering, 1977, II; 3rd Int. Symp. IRIA, eds. R. Glowinski and D. L. Lions, Springer Lecture Notes in Physics, 91, 3-12. 1980 [91] (with R. S. Phillips) Scattering theory for automorphic functions (AMS), Bull. of C.N.S. 2, 161-195. 1981 [92] On the notion of hyperbolicity, Comm. Pure Appl. Math. 33, 395-397. [93] (with A. Harten) A random choice finite difference scheme for hyperbolic conservation laws, SIAM J. Numer. Anal. 18, 289-315.

247 [94] (with R. S. Phillips) The translation representation theorem, Integral Equations and Operator Theory 4, 416-421. [95] (with R. S. Phillips) Translation representations for the solution of the nonEuclidean wave equation, II, Comm. Pure Appl. Math. 34, 347-358. [96] Applied mathematics 1945 to 1975, Amer. Math. Heritage, Algebra & Appl. Math., 95-100. [97] Mathematical analysis and applications, Part B, essays dedicated to Laurent Schwartz, ed. L. Nachbin (Academic Press) Advances in Math. Suppl. Studies, 7B, 483-487.

1982 [98] The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces, J. Func. Anal. 46, 280-350. [99] The multiplicity of eigenvalues, AMS Bull, 213-214. [100] (with R. S. Phillips) A local Paley-Wiener theorem or the Radon transform of L2 functions in a non-Euclidean setting, Comm. Pure Appl. Math. 35, 531-554.

1983 [101] Problems solved and unsolved concerning linear and nonlinear partial differential equations, Proc. Int. Cong. Mathematicians, 1 (North-Holland), 119-137. [102] (with A. Harten and B. van Leer) On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SI AM Rev. 25, 35-61. [103] (with C. D. Levermore) The small dispersion limit of the Korteweg-de Vries equation, Comm. Pure Appl. Math. 36, I 253-290, II 571-593, III 809-930.

1984 [104] On a class of high resolution total-variation-stable finite difference schemes, Ami Harten with Appendix by Peter D. Lax, SIAM 2 1 , 1-23. [105] (with R. S. Phillips) Translation representations for the solution of the non-Euclidean wave equation, Comm. Pure Appl. Math. 37, I 303-328, II 779-813, III 38 (1985), 179-813. [106] Shock Waves, Increase of Entropy and Loss of Information, Seminar on Nonlinear Partial Differential Equations, ed. S. S. Chern (Math. Sci. Res. Inst. Publ.), 129-171.

248 1985 Large Scale Computing in Science, Engineering and Mathematics, Rome. (with R. J. Leveque and C. S. Peskin) Solution of a two-dimensional cochlea model using transform techniques, SIAM J. Appl. Math. 45, 450-464. (with R. Phillips) Translation representations for automorphic solutions of the wave equation in non-Euclidean spaces; the case of finite volume, Trans. AMS 289,715-735. (with P. Constantin and A. Majda) A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math. 38, 715-724. 1986 On Dispersive Difference Schemes, 1985, Kruskal Symposium, Physica 18D, 250-255. Mathematics and computing, J. Stat. Phys. 43, 749-756. (with A. Jameson) Conditions for the construction of multipoint total variation diminishing difference schemes, App. Numer. Math. 2, 335-345. Mathematics and its applications, Math. Intelligencer 8, 14-17. Hyperbolic systems of conservation laws in several space variables, Current Topics in Partial Differential Equations, papers dedicated to Segeru Mizohata (Tokyo Press), 327-341. The Soul of Mathematics, Studies in Mathematics and its Applications 16, Patterns and Waves (North-Holland). 1988 Oscillatory solutions of partial differential and difference equations, Mathematics Applied to Science (Academic Press), 155-170. (with R. J. Leveque and C. Peskin) Solution of a two-dimensional cochlea model with fluid viscosity, SIAM J. Appl. Math. 48, 191-213. The flowering of applied mathematics in America, AMS Centennial Celebration P r o c , 455-466; SIAM Rev. 3 1 , 65-75. (with J. Goodman) On dispersive difference schemes I, Comm. Pure Appl. Math. 4 1 , 591-613. 1989 Science and computing, Proc. IEEE 77. Writing mathematics well, Leonard Gillman review, Amer. Math. Monthly 96, 380-381.

249 [123] From cardinals to chaos: Reflections on the life and legacy of Stanislaw Ulam, reviewed in Phys. Today 42, 69-72; Bull. AMS 22 (1990) 304-310, St. Petersburg Math. J. 4 (1993) 629-632. [124] Deterministic turbulence, Symmetry in Nature, Volume in Honor of Luigi A. Radicati di Brozolo II (Scuola Normale Superiore), 485-490. 1990 [125] (with R. Phillips) Translation representations for automorphic solutions of the wave equation in non-Euclidean spaces, IV, Comm. Pure Appl. Math. 43. [126] Remembering John von Neumann, Proc. Symp. Pure Math. 50. [127] The ergodic character of sequences of pedal triangles, Amer. Math. Monthly 97, 377-381. 1991 [128] Deterministic analogues of turbulence, Comm. 1047-1055.

Pure Appl.

Math.

44,

[129] (with T. Hou) Dispersive approximations in fluid dynamics, Comm. Pure Appl. Math. 44. 1993 [130] (with C. D. Levermore and S. Venekides) The generation and propagation of oscillations in dispersive IVP's and their limiting behavior, in Important Developments in Soliton Theory 1980-1990, eds. T. Fokas and V. E. Zakharov (Springer- Verlag). [131] The existence of eigenvalues of integral operators, in Honor of C. Foias, ed. R. Teman, Ind. Univ. Math. J. 42, 889-991. 1994 [132] Trace formulas for Schrodinger operator, Comm. Pure Appl. Math. 47, 503-512. [133] Cornelius Lanczos and the Hungarian phenomenon in Science and Mathematics, Proc. Lanczos Centennary Conf. (N.C. State University Press). 1995 [134] Computational fluid dynamics at the Courant Institute 1-5, Computational Fluid Dynamics Review, eds. M. Hafez and K. Oshima (John Wiley & Sons). [135] A short path to the shortest path, Amer. Math. Monthly 102, 158-159.

250

1996 [136] Outline of a theory of the KdV equation, Lecture Notes in Mathematics, Recent Mathematical Methods in Nonlinear Wave Propagation (Springer) 1640, 70-102. [137] (with Xu-Dong Liu) Positive schemes for solving multidimensional hyperbolic conservation laws, Comp. Fluid Dynamics J. 5, 133-156. [138] The Old Days, A Century of Mathematical Meetings (AMS), 281-283. [139] (with A. Harten, C. D. Levermore and W. J. Morokoff) Convex entropies and hyperbolicity for general Euler equations, SIAM J. Numer. Anal. 1997 [140] Linear Algebra, Pure and Applied Math. Series (Wiley-Interscience). 1998 [141] (with Xu-Dong Liu) Solution of two-dimensional Riemann problem of gas dynamics by positive schemes, SIAM J. Sci. Comput. 19, 319-340. [142] Jean Leray and Partial Differential Equations, Introduction to Volume II, Selected Papers of Jean Leray (Springer-Verlag). [143] On the discriminant of real symmetric matrices, Comm. Pure Appl. Math. 51, 1387-1396. [144] The beginning of applied mathematics after the Second World War, Quart. Appl. Math. 56, 607-615. 1999 [145] A mathematician who lived for mathematics, Book Review, Phys. Today, 69-70. [146] The mathematical heritage of Otto Toeplitz, in Otto Toeplitz, Bonner Mathematische Schriften, 319, 85-100. [147] Mathematics and computing, in Useful Knowledge, ed. A. G. Beam (Amer. Philos. Soc), 23-44. [148] Change of variables in multiple integrals, Amer. Math. Monthly 105, 497-501. 2000

[149] Mathematics and computing, in IMU, Mathematics: Frontiers and Perspectives (AMS), 417-432.

251 2001 [150] Change of variables in multiple integrals II, Amer. Math. Monthly 108, 115-119. [151] On the accuracy of Glimm's scheme, Math. Appl. Anal, to appear. [152] Functional Analysis (John Wiley and Sons), to appear. [153] The Radon transform and translation representation.

252 Proceedings of Symposia in Pure Mathematics Volume 65, 1999

A Sample of Lax's Contributions to Classical Analysis, Linear Partial Differential Equations and Scattering Theory Peter Sarnak

1. Classical Analysis and P D E In this lecture I will describe a small subset of Peter Lax's profound contribution to analysis and partial differential equations. His works, especially on hyperbolic equations, constitute some of the most influential works in the subject in the last 50 years. My own interests have been so to speak at the boundary of his work (though Peter has certainly influenced my work both directly and indirectly) so my angle on some of the topics below may be somewhat different from the mainstream views. My aim is to give you a glimpse of Peter's brilliance and originality. Let me begin naturally enough with Peter's first paper [LI] written in his teens, in which he resolved a conjecture of Erdos. If P(z) is a polynomial of degree n, then a well-known inequality of Bernstein asserts that (1)

max|P'(z)|
|z|
equality holding if and only if P{z) = az . Erdos conjectured that if P(z) has no zeros in \z\ < 1, then (1) should hold with n replaced by n/2 (and this is sharp, as (zn + l)/2 shows). In [LI] Peter gives an elegant proof of this conjecture. An impressive start for a kid, and I have no doubt that at that time the older Hungarian did his best to lure the younger one into a field like combinatorics. Luckily for us, Peter was drawn into the fields of analysis and partial differential equations. Indeed, some years later (after the war) we find that Peter is fully involved in PDE. In [L2] he establishes a very interesting extension of the Phragmen-Lindelof principle. Recall that a form of this principle asserts that if (2)

S={(x,t)

:0<x<

1,2/>0}

and if u{x, y) is harmonic in S, vanishes on x = 0 and x = 1, and is bounded, then u decays exponentially as y —> oo. It takes quite some insight to see that the root of this phenomenon is a spectral one which allows for the following generalization due to Peter. Let V0 be a compact subset of Rn and let V = V0 x [0, oo) C K n + 1 . Let L be an elliptic differential operator of order 2m with coefficients independent © 1999 American Mathematical Society

169

253 170

PETER SARNAK

of y. Suppose that u satisfies Lu ~ 0 ,

u = 0

on dV0 x [0, o o ) .

Then if the Dirichlet integral of u over V is finite (that is m derivatives are in Z,2 (£>)), then u decays exponentially as y —> oo (the decay rate being independent of u). His proof of this is ingenious and elegant (typical of much of Peter's work) and it introduces some themes which persist in his later works. These being; compactness and translation invariance with exponential decay as a consequence. These arise in his analysis of the last problem as follows: Let H be the set of functions uy{x), where y > 0 and uy(x) — u(x,y), u(x,y) satisfying the equation and the boundary conditions. For w > 0 the operator Tw : H —> H, with appropriate norms of H, defined by Twvy = vy+w satisfies: Tw is compact for w > 0, ||7i|| < 1, ||T„|) 1 / / n = ||rpn||l/n _^ a < i ; as n —• oo. The theorem follows easily from these. The above led Peter to think about translation invariant spaces in general. Much had been written about these for scalar valued functions. In [L3] he found the following striking and simple description of such spaces for vector valued functions. Let S be a finite dimensional inner product space. Let H$ be the space L 2 ( R > 0 , S) of square integrable functions from R>o to S. If T is a closed translation invariant subspace of L2(M.>0, S) (i.e., g & T ==> gt G r for t > 0 where gt(x) = g(x +1)) and R is the space of Fourier Transforms of members of r , then R = FHT, where T is an inner product space of dimension at most dim S, F(z) : T —* S is defined in 0>(z) > 0, ||F(z)|| < 1, F(z) is an isometry onto its image if z £ M. and F(z) is holomorphic for $s(z) > 0. Moreover this representation is unique except for multiplication of F on the right by a constant unitary matrix. In 1956 Peter wrote a paper [C-L] with Courant on propagation of discontinuities in wave motion. This was followed by one of Peter's seminal papers "Asymptotic solutions of oscillatory initial value problems" [L4]. This paper is the genesis of the theory of Fourier Integral Operators (see [HI] and [H2]). Even with all the excellent modern treatments of this subject, I still recommend to t h e new-comer t o this theory to first read Peter's paper. Moreover, as I indicate in more detail below, for certain purposes, the techniques introduced in this paper are still at the forefront of what we know today. The paper is concerned with the strictly hyperbolic system (3)

Mu = 0 .

M being a first-order differential operator acting on vectors u(t,x)

= I • I

(t,x),

x € R m , by the rule m

(4)

Mu = ut + Y^ Aj uXj + Bu

Aj and B being n x n matrix functions. Assume that in some neighborhood of t = 0 (which is assumed to be a space-like hypersurface) (5)

Pi M + VI M H

+PmA,

A SAMPLE OF LAX'S CONTRIBUTIONS TO CLASSICAL ANALYSIS

171

has real and distinct eigenvalues for all p e R m , this being the condition of strict hyperbolicity. One seeks oscillatory solutions to (3) of the form (6)

4>fr) = elie{x
+ v, /£ + •••}

for a suitable phase function £(x,t). Substituting (6) in (3) and solving recursively for the coefficients of £ ~ m one is led to the well-known eikonal equation from geometric optics. If X(t, x,pi,... ,pm) is one of the eigenvalue branches of (5) then we get the nonlinear first-order (scalar) equation d£ { ')

rv,

A\t,X\,

. . . , Xm,

t X j , . . . , tXrn ) -

One solves this equation for small t, \t\ < to (in particular before any caustics can develop) by the method of characteristics — these being the bi-characteristics of the original equation (3). The Vj's are determined by solving the corresponding 'transport equations.' In this way one gets a formal asymptotic series in f for such a 4>{x) with highly oscillatory initial data. Most importantly, Peter goes on to make this analysis precise and rigorous. In particular he develops a Fourier synthesis in £ which allows him to develop an explicit approximate geometric construction of the fundamental solution to (3), for \t\ < to- His 'parametrix' differs from the fundamental solution to any order of smoothness that one desires. Before this paper there were the constructions of parametrices by Hadamard [HA] and Riesz [RI] for second-order equations. As mentioned above, Peter's general method above led to the (local) theory of Fourier Integral Operators. To illustrate the power of the above method, consider the wave equation on a domain or a Riemannian manifold. The construction yields, for \t\ < to and any integer JV, an explicitly described distributional kernel KN(t,x,y) which differs from the fundamental solution K(t,x,y) by a CN+1 function, in all the variables. It can be used among other things to study the spectrum of the Laplacian. For example, if X is a smooth compact Riemannian manifold of dimension v and A is the Laplace-Beltrami operator acting on functions on X, then there is an orthonormal basis of eigenfunctions <j>j and eigenvalues kj such that (8)

- A ^ = ^ . .

For the (hyperbolic) wave equation (9)

utt = Aw

on 1 x I , with initial conditions w(0, x) = 6(x,y) fundamental solution

and ut{0,x)

= 0, we have the

oc

(10)

K(t,x,y)

= ^coB(fcjt)0j(i)0j(j/). 3=0

The Lax construction gives the singular part of K(t,x,y) one to investigate asymptotically the sums (11)

^

4>j(x)4>j(y)

as

for small t. This allows

A -* oo.

A
This range, X < kj < A + 1, might be called the "Lax range" since it comes from his parametrix. I mention a couple of my favorite applications:

172

PETER SARNAK

(A) Weyl's Law with Remainder (Hormander [HI])

N(X) := J2 1 = (27r)"" c" Vo1 (X) X" + ° (A"_1) fcj
where c„ is the volume of the unit ball in W'. (B) Bounding the eigenfunctions in Lp (Seeger and Sogge [S-S]), If 6(p) = max

{"

1 1 p~2

UA^
\, o | for

then ?i^±il
These results are sharp for the i/-sphere with its round metric. However if X has negative curvature for example (in which case the bicharacteristic Hamilton flow which is the geodesic flow cmT*X, is "chaotic") then (A) and (B) are no doubt very far from being sharp. Recently there has been a lot of interest in such fine spectral questions for such X's (the subject going by the name Q u a n t u m Chaos, see [G-V-Z], [SA]). One problem being to understand the sums in (11) but in shorter ranges X < Kj < \ + 77(A), with 77(A) of size A _ Q for some a > 0. Unfortunately, in general no progress has been made on such short sums. The global theory of Fourier Integral Operators allows one to deal with 77(A) is small as 1/ log A and this already is quite deep, see Duistermaat and Guillemin [D-G] and Berard [BE]. The exponential proliferation of periodic orbits of the bicharacteristic flow for such an X creates a fundamental barrier to analyzing sums substantially shorter than the Lax range. It seems fair to say that this is where we are stuck on such problems today. The fine structure of the spectrum of the Laplacian has been attacked by many people and with varying approaches such as the heat equation, the Schrodinger equation, and the wave equation as above. In this connection Peter once remarked that it is surely better to throw light rather than heat or uncertainty on the matter. Indeed, today the wave equation offers the best approach to these problems. To end Section 1, let me mention what appears to be the unique joint paper by Lax and Nirenberg [L-N]. It concerns the sharp Garding inequality. Recall that in the theory of pseudo differential operators, if a(x,£) is a zeroth order nonnegative symbol, then Garding showed that for e > 0 there is K{e) such t h a t (12)

Re(Au, u) > -e\\u\\20 - K{e) \\u\\\/2

.

For some applications of this inequality one would like to take e = 0 and still have K finite. Such an inequality is known as a sharp Garding inequality and was established by Hormander [H3], Fefferman and Phong [F-P], and others. It is natural to ask whether such sharp inequalities hold for matrix symbols a(x,^). Indeed, such a result has applications to some numerical difference schemes, see [L-N]. Lax and Nirenberg show that there is a matrix version of the sharp inequality. Precisely if a(x,£) is nonnegative and Hermitian, then (12^ holds with e = 0 and a suitable K. It appears that in this matrix case their result is still the best known. 2. S c a t t e r i n g T h e o r y I turn now to Peter's joint papers with Ralph Phillips (who unfortunately could not be present on this special occasion). This collaboration is the only one that I can think of that rivals that of Hardy and Littlewood. They, Peter and Ralph,

A SAMPLE OF LAX'S CONTRIBUTIONS TO CLASSICAL ANALYSIS

173

that is, have developed a general time dependent approach to scattering theory and applied it in a number of settings. I discuss two cases, one of the exterior of an obstacle in R n and the other a hyperbolic manifold with ends. Let CI C K" be a compact smooth subdomain. Consider the wave equation (13)

u „ - Au = 0 n

on R \ $1 with u satisfying either Dirichlet boundary conditions, U\QQ = 0, or Neumann conditions dnu\an = 0. Scattering theory is concerned with the relation between the behavior of the solutions u(x, t) to (13) as t —* ±oo and in particular the effect of the obstacle on waves' which hit the obstacle. Now as Rellich showed that the spectrum of A on L2 (R n \Q) is continuous. So an analysis of the scattering via A requires analyzing the resolvent (A — A ) - 1 and its meromorphic continuation. This is one approach to scattering theory. The Lax-Phillips theory is time dependent and proceeds much more geometrically. For simplicity, assume that n is odd so that there is a Huygens principle (i.e., sharp propagation of signals). There are solutions u(x,t) which vanish for all x, for which d(x,Q.) < t, (t > 0). For positive times these do not interact with fi and are called outgoing solutions. Similarly, if u(x, t) vanishes for all x with d(x, t) < —t for t < 0, we call u incoming. Let U(t) be the unitary group which gives the solution operator to the wave equation (13) for all t. T h a t is for initial conditions, / = ( y 1.

where u(x,0) (15)

= f\ and u t ( x , 0) = fa- The energy E(f)

:= [

( ( / 0 £ + (h)2x)

dx

is preserved by the evolution operator U(t). Abstracting this set up (it is clear that Peter likes to abstract to a point that is elegant and insightful but without straying from the concrete — he seems to be able to find the perfect balance) we get the Lax-Phillips axioms [ L - P l ] : U(t) acts unitarily on a Hilbert space H and assume t h a t there are orthogonal subspaces £>_ and U+ of H such that !/(<)£>_ C D _ U(t) V+ c V+

n W

for for

(ii)

()U(t)V-

(iii)

\JU(t)V-=\JU(t)T>+ = H. t

= {0} =

t<0 t > 0. {)U(t)V+

t

n

For K \ f2 it is (iii) that is the most difficult to establish (with D_ and V+ the incoming and outgoing subspaces respectively). Indeed, establishing it is what put the theory in motion. Given U(t), X>_, X>+ as above, they prove a representation theorem (much like the usual spectral theorem) which asserts that there is a Hilbert space ./V and isometries / and O from H to L 2 (R, N) such t h a t (I)

7(D_) = L 2 ( R _ , A 0 ,

R- =

{x\x<0}

174

PETER SARNAK

(II)

0(V+) =

L2(R+,N)

(III)

U(t) is conjugated to translation by t.

In particular the diagram defines S : L2(R,N)

L2{R,N)

* S

-> L2(R,N)

and is called the

L2(R,N)

scattering operator. Clearly S commutes with translation and hence acts by convolution by a function denoted by S(t). Furthermore S is unitary and S(t) = 0 for t > 0 (causality). Its Fourier transform S(z) is a multiplication operator (giving the spectral representation of 5) and is holomorphic in $s(z) < 0. S(z) is called the scattering matrix. The abstract theory revolves around a semigroup and its infinitesimal generator. Let P+ and P_ be the projections which annihilate 23+ and 23_ respectively. For t > 0 set Z(t) := P+ U(t) P_ So Z{t) : K —• K where K = H 0 (23- © 23+). It satisfies ||Z(t)|| < 1 for t > 0 and forms a semigroup. Denote its infinitesimal generator by B. They show that B has discrete spectrum in C (in fact they prove that Z(\)(\Q — B ) - 1 is compact) and that its eigenvalues A* satisfy (16)

Re(Xk)<0.

The importance of B lies in the following fundamental result [L-Pl]: S(z) has a menomorphic continuation to C with poles at i\k, where {A*} is the spectrum of B. With this data (i.e., the poles) coming from M.n \ H they ask the fundamental question about the location of the poles of S(z) (or the eigenvalue of B) and its relation to the classical mechanics on R n \ U (i.e., rays ... ).. It is remarkable that Lax-Phillips are asking these questions about the influence of periodic or trapped rays on the poles, as early as 1962. This is long before such questions were addressed in connection with the eigenvalues of Laplacians by physicists such as Gutzwiller [GU] or mathematicians Chazarain [CH] and Colin de Verdiere [CVl]. The first result relating the geometry of K" \ fi and the poles is due to Lax, Morawetz, and Phillips [L-M-P]. It asserts that if SI is star shaped then there is an £o > 0 such that Re(Xk) < —£o for all k. A striking corollary to this result is that when SI is star shaped then locally solutions to (13) decay exponentially with a fixed rate as |£| —• oo. Such a result is ample proof of the power of the Lax-Phillips method. After this they were led to a fundamental conjecture. They say f2 is nontrapping, if there is a ball B D ft and an £ < oo such that any unit speed ray starting in

A SAMPLE OF LAX'S CONTRIBUTIONS TO CLASSICAL ANALYSIS

175

B \ Q and obeying the law of reflection on dCl, will remain in B \ CI for a time of at most L If 0, does not satisfy the above then it is called trapping. The conjecture mentioned above (see [ L - P l ] , 1967) asserts that a) If O is nontrapping then Z(t) is eventually compact (which then has many implications on the poles of B). b) If Q. is trapping then \\Z{t)\\ = 1 for all t > 0. The above far-reaching insight led to lovely results by a number of different authors. Today, the conjecture is essentially settled. In 1979, Ralston [RA] showed that if fl has a trapped ray (and he makes a certain technical assumption about the order of tangency not being infinite) then (b) holds. The proof of (a) had to await developments in microlocal analysis. Its proof by Melrose [Ml] is one of the major results in scattering theory. The study of the distribution of the poles was of course also initiated by Lax and Phillips. For example, they show that if Z(to) is compact for some to then for some b > 0 Re(Xk) < a b\og\Q{\k)\. In connection with the behavior of the continuous spectrum, Majda and Ralston [M-R] developed an analogue of the Weyl law for the winding number of the scattering matrix determinant, which gives the asymptotics of J_^ log det S(a) da, as A —» oo. Other results on the distribution of poles have been established especially by Melrose [M2] and Zworski [Z]. This is still a very active area of research. The second setting in which I describe the Lax-Phillips scattering theory, is for hyperbolic manifolds. T h a t their approach is well suited to this setting was first noted by Faddev and Pavlov [FA-P]. Let H n + 1 be hyperbolic n + 1 dimensional space, that is {{y,x)\y > 0, x G R™} equipped with the line element (17)

d

ds> =

-^±^ . y2 Let A be the corresponding Laplacian. Also let T be a discrete subgroup of the isometry group of H n + 1 ( = 0^(n + 1,1)) and let Xr = r \ H n + 1 be the corresponding quotient. Xr is a hyperbolic manifold and we let A r denote the corresponding Laplacian on L2(Xr)Lax and Phillip's work is concerned with the case that Xp is geometrically finite. T h a t is to say that there is a fundamental domain for Xr in H n + 1 bounded by a finite number of geodesic hyperplanes. For scattering to take place we should have either: (A) Xp has finite volume (but not compact). (B) Xr has infinite volume. The study of the spectrum of A r in the case (A) was pioneered by Selberg in the 50's [SE]. His two major achievements were the analytic continuation of the Eisenstein series (which furnish the continuous spectrum for X r ) and the celebrated trace formula. In this finite volume case, Xr looks like a compact piece K together with a finite (disjoint) union of cusps C i , . . . ,CT. The Lax-Phillips axioms may be modified to apply to the wave equation on Xr'u\

=

Kut)t

(18)

where

Lu

=

[0 \L (

A

I\

(u 0)\ut/

+ (?)

)

259 176

P E T E R SARNAK

FIGURE 1.

The shift by (^) ensures a Huygens principle in odd dimensions but it also renders the corresponding energy form to be indefinite. However, if one projects out the finite number of positive eigenvalues, then the energy form is nonnegative. There may also be an infinite dimensional space of bound states (i.e., square integrable eigenfunctions) within the continuum. The scattering theory takes place on the orthogonal compliment of all bound states. The spaces X>_ and T>+ correspond to special functions supported in the cusps. Examples of members of V+ are solutions to the wave equation initially supported in C\ which are functions of y alone and which move out into the cusp as t increases (t > 0). The scattering process corresponds to what comes back into the cusps when sending a wave into the compact part. The theory is modified to take these new features into account. One defines a corresponding semigroup with infinitesimal generator B. Again the eigenvalues Afc of the operator B (whose resolvent is compact) for which Re(Xk) < 0, correspond to poles of the scattering operator S by the relation Afc —»iX).. These poles can also be identified with poles of the Eisenstein series. In fact, their set up yields a natural and elegant new proof of the analytic continuation of the Eisenstein series [L-P2]. A variation of their proof using their "cut-off Laplacian" was given by Colin de Verdiere [CV2]. His proof was exploited very profitably in Mueller's proof of the trace class conjecture for higher rank symmetric spaces [MU]. The Lax-Phillips set up also leads very naturally to a derivation of the trace formula [L-P2]. One might ask if their approach yields further insights or techniques not offered by Selberg's method. There is at least one fundamental problem for which (at the present time) their approach is indispensable. This concerns deformations of Xr or of T. The spectrum of B in C identifies both the poles of the Eisenstein series and the discrete spectrum of L. It allows one to study the behavior of these quantities as we deform Y along a curve Tt in the Teichmuller space of such T's. This was used crucially by Phillips and myself [P-S] in our work which showed that the bound states are very fragile and are probably present in abundance only for arithmetical T's. In this connection I would like to acknowledge Peter, who pointed us to the advantage of using real analytic curves in the parameter t. We turn to the case (B) above when Xp has infinite volume. In this case, before their work, much less was known even about the nature of the spectrum of

A SAMPLE OF LAX'S CONTRIBUTIONS TO CLASSICAL ANALYSIS

177

Xr- Patterson [PA] had some definitive results but only for n = 1 (i.e., Xr of dimension 2). The two basic results which Lax and Phillips [L-P3] established are that the spectrum of A r in [ ( f ) 2 , oc) is continuous (i.e., there are no bound states with eigenvalues in this range) and that in [0, ( f ) 2 ) the spectrum of Ap consists of a finite number of points. The proof of there being no bound states in [ ( f ) 2 , oo) is based on a clever use of Volterra integral equations to show that there are no such square-integrable eigenfunctions in any sector of H n + 1 which contains a piece of dW1+x (note that infinite volume, geometrically finite Xp's contain such a piece). Using the non-Euclidean Radon transform Lax and Phillips go on to develop a translation representation for the wave operator on an Xr of infinite volume. This in turn leads to a Fourier decomposition of the general function of Xr- Their analysis includes such Xp's which have cusps of intermediate ranks. A development of a complete trace formula for such Xp's remains an open problem, though Patterson and Perry [P-P] develop properties of the Selberg Zeta Function for such X r , which are closely related. Lax and Phillips have applied their analysis of the wave equation on Xr to the problem of counting lattice points in H n + 1 [L-P3]. Their method is simple and effective and it yields the sharpest results known. Precisely let T be as above. For z,w e Mn+1 and R>0 let N(z, w, R) = # { 7 £ T| din, z, w) < R} where d(z, w) is the non-Euclidean distance from z to w. The problem is to determine the asymptotics of N as R —> oo. Let bo = ^ / ( f ) 2 — Ao where Ao is the smallest eigenvalue of Ap. For simplicity, we assume t h a t 0 < Ao < ( f ) 2 with corresponding eigenfunction o(z) and that there are no other eigenvalues of Ap in [0, ( f ) 2 ) (otherwise these enter in a simple way). Their result asserts that Nr(z,w,R)

andc(A0)

=

=

c(\0)0(z)M™)e^+b^R

^ ( ^ , ( ^ 0 - 1 ) !

This result is the best known even when Xr is compact (in which case Ao = 0), when the main term in the asymptotics goes back to Delsarte [DE] and the above sharp remainder term is due to Selberg. To end let me mention a couple of papers which do not strictly fall under the title of this lecture. These are papers by Peter — on sums of squares. In [L-L] he and Anneli Lax show that F{x\,... ,15) = J2i=\ YljM(xi ~~ xj)' which while 5 being nonnegative for x £ R is nevertheless not a sum of squares of polynomials in x. Examples of such polynomials were already found in Hilbert [HI] disproving a conjecture of Minkowski. However the above is a very pleasing example which, by the way, arose in a mathematical olympiad where it was asked to show that F(x\,X2,- • • ,£5) > 0. Recently in [L5] (between the time of the Florence conference and the writing of this lecture) Peter gave a beautiful proof of a result of Ilyushechkin [IL], that the discriminant D(a,ij) = rL<j(^> ~ ^ J ) 2 °f a r e a ' symmetric matrix A = ( a ^ ) (where A ; . . . An are its eigenvalues) which is plainly nonnegative, is a sum of squares of polynomials in a , j . He further discusses the

178

P E T E R SARNAK

interesting question as to the minimal such representation. Clearly the master has not lost his touch! References [BE]

P. H. B e r a r d , On the wave equation on a compact Riemannian manifold without conjugate points, M a t h Z. 1 5 5 (1977), p p . 2 4 9 - 2 7 6 . [CH] J. C h a z a r a i n , Formules de Poisson pour les varietes riemannienes, Inent. M a t h . 2 4 (1974), p p . 6 5 - 8 2 . [C-L] R. C o u r a n t a n d P. Lax, The propagation of discontinuities in wave motion, P r o c . N a t . Acad. Sci. U.S.A. 4 2 (1956), p p . 8 7 2 - 8 7 6 . [CV1] Y. Colin-de-Verdiere, Sur le spectre des operateurs elliptiques a bicharacteristique toutes periodiques, C o m m e n t . M a t h . Helv. 5 4 (1979), p p . 5 0 8 - 5 2 2 . [CV2] Y. Colin-de-Verdiere, Une novelle demonstration du prolongement meromorphe des series de Eisenstein, C. R. A c a d . Sci. P a r i s Serie I, 2 9 3 (1981) p p . 3 6 1 - 3 6 3 . [DE] J. Delsarte, Sur le gitter fuchsien, C. R. A c a d . Sci. P a r i s 2 1 4 (1942), p p . 147-149. [D-G] H. D u i s t e r m a a t a n d V. Guillemin, The spectrum of positive elliptic operators and periodic ^characteristics, Invent. M a t h . 2 9 (1975), p p . 3 9 - 7 9 . [FA-P] L. Faddeev a n d B . Pavlov, Scattering theory and automorphic functions, P r o c . Steklov Inst. M a t h . 2 7 (1972), p p . 1 6 1 - 1 9 3 . [F-P] C. Fefferman a n d D. P h o n g , The uncertainty principle and sharp Garding inequalities, C o m m . P u r e A p p l . M a t h . 3 4 (1981), p p . 2 8 5 - 3 3 1 . [GU] M. Gutzwiller, Phase integral approximation in momentum space and the bound states of an atom, J. M a t h e m a t i c a l P h y s . 8 (1967), p p . 1 9 7 9 - 2 0 0 0 . [G-V-Z] M. G i a n n o n i , A. Voroz, a n d J. Z i n n - J u s t i n , eds., Chaos and Quantum Physics, Les Houches, 1991. [HA] J. H a d a m a r d , Le Probleme de Cauchy, H e r m a n n , P a r i s , 1932. [HI] D. Hilbert, Uber die Darstellung definiter Formen als Summen von Quadraten, Math. A n n . 3 2 (1888), p p . 3 4 2 - 3 5 0 . [HI] L. H o r m a n d e r , Fourier intergral operators I, A c t a M a t h . 1 2 7 (1971), p p . 7 9 - 1 8 3 . [H2] L. H o r m a n d e r , The specturm of a positive elliptic operator, A c t a M a t h . 1 2 1 (1968), p p . 193-218. [H3] L. H o r m a n d e r , Pseudo-differential operators and non-elliptic boundary problems, Ann. of M a t h . 8 3 (1966), 129-209. [IL] N. V. Ilyushechkin, The discriminant of the characteristic polynomial of a normal matrix, M a t . Z a m e t k i 5 1 (1992), p p . 1 6 - 2 3 . [LI] P. Lax, Proof of a conjecture of P. Erdbs on the derivative of a polynomial, Bull. Amer. M a t h . Soc. 5 0 (1944), p p . 5 0 9 - 5 1 3 . [L2] P. Lax, A Phragmen-Lindelof theorem in harmonic analysis and its application to some questions in the theory of elliptic equations, C o m m . P u r e Appl. M a t h . 1 0 (1957), p p . 361-389. [L3] P. Lax, Translation invariant spaces, A c t a M a t h . 1 0 1 (1959), p p . 163-178. [L4] P. Lax, Asymptotic solutions of oscillatory initial value problems, Duke M a t h . J. 2 4 (1957), p p . 6 2 7 - 6 4 6 . [L5] P. Lax, On the discriminant of real symmetric matrices, p r e p r i n t , 1997. [L-M-P] P. Lax, C. M o r a w e t z , a n d R. Phillips, Exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle, C o m m . P u r e A p p l . M a t h . 1 6 (1963), p p . 4 7 7 486. [L-Pl] P. Lax a n d R. Phillips, Scattering Theory, 2nd ed., A c a d e m i c P r e s s , San Diego, 1989. [L-P2] P. L a x a n d R. Phillips, Scattering Theory for Automorphic Functions, A n n a l s of M a t h e m a t i c a l Studies, No. 8 7 , P r i n c e t o n , 1976. [L-P3] P. L a x a n d R. Phillips, The asymptotic distribtuion of lattice points in Euclidian and [L-L] [L-N] [Ml]

non-Euclidian spaces, J. F u n c t . A n a l . 4 6 (1982) p p . 2 8 0 - 3 5 0 . A. Lax a n d P. Lax, On sums of squares, Linear A l g e b r a a n d A p p l . 2 0 (1978), p p . 7 1 - 7 5 . P. Lax a n d L. N i r e n b e r g , On stability for difference schemes; A sharp form of Garding's inequality, C o m m . P u r e Appl. Math- 1 9 (1966), p p . 4 7 3 - 4 9 2 . R. Melrose, Singularities and energy decay in acousticle scattering, Duke M a t h . J. 4 6 (1979), p p . 4 3 - 5 9 .

A SAMPLE OF LAX'S CONTRIBUTIONS TO CLASSICAL ANALYSIS [M2] [MU] [M-R] [PA] [P-P] [P-S] [RA] [RI] [SA] [SE] [S-S] [Z]

179

R. Melrose, Polynomial bound on the number of scattering poles, J. Funt. Anal. 53 (1983), pp. 287-303. W. Muller, The trace class conjecture in the theory of automorphic forms, Ann. of Math. 130 (1989), pp. 473-529. A. Majda and J. Ralston, An analogue of Weyl's formula for unbounded domains, Duke Math. J. 45 (1978), pp. 183-196. S. Patterson, The Laplace operator on a Riemann surface I, Compositio Math. 31 (1975), pp. 83-107. S. Patterson and P. Perry, Divisor of the Selberg zeta function for Kleinian groups, I, preprint, 1997. R. Phillips and P. Sarnak, Perturbation theory for the Laplacian on automorphic functions, J. Amer. Math. Soc. 5, (1992), pp. 1-32. J. Ralston, Solution of the luoue equation with localized energy, Comm. Pure Appl. Math. 22 (1969). pp. 807-823. M. Riesz, L'integrale de Riemann-Liouville et le probleme de Cauchy, Acta Math. 81 (1949), 1-223. P. Sarnak, Arithmetic quantum chaos, The Schur Lectures, Israel Math. Conf. Proc. Vol. 8, Ramat Gan, 1995, pp. 183-236. A. Selberg, Collected Works, Vol. 1, Springer-Verlag, 1988. A. Seeger and C. D. Sogge, Bounds for eigenfunctions of differential operators, Indiana Univ. Math. J. 38 (1989), pp. 669-682. M. Zworski, Sharp polynomial bounds on the number of scattering poles, Duke Math. J. 59 (1989), pp. 311-323.

PRINCETON UNIVERSITY, PRINCETON, NJ 08544

E-mail address: sarnakOmath.princeton.edu

Wolf Prize in Mathematics, Vol. 2 (pp. 263-310) eds. S. S. Cfaem and F. Hirzebruch © 2001 World Scientific Publishing Co.

I»

Courtesy of Mrs Helen Lewy

264 HANS LEWY

1. Curriculum Vitae Mathematicae by David Kinderlehrer

Hans Lewy was born on October 20, 1904 in Breslau, Germany (now Wroclaw, Poland), the first son and second of three children of Max and Greta Lewy. As a child he was drawn to science and to music. Gottingen, in 1922 when Hans Lewy matriculated, was among the premier mathematical establishments in the world. With Klein and Hilbert still in residence, as well as Runge, Prandtl, Landau, Herglotz and Emmy Noether, Courant, of course, and many others it radiated irresistable scientific excitement. We are told that Lewy's gymnasium professor suggested to his father that the young man should travel to Gottingen. In this the professor was not only prescient but, perhaps, well informed. Not only Courant, but also Born, Hellinger and Toeplitz had come to Gottingen via Breslau. Friedrichs also arrived in 1922 and their life-long friendship took hold immediately. Unfortunately, 1922 marked the beginning of the Weimar hyper-inflation period, which ended at the end of 1923. The situation led to considerable conflict between Lewy and his family, who understandably wanted the eighteen-year-old to return to the security of his home, and eventually led to a temporary rupture of relations. In this period he took a railroad job, doing maintenance on the tracks and ties; neither his personality nor his physique made this easy for him. Courant wrote several times to Lewy's parents expressing great satisfaction and extolling the achievements of his student, perhaps in part to relieve this family pressure. He completed his thesis in 1926 with Courant and became, together with Friedrichs, Courant's Assistant and a Privatdozent. In the next few years he, and Friedrichs, pursued new methods for understanding elliptic and hyperbolic problems. He solved the initial value problem for general nonlinear hyperbolic equations in two independent variables [5]. He used this to give a new proof of the analyticity of solutions of elliptic equations in two variables which far exceeded the known proof in its elegance and simplicity, [9]. He and Friedrichs developed energy methods for wave equations. Most well known, in 1928, the joint work of Courant, Friedrichs, and Lewy [7] was published. About this, Lax has written, the onset of the Second World War brought an unprecedented (and unabated) pressure on the technological centers to provide numerical solutions of partial differential equations; here the ideas introduced by C-F-L; the famous stability condition, energy inequalities, and the leap frog difference method, turned out to be basic. This is an outstanding example of research undertaken for purely theoretical purposes turning out to be of immense practical importance. Lewy's work attracted the attention of Hadamard who included a long special appendix about it in the second edition, which was the later To be published in Collected Works of Hans Lewy (Birkhauser).

265 French edition, of his book. 1 To paraphrase Leray, he opened a world to us whose richness continues and will continue to amaze us. These years in Gottingen also marked the period of the "boy physics". In 1929-1930, under the sponsorship of the Rockefellar Foundation, he studied in Rome where he was especially influenced by Tullio Levi-Civita. He also met many younger Italian mathematicians, and in particular Beniamino Segre who would later have an influence on his work. The fellowship then took him to Paris, 1930-31, where he was active in Hadamard's seminar. Lewy left Germany quite soon after Hitler assumed power in 1933. He went first to Italy and then to Paris where Hadamard had managed to obtain for him a year's support. At this time, Lewy introduced Jean Leray and Juliusz Schauder. Later, during the war when Leray was taken prisoner, Lewy communicated with him via the Red Cross, and after the war the families remained quite close. In Paris, on the recommendation of Hadamard, he was offered a one to two year position at Brown, funded by the Duggan Foundation. In the fall of 1933, Lewy was in Providence, where he completed his first work on the Monge-Ampere Equation [14, 16, 17]. On the invitation of G.C. Evans, Lewy went to Berkeley in 1935. Courant, visiting Berkeley in 1932, had spoken enthusiastically of Lewy's work on that occasion. Courant himself was, in fact, offered a position at Berkeley. Evans arrived in Berkeley in 1934, chosen to build up the department. C.B. Morrey arrived in 1933 and Lewy in 1935. A.P. Morse, Jerzy Neyman and Raphael Robinson came at the end of the 1930s. Lewy's work on Minkowski's Problem was completed here [18, 19]. Erhard Heinz remarked that, "His estimates are of local type. It is remarkable that he did not make use of the Weyl estimates. Thus he went far beyond the original goal." Although the events in Europe suggest that it was natural for America to welcome scientists from the fascist countries, it is well to keep in mind that the country remained in the Depression and that there was considerable concern, by all parties, that emigres might be assuming positions that could be held by natives and that this could lead to a negative environment. With the outbreak of the war, Lewy took flying lessons and obtained a solo license in the hope of offering his services, but was soon called to Aberdeen Proving Grounds as part of the University of California contingent. He also worked halftime with the Office of Naval Research in New York. Here he became interested in water waves and the Dock Problem, resuming his collaboration with Friedrichs, [24]. The explicit solution in [23] also gave rise to a proof of the Law of Quadratic Reciprocity. At the war's conclusion, Lewy, then a Full Professor, was invited to return to Gottingen as Privatdozent! In 1947 he and Helen Crosby, an artist, writer and translator, were married. Their honeymoon included a trip around the world, beginning with a return to Europe. Also included was a two-month stay in Chengtu, Szechuan (Chengdu, Sichuan) where Lewy gave a course on water waves and a month visiting other 1

Le probleme de Cauchy et les equations aux de.rive.es partielles lineaires hyperboliques. & Cie, Paris 1932.

Hermann

266

institutions in China. Still under Nationalist rule, China was rarely visited by westerners in these years. The analytic continuation of minimal surfaces bounded by analytic arcs or satisfying free boundary conditions was given in the early 1950s, [28, 29]. These and related results had a formative impact on the study of free boundary and regularity problems in variational principles. Also in this decade he devoted considerable attention to the issue of the smoothness of solutions of equations and systems at the boundary and their asymptotic development near change of regime boundary points [26, 27, 32]. During the Loyalty Oath controversy in the state of California, Lewy was part of the group dismissed for refusal to sign the oath. Having seen Fascism at first hand in Italy, and watched its rise in Germany, he was wary of cooperating with any totalitarian tendencies in his new country. The details of those events, insofar as he is involved, are chronicled in Constance Reid's article. 2 He was on the faculty at Harvard in the fall of 1952 and then at Stanford in 1952 and 1953. This was the genesis of the work on Riabouchinsky flow with Garabedian and Shiffer, [33]. In the settlement of the Loyalty Oath dispute by the courts, the professors were reinstated and Lewy returned to Berkeley. In 1955, the Lewys' son Michael was born. Lewy's completely unexpected example of a linear equation with no solution, and with it a homogeneous equation with only the trivial zero solution, appeared in 1957, [38], one of the major events in the modern development of PDE theory. This example arose in the context of Lewy's study of Cauchy-Riemann structures in several complex variables and pseudoconvex domains, [36, 37, 43]. Both simple and general, it illustrates many different aspects of differential equations. We teach, for example, that real first order linear equations admit solutions by the elementary method of characteristics. But complex ones may not admit any solutions. It put an end to speculation about finding a version of the fundamental solution for a general linear equation, even with analytic coefficients, although this had recently been established for equations with constant coefficients. He received the Steele Prize of the American Mathematical Society for this work in 1979. In 1959-1960 Lewy accepted an invitation from the Scuola Normale Superiore and the University of Pisa. In these years, Pisa was achieving stature as a vigorous world center with Andreotti, De Giorgi, and Stampacchia among its leaders. Here, Lewy helped creating the emerging area of variational inequalities with Stampacchia. Their work was fundamental to the growth of the subject, introducing new types of free boundary problems. Lewy published in this area through the 1980s. In 1969-1970, he returned to Rome on the invitation of the Accademia dei Lincei. He was elected a Foreign Member in 1972. He also retired in 1972, continuing his research with undiminished vitality. On retirement he insisted that available funding should be dedicated to the support of younger scientists and declined to apply for any more grants. In 1973 he joined Ahlfors as one of the first two Ordway Professors at the University of 2

Hans Lewy, 1904-1988, Miscellanea mathematica (P. Hilton, F. Hirzebruch, R. Remmert, editors) Heidelberg, Springer, 1991.

267 Minnesota. Always attentive to the encouragement of young people, he returned to Minneapolis to participate in the 1984/1985 Institute for Mathematics and its Applications Program on Continuum Physics and Partial Differential Equations. In 1985 in Trento there was a meeting in his honor. He shared the 1984/1985 Wolf Prize with K. Kodaira for initiating many, now classic and essential, developments in partial differential equations. Subsequently he wrote frequently to the Wolf Foundation in support of other scientists, especially De Giorgi, who received the 1990 prize. He was awarded an honorary doctorate from Bonn University in 1986. In his acceptance speech Krisen in der Mathematik he spoke, characteristically, about the future. Hans Lewy died on August 23, 1988 in Berkeley.- He and Helen had recently returned from a trip to Europe where he delivered his last paper, [74], in honor of Ennio De Giorgi. There are many scientists whose objective is to bring order to chaos, to organize a disparate subject and bring it unity of theme and method. In an arcane way, Lewy was their antithesis: with a profoundly innovative point of view, he illustrated and revealed to us fundamental principles and ideas with the simplest of tools. Hans Lewy was a pillar of science in the twentieth century. His work touched nearly every significant area of analysis and endures in the canon that we pass on to our children.

268

HANS LEWY (gezeichnet von Helen Lewy)

269 2. The Music in Hans Lewy's Life by Helen Lewy

Hans Lewy always shunned the intrusion of the personal into his professional life. His private life was his own domain. Yet he was an eminently social being with so many friends and colleagues across Europe and Asia, and in the U.S., that a great deal about his personal life, - his work habits, his personality, his roots, his friends and "hang-outs" - is already well known. During his extended stays in Europe, especially in Italy, it was his custom to spend a great deal of his time socially, as well as mathematically (if the two can be separated) with the students. Like all scholars, he was always energized by the marvelous curiosity of the young. I have been asked to write a few words about Hans Lewy for this volume. His family and I are, for the most part, content to let his legacy rest on his mathematical works, for that's where it should be. I will not try to evoke the man behind the math. But, I do not feel he would object, if, to honor the students he loved to be with, I pass along some facts about the music which was always the companion of the math in his life. The combination of mathematics and music, while not uncommon, especially in the Europe of Hans Lewy's time, was, in his case, serious enough to force him to make, in his teens, a difficult career choice. He played the violin, the viola and the piano with mastery, and, at times in his life also the clarinet and the cello. He composed many string quartets, which unfortunately were lost, and at least one string trio (this was found by Lori Courant after the death of her mother; it was composed when Hans Lewy was 20 years old, and all three parts had been laboriously hand-copied by Nina Courant (this was long before photocopies - or even mimeographs). Over a period of years during the 1990s, Lori Courant, with fellow musicians, in a labor of love, edited, corrected, copied and brought to life the faded sheets of the score). According to his mother, Hans Lewy's love for music began with the piano lessons of his older sister when she was about 10 years old. At that particular time, she had little interest in learning to play the piano, but her little brother, five years her junior, did; he managed to be present during the lessons and soon he was outperforming his sister. This situation led to so much "mischief in the family", according to his mother, that a music teacher for little Hans was promptly acquired, for lessons not on the piano, but on the violin. Hans took to his lessons with passion, ("ferocity", was the word his mother used): he soon had outgrown his first two teachers, and, at length, as a pre-teen he was studying under a well-known concert violinist. By then the violin had become the mode through which Hans' native ebullience and independence found expression. He told me, sixty years later, of his joy in playing even as a child (but added that he had not always enjoyed "being produced" for family occasions). By the time he reached his teens, he was already playing regularly with chamber-music groups. To be published in Collected Works of Hans Lewy (Birkhauser).

270 His parents would not permit the unusual skills of their son to be exploited, until, under pressure from his teacher, they agreed when he was 16 to allow him to perform in public, for one time only: the concert took place in Bautzen in July 1920. One can sense the excitement of the young man in this partial translation of the letter he wrote to his parents two days later: My dear ones: Saturday was THE day: an orchestra, a male chorus, a woman soloist and me. I played the Chaconne by Bach and Devil's Trills by Tartini, accompanied by a very nice elderly gentleman, Prof. Crusl. During the first piece I still felt a little tight, but the second I could hardly wait to start playing. The reviews are here today; I quote them verbatim: Bautzner Tageblatt "The talented young Hans Lewy did the ... "Devils Trills" sonata by Tartini. The youthful musician amazed the audience with his splendid technique. Doublestopping, octaves and staccato don't seem to give him any trouble. That he also has a mature understanding of the character of the music was apparent when he played Bach's "Chaconne" for violin only. The audience thanked him with thundering applause." Bautzner Nachrichten "The youthful violinist, Hans Lewy, from Breslau, a very promising high school student there, played the "Chaconne" by Bach and the "Devils Trills" sonata by Tartini with admirable precision and technical expertise." Actually the teenager was allowed another performance before the public; it took place also in Bautzen, in October of the same year. This time he was soloist for Mozart's D-sharp violin concerto. The reviews were, to say the least, enthusiastic. Here are excerpts, in translation, from two of the reviews: Bautzner Tageblatt "The spirit and solid technique so important in Mozart's works were present. The bowing is light and elegant, and, on the other hand, does not lack energy. The left hand is excellently schooled and establishes full success in passage-work in all ranges. Nevertheless, Mozart's splendid work was interpreted with classic poise and superior precision, and the youthful violinist received rich applause that came from the heart." The Bautzner Nachrichten review hailed him as a "Virtuoso " and "Wunderkind with true Mozartian charm" and predicted for him a great future as a musician. After these successes, the teenaged Lewy, of course, had to consider a career as a professional musician. The choice would have been easy if it had not been for that other passion, mathematics. His father favored mathematics; his mother was not sure; and he, of course, wanted both. In the end, mathematics won, and he soon left home for his studies in Gottingen. There he played with the city orchestra and, during all his years as student and Privatdozent, participated in weekly musical evenings at the home of Helene

271 Wintgen, the wife of a professor of Chemistry, and a friend and contemporary of Nina Courant. Frau Wintgen, as an octogenarian, described for me her first meeting with my husband. The quartet needed "a violin" and a newly arrived student had been recommended. It was raining heavily that evening when the new student knocked on her door, and "there he was", she said, smiling, "the 17-year-old Hans, standing there forlorn and drenched: he wore no hat, no raincoat, and carried no umbrella but the violin was carefully wrapped in a towel!" After his move to the U.S., Lewy's instrument of preference gradually became the piano. (It is my impression that the making of "salon music" in the Gottingen of Lewy's time, was very informal-improvisational, rollicking, if you will. In those days, the teaching of music in Germany placed more emphasis on sight-reading than was customary in the U.S., and perhaps this had something to do with the "freewheeling" character of the musical evenings in Gottingen described by my husband . . . . At any rate, from the stories that reached my ears, it was evident that the players in his circle enjoyed lustily, not only the challenges and the riches of the music - but also each others' company and wit - and, at times, even some nonsense; I remember my husband telling of one "session" that included a very accomplished Russian violist, and a good friend. They were about to begin playing, when his Russian friend put down his bow and said solemnly: "Let's try it first without sharps and flats!") But had mathematics won? That is, had music lost? My husband said later in his life that he had never regretted his decision. He made music almost every day of his life, (exceptions were extended foreign stays when no piano was at hand). But his seminal decision had made it possible for him to make music, and live, as a free spirit, - free of the constraints of a concert performer's life, especially, he told me more than once, the constant travelling, and the uprooted life style . . . Mathematics and music remained the twin passions of his life, and, in the end, I believe he felt he had been lucky enough to have had the best of both worlds.

272

3. Crises in Mathematics by Hans Lewy

Your Magnificence, Colleagues of the Science Faculty, Ladies and Gentlemen, I am very grateful to you for the honor you have conferred to me. The title of my lecture is "Crises in Mathematics". I will not define the word "crises"; I hope to make clear what is the point in question. Our century has seen scientific discoveries in abundance, and in many cases mathematics provided essential assistance. Often these discoveries have overthrown older notions and conceptions and were replaced by new notions, expressing them in a mathematical way in order to formulate them as incontestable as possible. For, one may think, mathematics expresses eternal truths since it only states the logical consequences of a few principles, and these principles or axioms are of such simplicity and immediacy that to deny them would only lead to fruitless sophistry. But is mathematics truly always the "resting pole admidst the whirl of phenomena", or have there been bitter differences of opinion also in mathematics? Do they exist even today? Are there, possibly unspoken, opinions underlying our mathematical thinking, views which cannot be defended by logic alone? The first attempt towards an axiomatization of mathematics goes back to the ancient Greeks and is presented in Euclid's books, which were thought to be basic for almost two thousand years. Why? It is known that many mathematicians of the previous generations had studied in Babylon where they had learned their algebra. In Babylon one considered as numbers the so-called natural numbers; 1, 2, 3, etc., and the fractions of these as quantities which we nowadays denote as positive rational numbers. But then came the discovery that there are geometric quantities which are not fractions of integers. Namely, in a rectangular triangle with two sides of length one, the third side has a length which is not a fraction of integers. For this fact we say today that the square-root of two is irrational. How exciting this discovery must have been for the natural philosophers of that time who had learned to consider the natural numbers as a very successful foundation for all contemporary descriptions of nature! The echo of this crisis is reflected in Plato's utterance that a human being who is not shaken in his heart by learning that the square-root of two is not a rational number resembles a stupid beast. And undoubtedly the geometric expression of all axioms in Euclid's books can be attributed to the desire to give these axioms a generality which could not be achieved by the algebra of the time. Since Descartes, the modern conception takes the opposite way and reduces all geometric notions to algebraic ones. But the endeavor to formulate a sufficiently general concept of quantities algebraically requires the notion of the so-called actual infinity, thereby generating a deeper rooted crisis the offshoots of which have had an influence up to the recent past. Today we learn in school that each quantity is an infinite decimal fraction, English translation of Hans Lewy's lecture Krisen in der Mathematik at Bonn University on 28 November 1986 when Hans Lewy was awarded an honorary doctorate by the Science Faculty.

273 say 1.2357 . . . ad infinitum. This concept of quantities requires the specification of infinitely many numbers from 0 to 9. Such a definition would have been unbearable for Euclid. If he speaks of the infinity of the set of all prime numbers, that is, of all those numbers having no divisors except themselves and the unit, he expresses and proves this by: There exist more prime numbers than in any given set. The need to include infinity into mathematics became of immediate interest only with the invention of the infinitesimal calculus. The success of the infinitesimal calculus and its overwhelming fertility in solving old problems and creating new questions led in the 18th century to the well-known slogan: "En avant, toujours en avant, et la foi vous viendra." (Literally translated: "Forwards, always forwards, and belief will follow.") But was it really easy to believe? For example, the infinite series 1 — 1/2 + 1/3 — 1/4 + . . . converges to natural logarithm of 2. However, if one orders the terms of this series in a different way, then one can obtain any positive or negative number as sum, or we can let the series diverge, just as we wish. This example must have made a devastating impression upon the contemporaries. In fact, the foundation of the infinitesimal calculus was, even in the 19th century, a topic of vivid discussion, and this not only in mathematical circles. So for instance Karl Marx wrote an essay about the foundation of the infinitesimal calculus, and his concept of this field has been taught in the schools of the Soviet Union and of China up to our time. I myself witnessed the lecture of a Russian mathematician entitled "Marx's foundation of the calculus" held in the course of the International Congress of Mathematicians in Zurich, 1932. The inclusion of this lecture of 45 minutes in the program of the meeting was a precondition for the participation of the Russian colleagues in the congress. The speaker characterized Weierstrass's e — (5-definition of convergence as a static and bourgeois concept, whereas Marx's definition was seen to be something fluent and vivid. Weierstrass's concept of convergence rested on preparatory work by Cauchy and Abel and allowed a clarification of all seeming contradictions of the infinitesimal calculus. But this long desired goal had scarcely been attained when in the secondhalf of the 19th century there arose a completely new conception of infinity in the set theory of Georg Cantor. Cantor introduced the notion of denumerability of a set and showed that there are infinite sets which are not denumberable, i.e. sets where it is impossible to furnish each of its elements with a natural number in such a way that two different elements obtain different numbers. The simplest example is the set of all decimals between zero and one. For us, having grown up with the notions of denumerability and nondenumerability of infinite sets, it is hard to understand any longer why so many outstanding mathematicians of those days fought with such hostility against these new concepts. Weierstrass's point of view is interesting: The famous Gottingen mathematician David Hilbert, at that time a young man, visited Weierstrass in Berlin and asked him for his opinion about the new set theory. Weierstrass told him that he would accept this theory as soon as it would be able to prove the existence of transcendental numbers, that is, the existence of numbers which do not satisfy any algebraic equation with integers as coefficients. However, this was proved by Cantor in the following year, actually by verifying a stronger result:

274 The set of transcendental numbers is not only infinite, but even nondenumerable. Further successful applications of the set theory to other branches of mathematics which previously seemed to have nothing to do with it, secured a firm place for set theory in mathematics. But with the new attitude against the actual infinity new clouds arose in the firmament of mathematics. Certain paradoxes such as the set of all sets containing themselves showed that even the axioms of logic have to be revised if infinite sets are taken into consideration. In particular the so-called axiom of choice by Zermelo in its unrestricted application to infinite sets made many mathematicians feel ill at ease. A radical cure for all doubts was suggested by so-called intuitionism. Essentially, intuitionism expelled all statements from mathematics containing notions which cannot be defined in a finite number of steps. Particularly, for intuitionists the principle of the tertium non datur is not applicable to infinite sets. For example, let us take the statement: In the decimal representation of it = 3.1415... there is an uninterrupted sequence of ten Nines. The intuitionist says: As long as there is no finite proof of this statement or of its reverse, one is not allowed to say that this statement be either correct or false. This was the example cited by the well-known topologist L.E.J. Brouwer in a great lecture at Gottingen, about 1930. According to Brouwer most of the mathematics taught at the time was false. In the discussion following his lecture he was asked as to whether this statement should also be extended to his own papers that had made him famous. Yes, he answered, it's all false. The majority of mathematicians rejected intuitionism, mainly for the reason that intuitionism was not only extensively restricting the area of allowed questions, but would also make all proofs uncommonly complicated, even in those cases where it led to the same results as ordinary mathematics. Still one has to admit that the problem of consistency within mathematics is, in the broadest sense, unsolved. Intuitionists would cripple mathematics in order to make sure that there is no contradiction. In any case, logicians had step by step erected a building of mathematics, beginning with the natural numbers, when in the thirties Goedel's discovery struck like a lightening bolt that, expressed briefly, for any given finite system of axioms there is always an assertion formulable by means of the notions of this system whose correctness cannot be decided by these axioms alone, and also that the consistency cannot be proved with this system of axioms alone. Now I turn to another crisis in which present-day mathematics finds itself. We older mathematicians and the majority of our younger colleagues believe it to be absolutely unavoidable to convince ourselves personally of the correctness of the results of others if we want to use them. But today there exist published papers based on results whose examination would require the study of more than 5,000 printed pages, a task which is admittedly beyond human capability. In all experimental sciences it is of course impossible to check personally the validity of all experiments preceding one's own research and forming its foundation. But should we mathematicians, so far having been proud to recognize no other authority than our own intellect, also resign ourselves to accept upon trust the printed word as

275 the truth? And here is another source of discomfort - the reliance placed on the computer. A celebrated, more than a hundred years old problem is the so-called four-color-problem. This is the question: whether it is always possible to paint a globe with many countries by only four colors in such a way that any two countries with a common border line have different colors. Two American mathematicians have reduced this problem to another one, approachable by the computer, and after the work of many hours the computer had decided: Yes, it is possible. But do I have to accept it thereby to be proved that four colors suffice? What is a mathematical proof? The answer to this question contains a personal element, even if one rejects placing trust in the computer and the acceptance of unchecked results. I know of a paper which was published prior to the invention of the computer, and whose aim was to prove a certain conjecture by Leonhard Euler. This paper makes a distinction of several hundred cases and investigates each case separately, thereby arriving at the desired result. Even if I had succeeded in examining all these cases, I would not take this as a proof of the correctness of this assertion, but only as a confirmation of my inability to discover a mistake. It seems to me that one has to ask more of a proof, namely, an understanding of the connections which underlie the truth of an assertion. The nonmathematician envies us for the incontestability of our results but the questions of conscience may well escape his notice that, as in the past, so also today have contributed to a crisis in mathematics.

276 4. Further Reading

1. C. Reid, Courant in Gottingen and New York (Springer, 1976). 2. M. Protter, J.L. Kelley, T. Kato and D.H. Lehmer, in Memoriam Hans Lewy (University of California Obituary 1988). 3. C. Reid, Hans Lewy, in More Mathematical People, eds. D.J. Albers, G.L. Alexanderson and C. Reid (Harcourt Brace Jovanovich Publ., 1990), pp. 180-194. 4. C. Reid, Hans Lewy. 1904-1988, in Miscellanea Mathematica, eds. P. Hilton, F. Hirzebruch and R. Remmert (Springer, 1991), pp. 259-267. 5. R.E. Rider, An opportune time: Griffith C. Evans and Mathematics at Berkeley, in A Century of Mathematics in America, Part II, ed. P. Duran (AMS, 1989). 6. S. Hildebrandt, F. Hirzebruch, F. John, R. Leis, H. Lewy and J. Tits, Mathematische Betrachtungen, Bonner Akademische Reden, Nr. 68 (Bouvier Verlag, 1988).

277 5. Bibliography by David Kinderlehrer

1925 [1] Uber einen Ansatz zur numerischen Losung von Nachrichten Ges. Wiss. Gottingen, 18-21.

Randwertproblemen,

1927 [2] Verallgemeinerung der Riemannschen Methode auf mehr Dimensionen, Nachrichten Ges. Wiss. Gottingen, 118-123. [3] Uber den analytischen Charakter der Losungen elliptischer Differentialgleichungen, Nachrichten Ges. Wiss. Gottingen, 178-186. [4] Uber die Methode der Differenzengleichungen zur Losung von Variations- und Randwertproblemen, Math. Ann. 98, 107-124. [5] Uber das Anfangswertproblem bei einer hyperbolischen nichtlinearen partiellen Differentialgleichung zweiter Ordnung mit zwei unabhangigen Veranderlichen, Math. Ann. 98, 179-191. [6] (with K. Friedrichs) Uber die Eindeutigkeit und das Abhangigkeitsgebiet der Losungen beim Anfangswertproblem linearer hyperbolischer Differentialgleichungen, Math. Ann. 98, 192-204. 1928 [7] (with R. Courant and K. Friedrichs) Uber die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann. 100, 32-74. [8] (with K. Friedrichs) Das Anfangswertproblem einer beliebigen nichtlinearen hyperbolischen Differentialgleichung beliebiger Ordnung in zwei Variablen. Existenz, Eindeutigkeit und Abhangigkeitsbereich der Losung, Math. Ann. 99, 200-221. 1929 [9] Neuer Beweis des analytischen Charakters der Losungen elliptischer Differentialgleichungen, Math. Ann. 101, 609-619. 1930 [10] Sulla unicita della soluzione del problema di Cauchy per un'equazione ellitica del secondo ordine in due variabili, Rend. Accad. Lincei, XI.6, 162-165.

To be published in Collected Works of Hans Lewy (Birkhauser).

278 1931 [11] Eindeutigkeit der Losung des Anfangswertproblems einer elliptischen Differentialgleichung zweiter Ordnung in zwei Veranderlichen, Math. Ann. 104, 325 339. 1932 [12] (with K. Friedrichs) Uber fortsetzbare Anfangsbedingungen hyperbolischer Differentialgleichungen in drei Verander lichen, Ges. Wiss. Gottingen, Math.-Phys. Klasse., 135-143. 1935 [13] (with C.R. Adams) On convergence in length, Duke Math. J. 1, 19-26. [14] A priori limitations for solutions of the Monge-Ampere equations, Trans. A.M.S. 37, 417-434. 1936 [15] Generalized integrals and differential equations, Proc. Nat. Acad. Sci. 22, 377-381. [16] On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. A.M.S. 42, 689-692. 1937 [17] A priori limitations for solutions of the Monge-Ampere equations II, Trans. A.M.S. 41, 365-374. 1938 [18] On the existence of a closed convex surface realizing a given Riemannian metric, Proc. Nat. Acad. Sci. 24, 104-106. [19] On differential geometry in the large I (Minkowski's problem), Trans. A.M.S. 43, 258-270. [20] Generalized integrals and differential equations, Trans. A.M.S. 43, 437-464. [21] A property of spherical harmonics, Amer. J. Math. 60, 555-560. 1939 [22] Aspects of the Calculus of Variations (University of California Press, 96 pp.), notes by J.W. Green.

279 1946 [23] Water waves on sloping beaches, Bull A.M.S. 52, 737-775. 1948 [24] (with K. Friedrichs) Solution of the dock problem, Comm. Pure Appl. Math. 1, 135-148. [25] On the convergence of solutions of difference equations, Studies and Essays. Courant Anniversary Volume, 211-214. 1950 [26] Developments at the confluence of analytic boundary conditions, in Int. Cong. Mathematicians, Vol. 1, 601-605. [27] Developments at the confluence of analytic boundary conditions, in Univ. Calif. Publications New Series Vol. 1, 247-280. 1951 [28] On the boundary behavior of minimal surfaces, Proc. Nat. Acad. Sci. 37, 103-110. [29] On minimal surfaces with partially free boundary, Comm. Pure Appl. Math. 4, 1-13. 1952 [30] A note on harmonic functions and a hydrodynamical application, Proc. A.M.S. 3, 111-113. [31] A theory of terminals and the reflection laws of partial differential equations, Technical Report. [32] Asymptotic developments at the confluence of boundary conditions, Nat. Bureau Standards, Ser. 18, 255-256. [33] (with P. Garabedian and M. Schiffer) Axially symmetric cavitational flow, Ann. Math. 56, 560-602. [34] On steady free surface flow in a gravity field, Comm. Pure Appl. Math. 5, 413-414. 1955 [35] Extension of Huyghens's principle to the ultrahyperbolic equation, Ann. Mat. Pura Appl. (4) 39, 63-64.

280 1956 [36] On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables, Ann. Math. 64, 514-522. [37] On the relations governing the boundary values of analytic functions of two complex variables, Coram. Pure Appl. Math. 9, 295-297. 1957 [38] An example of a smooth linear partial differential equation without solution, Ann. Math. 66, 155-158. [39] On linear difference-differential equations with constant coefficients, J. Math. Mech. 6, 91-108. 1958 [40] Introduction, in the Collected Works of Bernhard Riemann, 8 pp. (Dover). 1959 [41] Composition of solutions of linear partial differential equations in two independent variables, J. Math. Mech. 8, 185-192. [42] On the reflection laws of second order differential equations in two independent variables, Bull. A.M.S. 65, 37-58. 1960 [43] On hulls of holomorphy, Coram. Pure Appl. Math. 13, 587-591. 1961 [44] Atypical partial differential equations, in Partial Differential Equations and Continuum Mechanics, ed. Rudolph E. Langer (Univ. of Wisconsin Press), 171-175. [45] (with R.S. Lehman) Uniqueness of water waves on a sloping beach, Coram. Pure Appl. Math. 14, 521-546. 1964 [46] On the analytic continuation of minimal surfaces and similar problems, in Proceedings of the Fourth All Union Mathematics Congress, Vol. 2, 233-236 (in Russian). [47] Sul prolungamento delle funzione armoniche, Rend. Sem. Mat. Torino 24, 27-29.

281 1965 [48] On the definiteness of quadratic forms which obey conditions of symmetry, Ann. Scuola Normale Superiore Pisa 19, 577-591. [49] On the extension of harmonic functions in three variables, J. Math. Mech. 14, 925-928. [50] The wave equation as limit of hyperbolic equations of higher order, Comm. Pure Appl. Math. 17, 5-16. 1966 [51] Sulla riflessione delle funzione armoniche di tre variabili. Convegno sulle Equazioni alle Derivate Parziali, (Nervi). 1968 [52] On a variational problem with inequalities on the boundary, J. Math. Mech. 17, 861-864. [53] On the nonvanishing of the Jacobian of a homeomorphism by harmonic gradients, Ann. Math. 88, 518-529. 1969 [54] About the Hessian of a spherical harmonic, Amer. J. Math. 9 1 , 505-507. [55] (with Guido Stampacchia) On the regularity of the solution of a variational inequality, Comm. Pure Appl. Math. 22, 153-188. 1970 [56] On a minimum problem for superharmonic functions, in Proc. Int. Conf. on Functional Analysis and Related Topics (Univ. of Tokyo Press), 283-284. [57] On a refinement of Evans' law in potential theory, Rend Accad. Naz. Lincei 48, 1-9. [58] On the partial regularity of certain superharmonics, in Studies and Essays presented to Yu-Why Chen on His Sixtieth Birthday, 169-171. [59] (with Guido Stampacchia) On the smoothness of superharmonics which solve a minimum problem, J. d'Anal. Math. 23, 227-236. 1971 [60] Obobscenie teoremy o prostranstvennom ugle (Generalization of the spatial angle) (translated from the English by Ju. V. Egorov), Usp. Mat. Nauk 26, 199-204. [61] (with Guido Stampacchia) On existence and smoothness of solutions of some non-coercive variational inequalities, Arch. Rational Mech. Anal. 4 1 , 241-252.

282 1972 [62] On the coincidence set in variational inequalities, J. Diff. Geom. 6, 497-501. 1975 [63] On the nature of the boundary separating two domains with different regimes, Rend. Accad. Naz. Lincei 217, 181-188. 1976 [64] On analyticity in homogeneous first order partial differential equations, Ann. Scuola Norm. Sup. Pisa 4, 719-723. 1977 [65] On the boundary behavior of holomorphic mappings, Accad. Naz. Lincei. Contributi del Centro Linceo Interdisciplinare di Scienze Matematiche e Loro Applicazioni, No. 35, 1-5. [66] On the minimum number of domains in which the nodal lines of spherical harmonics divide the sphere, Comm. P.D.E. 2, 1233-1244. 1978 [67] On the relation between analyticity in one and in several complex variables, in Complex Analysis and its Applications (Russian). 1979 [68] An inversion of the obstacle problem and its explicit solution, Ann. Scuola Norm. Sup. 6, 561-571. [69] Expansion of solutions of 't Hooft's equation. A study in the confluence of analytic boundary conditions, Manuscripta Math. 26, 411-421. 1980 [70] On conjugate solutions of certain partial differential equations, Coram. Pure Appl. Math. 33, 441-445. 1981 [71] Uber die Darstellung ebener Kurven mit Doppelpunkten. Nachrichten der Akad. der Wiss. Gottingen, Mathematisch-Physikalische Klasse Nr. 4, 109-130.

283 1985 [72] (with Tang Zhiyuan) On free boundary problems in two dimensions, Rend. Circ. Mat. Palermo (Ser. II) 34, 325-336. 1987 [73] (with U. Dierkes and S. Hildebrandt) On the analyticity of minimal surfaces at movable boundaries of prescribed length, J. Reine Angew. Math. 379,100-114. 1989 [74] On atypical variational problems, in Partial Diff. Equ. and Calc. of Variations. Essays in Honor of Ennio De Giorgi, eds. F. Colombini, A. Mariano, L. Modica and S. Spagnolo (Birkhauser), 697-702.

6. Publications Dedicated t o Hans Lewy

1. In 1978 the Scuola Normale Superiore di Pisa published a collection of 27 papers dedicated to Hans Lewy which has appeared in the Annali della Scuola Norm. Sup. di Pisa in the same year. The title of the collection is: Raccolta degli scritti dedicati a Hans Lewy. 2. Manuscripta mathematica (Vol. 28, Fasc. 1-3, 1978) published 15 papers dedicated to Hans Lewy and C.B. Morrey. (The drawing of Hans Lewy by his wife Helen is taken from this issue of Manuscripta Math., frontispiece.) 3. In 1986 a conference on Calculus of Variations and Partial Differential Equations was held in Trento in Honor of Hans Lewy, and 23 papers covering the lectures of this conference are published in Vol. 1340 of Lecture Notes in Mathematics, eds. S. Hildebrandt, D. Kinderlehrer and M. Miranda (Springer, 1988).

Neuer Beweis des analytischen Charakters der Losungen elliptischer Differentialgleichungen*). Von

Hans Lewy in Gottingen.

In einer Note in den Gottinger Nachrichten 1927 Heft 2 habe ich einen neuen Beweis des analytischen Charakters der Losungen elliptischer analytischer Differentialgleichungen in einem Spezialfall ausgefiihrt. Hier folgt der Beweis des allgemeinen Theorems fiir Differentialgleichungen zweiter Ordnung in zwei unabhangigen Variablen, der wie jener darauf beruht, die vorgelegte Losung u einer elliptischen analytischen DifEerentialgleichung zunachst fiir komplexe Werte der unabhangigen Variablen x und y zu ermitteln. Diese Fortsetzung ins Komplexe wird hier aufgefaBt als Losung eines Anfangswertproblems eines „kanonisch-hyperbolischen Systems" von Differentialgleichungen, d. h. eines solchen, wie es z. B. auch zur Losung des Anfangswertproblems einer hyperbolischen Differentialgleichung verwendet wird. Es wird sodann nachgewiesen, dafi in einem vierdimensionalen Gebiet der Variablen x1,x^,yl,y.i (Real- und Imaginarteile von x und y), das das reelle Ebenenstiick x., = 0, y.2 = 0, in welchem die Losung u urspriinglich vorliegt, im Inneren enthalt, die CauchyRiemannschen Differentialgleichungen beziiglich x1, a;3 und bezuglich yx, y„ erfiillt sind. Das bedeutet aber, daB u analytisch von x und y in jenem Gebiet, also auch schon in seinem urspriinglichen Definitionsbereich abhangt. §1Charakteristisclie Differentialgleichungen und Fortsetzung ins Komplexe. Sei F(x, y,u,p,q,

r,s, t) = 0

eine analytisch von den Argumenten abhangige DifEerentialgleichung zweiter J

) Beziiglich der alteren Beweise sei auf den Enzyldopadiebericht von Lichtenstein 2 C 12, S. 1320 und auf die jiingst erschienenen Arbeiten von T. Rado, Math. Zeitschr. 25, S. 514 ff., und S. Bernstein, Math. Zeitschr. 'M, S. 330 ff., verwiesen. Mathematische Annaien. 101.

H. Lewy.

610 Ordnung, in der P = ux,

? = «„, r = uxx,

s = uxy,

t = uyy

wie iiblich zur Abkiirzung gesetzt sind. Fiir eine vorgelegte Losung u sei die DifEerentialgleichung in einem Gebiet der x, y-Ebene vom elliptischen Typus, d. h. es sei dort or

dt

\dsj

woraus insbesondere — 4= 0, ^— 4= 0 folgt. Man verstehe nun unter den dr ' ' at ' ° stetigen Funktionen Q1(X, y) und Qi{x, y) die beiden sicher verschiedenen komplexen Wurzeln der Gleichung dF 2 a dF . dF n

^Q -^Q

+ -gI = °->

unter den Differentiationsoperatoren ' bzw. ' die folgenden linearen Kombinationen der Symbole — und — : J

dx

i

dy

£._(_„ J L

»

? i

~dx~f~Qldy'

JL

~dx^~Q*cy'

Dann folgt y' —

Q±X'

= 0

und

i/ — £>„ x' = 0.

Aus dem Bestehen dieser Gleichungen folgen jetzt in bekannter Weise rein algebraisch noch weitere charakterische Differentialgleichungen fiir die GroBen x, y, u,p, q, r, s, t. Diese sowohl wie die beiden schon genannten y'—gix'=0 und y —Q„ #' = 0 lassen sich nach geeigneter Auswahl in die folgende Form setzen: U^kVl^O

(» = 1 , 2 , . . . , 6 ) ,

iait
(.- = 7,8),

:i) 1=1

in der die 2, ...,
Analytischer Charakter der Losungen elliptisoher Diflerentialgleichungen.

611

den nach der Voraussetzung des elliptischen Charakters erfiillten Ungleichungen * dr dt

\ds)

+

U )

dr

+ U

'

dt

+ U

-

Um die in der Einleitung angekiindigte Fortsetzung ins Komplexe auszufuhren, werden wir alle Groflen cp als Funktionen von vier unabhangigen Variablen |,,f 9 , Vi> Vi angeben, die wir so wahlen, daB fiir f3 = rjz = 0 die Ebene | x , rj1 mit der x, y-Ebene zusammenfallt, d. h. so, daB * ( f l , 0 , i 7 1 , 0 ) = 9P 1 (| 1 , 0 , ^ , 0 )

=^,

y(Slt 0, Vi> ° ) = V. (fi. °> fi» ° ) = >?i ist. Dadurch sind in der Ebene £2 = »?2 = 0 die samtlichen 99, die vorher Funktionen von x, y waren, nun als dieselben Funktionen von S± und rjl bekannt. Um nun diese
2aik
(» = 7 , 8 ) ,

t

geben aber den Operatoren nun die Bedeutung 3 )

Man hat hier ein Anfangswertproblem fiir Funktionen der beiden unabhangigen Variablen £lt ?;2 vor sich, dessen Losbarkeit von mir in der zitierten Arbeit iiber das hyperbolische Anfangswertproblem bewiesen wurde. Allerdings werden bei dieser Fortsetzung die Funktionen cp moglicherweise komplex werden, da schon fiir £, = >/3 = 0 nicht alle aik reell sind. Die Bedeutung der ailc fiir komplexe Werte der cp anzugeben, hat keine Schwierigkeit, da die aik analytisch von den cp abhangen; wir erklaren die aik fiir komplexe Werte der cp einfach als analytische Fortsetzungen der 3 ) Wesentlich an dieser Festsetzung der Bedeutung der Operatoren ist nur, daB sie Differentiationen in zwei versehiedenen Bichtungen der fl( rj2-Ebene bedeuten und daB keine dieser Richtungen parallel zur „Anfangsgeraden" »?2 = 0 ist.

H. Lewy.

612

aiu fur reelle Werte der
) H. Lewy, loc. cit. I m folgenden werden noch die folgenden Eigenschaften v o n Losungen eines solchen „kanoniseh hyperbolischen" Systems benutzt, die ich hier zusammenstellen m o c h t e : 1. Man kann eine etwa aus zwei kongruenten Trapezen (vgl. Figur) bestehende beiderseitige Umgebung des Geradenstiickes »;2 = konst. angeben, i n d e r die Funktionen cp existieren u n d d e n obigen Differential/ S. gleichungen gehorchen; die GroCe der Trapeze hangt KrVt^nst/

/

\t*r,fkonst.

°

6

6

'

/

6

\7 /^i-V!Hl"'st-

dabei, w a s die Anfangswerte betnfft, n u r a b dtp ^r"%"Jmi^\ 2 «) v o n oberen Schranken der - — ,t —— auf dem 0Vi herausgegriffenen Geradenstiick; ft) v o n oberen u n d unteren Schranken fiir die cp,, ?>, selbst; y) v o n einer unteren Schranke fiir den Abstand d e r durch die Anfangswerte
) Denn sonst wiirde ein m i t d e r neuen Matrix gebildetes linear homogenes Gleichungssystem eine nicht triviale Losung besitzen, also auch ein linear homogenes Gleichungssystem mit der alten Matrix, was unmoglich ist, weil dessen Determinante nicht verscliwindet.

Analytischer Charakter der Losungen elliptischer Differentialgleichungen.

613

Das Verfahren der Fortsetzung, das soeben fur ein Geradenstiick rj1 = konst. der Ebene fa = tj9 = 0 als Anfangskurve beschrieben wurde, wende man nun auf alle parallelen Geradenstiicke rjx = konst. dieser Ebene, soweit sie in den Definitionsbereich der Losung u fallen, an. Man bestimmt so die Funktionen

r\x, i}„Raumes, namlich einem solchen, in dem | 9 = 0 gilt. Dies Gebiet enthalt das Ebenenstiick der Ebene f3 = »?9 = 0 im Inneren. Nunmehr geht man von den samtlichen Geraden £x = konst., rj2 = konst. des soeben betrachteten dreidimensionalen Gebietes | ? = 0 des f1} f4, i]1, rj^Raumes aus und setzt die dort bekannten Funktionen tit
(* = 1 , . . . , 6),

k

Uaik
(» = 7,8)

k

benutzt, jetzt aber den Operatoren die folgende Bedeutung gibt: , _

l**\

_d

d_

' =+ — + — -

" '

^ dg, ^ dVl §2. Riickkehr zu den alten Yariablen. Ein Hauptpunkt der folgenden Untersuchung ist der Nachweis, daB man anstatt £x, £>, TJ1, J;9 auch xx, a;3, yx, y2 als unabhangige Variabe einfiihren kann, wenn man sich auf eine hinreichend kleine vierdimensionale Umgebung eines Punktes x, y des urspriinglichen Definitionsbereiches von « beschrankt, was wir natiirlich tun wollen. Es geniigt dazu wegen der StetigkeitT), das Nichtverschwinden der Determinante

in dem betrefEenden Punkte der „Anfangsebene" nachzuweisen. Da dort die £x, f2, rj1, tj„ bzw. mit den xx, or3, yx, y„ iibereinstimmen, hat man dort die Gleichungen fur acht von den in Frage kommenden Ableitungen:

(2)

af,

l

'

££i = n Bit ' •) Vgl. die FuBnote 4). ') Vgl. Anna. *}.

dSi~u'

dgl-v'

i^ —o sVl~v>

^i_i dVl~ '

d^-"' ?y* — (\ dVl~"-

614

H. Lewy.

Spaltet m a n weiterhin die Funktionen Q1 u n d e 2 , vgl. S. 610, in dem betreffenden P u n k t e nach Real- u n d Imaginarteil, also Qi=ai+

e» = ai — * ai

i °2>

o, + 0,

so liefern die Gleichungen y'— Q1X'=Q u n d «/' — Q^X' = 0 in den verschiedenen Bedeutungen (*) u n d (**) der Differentiationsoperatoren die folgenden vier Gleichungen (unter Benutzung von (2))

4* + t - | * _ ( o + 1 - 0 a ) ( (3) 3%

dyi;

i + | * + ,•*&) = 0 ,

h - M - ' + S + 'S)- 0 '

dj? a

+ ••#-1-(«. " ^ + l '••». "") U® f , '+' 9'ft) * . = «.

3fa^*3fa

(4)

8fi Die ersten beiden liefern durch Subtraktion

2

dXs

«i + 2 »«.(l5 + » 8*13

0*, , . aa;2

. a,,

3xt

"T" l ~~ > oa

"5 T * "5— = 3i?2 9i? 4

"5— a^

0;

:

o, dx.,

al

Weiter erhalt m a n d

V\ _J_ i 1^? _ ( g i-r*' g a)( g « + *'gi)

»(g? + g|). oa '

*?_! — $• 8T]2 '

8y

* = 3<7a

"{+"•' a2

Ahnlich entsteht aus den Gleichungen ( 4 ) durch Subtraktion as,

1

c| 8

woraus dann <>Vi _i_ v 9% di-i o$3

-, , (Qi + tq a )-t _ • < » ! . 3?! _ oa <j3 3fa

n

3ya _ o, of,
folgt. Die Matrix der Ableitungen der x1 x„ yx y^ nach den | t f 9 ^ J?2 sieht also in dem ins Auge gefafiten P u n k t e der Anfangsebene | 9 = »?9 = 0 so aus: *1

X.,

2*1

y-i

f,

1

0

0

0

f-2

0

Vi

0

Vt

0

1 g

0

2

0

1

o.3

0

0 °2

°3

291 Analytischer Charakter der Losungen elliptischer Difierentialgleichungen. 6 1 5

Wie man sieht, ist die Determinante ungleich null, namlich gleich 1; man kann also in einer vierdimensionalen Umgebung des betreffenden Punktes der x, y-Ebene die x1, x2, y1, t/2 als neue Koordinaten einfuhren. Allgemein werden die DifFerentiationssymbole nach £±, fa, r}x, rj2 lineare Kombinationen derer nach xx, x 2 , yx, y2. Schreibt man

m wird .va ( i

+ 5 L)(, 1 +

, - j , , ) _ 8 l ( ^ + jL)( I l + ,-«,)-0 wegea

der Tatsache, daB der Operator

\- i —— null ergibt, wenn er auf eine

analytische Funktion von x1-\-ix2

(hier x1-{-ix2

selbst) ausgeiibt wird,

und wegen der entsprechenden Tatsache fur den Operator B1-glA1

f- i —

= 0,

und Ax ist ungleich null, weil sonst der Operator (5) nicht reell sein kann, aufier wenn Cx und Dx auch verschwinden; das ist aber unmoglich, weil nicht -r— = — —- ist. Man findet demnach (6)

-^ + ~ = A1^ + QlA1^-

+ Cx(^- + i^-)+D1(^

+ i^-),

und ahnlich

(7) K

'

-Jz+&-±A+^k+
c?i

"3*i

9

a

^2/i

- \5x t '

3x s /

3

\a2/i

cz/.2

Alle diese GroBen Ax, Gx, Dx, ..., H„ sind nichts weiter als lineare Kombinationen der Differentialquotienten der xx, x.,, yx, t/a nach den fi; £2, rjx, J?2; z. B. ist offenbar

*C,-( ir + Ah-

iD

>~( £ + £)*•

616

H. Lewy.

und es ist auBerdem wichtig, nochmals hervorzuheben, daB A1} A2, Et, E.2 samtlich ungleich null sind. Fiir dasFolgende wesentlich sind dieWerte der C,, Dt, C 3 , Z>3, Gx, G„ Hx, H2 in der Anfangsebene x<1 = yi = 0. Man hat dort unter Benutzung unserer Differentiationstafel 1

°2

°2

°2

(10)

°2

•». = ^ '

°2

»'»= = t -

i0

* = T> °2

§3. In der Anfangsebene gelten nun die charakteristischen Differentialgleichungen (1) in dreierlei Form, entsprechend den drei Bedeutungen des Operators ' bzw. ' . Die erste auf S. 610 erwahnte Bedeutung von ' bzw.' lautete

' = J-_L

J_

, __ J_ ,

g

die zweite war die von (*), die dritte die von (**). Man subtrahiere nun von dem System (1) in der zweiten Bedeutung das mit Ax bzw. At multiplizierte der ersten Bedeutung und erhalt so unter Beriicksichtigung von (6) und (7) und von (10) 8

wobei der Index i von 1 bis 8 lauft. Aus dem Nichtverschwinden der Determinante \aik[ in der Anfangsebene folgt nunmehr

(11)

{«(J- + iJJ)+°J±?i(J-

v

i_ai\cxl

I

ox^j

a„

+

\dyx

iJl-Y\
(* = 1,...,8). v

Ahnlich folgert man, wenn man das charakteristische System in seiner ersten Bedeutung mit EL bzw. E.s multipliziert und von dem charakteristischen System in der dritten Bedeutung (**) abzieht und wiederum die Werte (10) der Koeffizienten Gx, G.2> Hx, Hx in der Anfangsebene heranzieht:

293 Analytischer Charakter der Losungen elliptischer Differentialgleichungen.

617

Fur jedes einzelne
a

? + g2

folgt. (Das darf nicht verwundern, denn die Bedeutung der Operatoren ' bzw. ' war natiirlich dementsprechend festzulegen.) Somit ist das Erfulltsein der Cauchy-Riemannschen Differentialgleichungen in der Anfangsebene x„ — y2 = 0 , oder um in der Sprache der £i>£i> Vi> Vi z u reden, in der Ebene f9 — r\„ = 0 gesichert. Wir weisen zunachst ihre Giiltigkeit in jenem dreidimensionalen Raumstiick f3 = 0 nach, in das wir zuerst vermoge der Bedeutung (*) der Operatoren ' bzw.' fortgesetzt haben. Wir setzen zur Abkiirzung

Wir dividieren die Gleichungen des charakteristischen Systems in der besagten Bedeutung bzw. durch At und A2, die ja beide nicht verschwinden (vgl. S. 615) und wenden auf das so umgeformte System die Operatoren Vx und Vy an. Da die aik analytisch von den q>1,...,q>s abhangen, ist

*,*» = 2j*r,

(In diesen Gleichungen wird in der komplexen Differenzierbarkeit der aik zum einzigen wesentlichen Male die analytische Abhangigkeit der Differentialgleichung F = 0 von ihren Argumenten benutzt.) Weiter ist beispielsweise 'Al

* \axl ^ ^1 dy^ i-

Ai

>x ~h Ai yy)
8

'

*Al

xP

«

'

x

fk

A l y ^cyL

ZJ

8 Ai

\yx
. . ^ vx

Vl.

dVlxV<-

H. Lewy.

618

Wir sehen also, daB die Anwendung des Operators Vx auf das durch At bzw. A^ dividierte Difierentialgleichungssystem zu dem folgenden Differentialgleichungssystem fiihrt: 2aik(Vx
(12)

k

+ ciliVy(pk) = 0

(»=1,...,6),

+ cikVy
(» = 7 , 8 ) .

k

Eine ahnliche Anwendung des Operators Vy bewirkt die Gleichungen 2««(*>*)'+ (13)

2{dikVx
Uaik(Vy
+ eikVy
(»•= 1, . . . . 6), (*=7,8).

k

Dieses System von 16 Differentialgleichungen fur die 16 Eunktionen Vxk Losungen eines Anfangswertproblems dieses Systems vor uns, namlich eines solchen, dessen Anfangswerte samtlich null sind. E i n e Losung dieses Problems wird wie aus der Homogenitat der Differentialgleichungen folgt. Aus dem fiir diese Systeme giiltigen Eindeutigkeitssatze 8 ) folgt nunmehr, daB es nur diese Losung gibt, also daB die Cauchy-Riemannschen Differentialgleichungen in dem betrachteten dreidimensionalen Gebiet gelten. Ganz ebenso beweist man nunmehr die Gleichungen

in dem zu betrachtenden vierdimensionalen Gebiet, indem man sich erinnert, daB die Fortsetzung der
) Ein solcher Eindeutigkeitssatz steht in meiner zitierten Abhandlung. Es laCt sich die Eindeutigkeit schon unter schwacheren Voraussetzungen als den dort angegebenen beweisen, wenn man die in der Theorie der gewohnlichen Differentialgleichungen ublichen Methoden benutzt; vgl. hierzu auch O. Perron, „tJber Existenz und Nichtexistenz von Integralen partieller Differentialgleichungen im reellen Gebiet", Math. Zeitschr. 27, S. 549 ff., sowie Haar, C. R. 187 (2.6. 1928), und meine anfangs zitierte Abhandlung in den Gottinger Nachrichten 1927, S. 184 ff. Hier liegen die Dinge wegen der Linearitat der Differentialgleichungen ubrigens besonders einfach.

Analytischer Charakter der Losungen elliptischer Differentialgleichungen.

619

§4. Zum SchluB noch ein Wort iiber die Differenzierbarkeitsvoraussetzungen, denen die vorgelegte Losung u(x,y) gehorchen mufi. Zur Fortsetzung in den vierdimensionalen Raum ist es hinreichend, die Funktionen

104

MATHEMATICS:

H.

LEWY

PROC. N . A.

S.

ON THE EXISTENCE OF A CLOSED CONVEX SURFACE REALIZING A GIVEN RIEMANNIAN METRIC B Y HANS LEWY DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA AT BERKELEY

Communicated January 14, 1938

Let (To be the surface of the unit sphere and (dso)2 the first fundamental form of Differential Geometry on a0. Denote by (ds ) 2 another given positive definite form defined at each point of
G(dvy.

VOL.

24, 1938

MATHEMATICS:

H.LEWY

105

Suppose that the Gaussian curvature K\ of the metric (ds{)2 is positive. Weyl 1 has attacked the following problem: Does there exist a closed convex surface a\ which may be mapped in a one-to-one way on
+ 2Fxdudv + Gx(dv)2

in the parameters (u, v) on 0. Take P as the ou ov center of the largest sphere inscribed to
106

MATHEMATICS:

H.LEWY

P R O C . N . A. S.

i) I Pxi |> I Pxi |> • • • a r e uniformly bounded. This is a consequence of Bonnet's theorem which states t h a t the diameter of a closed convex surface is bounded from above b y a n u m b e r depending only on a (positive) lower bound of t h e Gaussian c u r v a t u r e .

dpx,

dpx,

dpx,

ii) —— I, ——- ,i ——^ , ... are uniformly bounded. This follows from 5M ou bv bv 5M (rfpx)2 ^ (dsx)\ 52Px 5 2 p x /52px\2 iii) T h e coefficient of - — 2 -——2 — I —— J in Darboux's equation can be H ' 5M 5Z/ \budvJ assumed to b e = 1 while t h e other coefficients are bounded for complex d/0x 5px values of u, v, px, ——, —— in a bounded neighborhood of their values under ou ov consideration. T h e c o n t r a r y assumption is ruled o u t b y a simple geometrical consideration showing t h a t the radii of the m a x i m u m spheres inscribed in
5?

+

. 52M

5^

, [fbuV

= h

T\ba)

+

.

fdu\2~\

\W J

.

, (bu

+ K

\bab-a

bv

+

. bu

bv\

WW

4(0+(S)"]+*<

x

+

b(u, v) 5(a, ff)

S+g-4Cs),+Cs)"]+-+i*^-"' where /fix, thx, • • • , &ix, &2x> • • • depend on (u, v), b u t not explicitly on px bp\ bp\ nor — nor — . I n our case a + iff and a — i/3 are parameters on the bit bv asymptotic lines a n d our equations (1) are identical with equations a l r e a d y found b y Darboux. 5 1

Hermann Weyl, "Uber die Bestimmung einer geschlossen konvexen Fluche durcli ihr Linienelement." Vierteljahrsschrift der naturf. Ges. Zurich, 61, 40-72 (1915). 5 1 have been in possession of this proof for several years, cf. Bull. Am. Math. Soc, 42, 824 (1936). The preparatory investigations of Monge-Ampere equations were undertaken mainly for this purpose and published in Trans. Am. Math. Soc, 37, 417-434 (1935) and 41, 365-374 (1937). 3 G. Darboux, Theorie des surfaces, vol. 3, p. 260 (Paris, 1894). 1 Loc. eit., 41, 373 (1937); also Bull. Am. Math. Soc, 42, 689-692 (1936). 5 Loc. eit., p. 290.

VOL.

ON

37, 1951

THE

MATHEMATICS:

BOUNDARY

H.

BEHAVIOR

LEWY

OF MINIMAL

103

SURFACES

B Y HANS LEWY UNIVERSITY OF CALIFORNIA

Communicated by G. C. Evans, December 16, 1950

This paper deals with the analytic extension of a minimal surface across its boundary. LEMMA 1: Let U (x, y) be harmonic and absolutely bounded in the half plane y > 0. Assume U (x, y) to possess1 continuous boundary values U(x, 0) on y = 0 which admit limits Z7(°°, 0) and U{— °°, 0). Assume furthermore that the total variation from x = — <& to x = » o/ U(x, 0), denoted by T1* [U(x, 0)], is bounded. Denote similarly by Tb[U(x, y)]the total variation of U (x, y) as x ranges from a to b where a and b > a are any two numbers including ± °o, and y is kept constant. Then, for y > 0,

T1„[U{x,y)\

<

T!„[U(x,0)],

and lim Tba[U(x,y)] y — 0

= Tba[U(x, 0)].

104

MATHEMATICS:

Proof:

H. LEWY

PROC. N. A. S.

U(x, y) = U(P) is for y > 0 expressed by the formula U(P) = T-'SQ"-

-~f(Q)d6(PQ)

where 6{PQ) is the angle from the negative y-axis to the vector from P to Q, and f(Q) = U(Q) for Q on the x-axis. Integration by parts yields U{P) = - * - ! fe(PQ)df(Q)

+ (/(co) _ / ( - » ) ) / 2 .

(1)

Differentiating with respect to a; we find (b/dx)U(x,

y) = T - ^ - £

0(PQ)df{Q),

(2)

which we also write with 0r instead of {b/bx)0(PQ) and where we note that — 8X is positive. Thus Tlm[U(x,y)] Now S-"

= y _ - . | d £ / ( * , y ) | < f f'm

- 0,
(3)

— Oxdx = x/2 — (— x/2) = x. Hence the desired result Tlm[U{x,y)]

<

nm[U(x,0)].

Similarly 7*[U{x, y)] < T^S'-WPIQ) - 8(P£)]\df{Q) \, P , = P{a,y), P 2 = P(b, y). The bracketed expression is bounded by x and tends uniformly to x as y -*• 0, for a + t < Q < b — t, and to 0 for Q > b + t or for Q < a — «if e > 0 is assigned sufficiently small. For the two missing intervals from a — e to a + e and from b — e to b + t we gather from the continuity of f(Q) which entails (as is familiar) that of Txa[f{Q)), that their contribution to the last integral becomes for ally > 0 less than an arbitrary small t' > 0 provided t is sufficiently small. Thus lim J"-,, [6(PiQ) — e(P*Q)]\df(Q)\

< r fa"+:\df(Q)\ b

+ « ' < Trfab\df(Q)\

lhn T 0[U(x, y)] ^

+ « ' , whence

T*[U(x,0].

y—0

But since also lower semicontinuity holds, i.e., J i m Tb[U(x, y)] > Tba + \{U(x, 0)] y—0

no matter how small e, hence also for e = 0, we have the desired continuity of Tb[U(x,y)]asy-*0. A theorem for the circle analogous to Lemma 1 was proved by Evans. 2 LEMMA 1':

/ / Ui(x, y), Ut(x, y), U3(x, y) are harmonic and absolutely

bounded in y > 0 and assume on y = 0 continuous boundary values of bounded total variation, then the length L(y) of the curve of coordinates Ui(x, y), Ui(x, y), U3(x, y) for y fixed and — =° < x < °° tends to the length L0 of the curve Uy{x, 0), U2{x, 0), U,(x, 0) = /,((?), MQ),MQ), asy^O. Proof: rL(y) = x f°„dx[(dUl(x,y)/dxy + (bU2/bXy + (bU3/bx)*]Vt

VOL.

37. 1951

MATHEMATICS:

H. LEWY

105

= y_". dxKS-- - 0* dMQ))2 + ••• ]Vt < f~„ dx[{f~„ - ex\dMQ)|y + . . . ] ' ' • since — 0X > 0. Now [(/_". - ex\dfi(Q)\)2 + • • - ] ' A < •/*-"- - MI4A(C)I2 + .. . ] v * because this formula expresses the fact that a straight line joining the origin to the point with coordinates yi"„ — Sx\dfi(Q)\, ./*_"„ — 9x\df2(Q)\, f-„ — Ox\dfi(Q) | is the shortest line of this property. Hence *L(y) < y j " . dx f-m

-

9,(\dfi\*

+ .. .)*

and by Lemma 1, lira rL(y)

< lim f-m

dx f°„

- ff,(|i/i(0) |» + .. - ) ' A = T / - " . [ | 4 f i ( 0 1 ! + • • • l' A = T^O.

But the semicontinuity of the length requires lim L(y) > La. Thus L(y) is continuous as y —*• 0, q. e. d. LEMMA 2: Under the hypotheses of Lemma 1, the total variation of U on segments perpendicular to the x-axis is at most T1„ [/((?) ]• Proof: It suffices to prove the case when the segment ends on the x-axis, at the origin. Write U = U+ + U-, where U+(P) = v-*fo"

- S(PQ)df(Q) + const., U- = x-iy_°„ - 6(PQ)df(Q) + const.

We have, with dP = dy > 0, dP8(PQ) = 0udy < 0,U Q > 0; 0vdy>0, if Q < 0; |
S°. dP0(PQ)\df(Q)\ Thus, integrating over the segment in question,

f\dU+(P)\

< r"» So" 8(OQ)\df(Q)\ = (V.) fo" \df{Q)\

and similarly

f\dU-(P)\ Hence f\du{P)\

< S\du+(P)\

< 'A/- 0 . \df{Q)\. + f\dU-{P)\

< (V 2 )r-„[/(0].

LEMMA 3: Let S be a minimal surface of the three-dimensional £, 17, f space, and let S be bounded by a rectifiable Jordan curve C of length L. Take the conformal mapping of S on the upper half plane y > 0 of an x, y-plane which establishes a continuous one-to-one correspondence a between C and the

106

MATHEMATICS: H. LEWY

PRoe. N. A. S.

x-axis and for which £, t\, f become bounded harmonic functions of x, y in y > 0 with boundary values according to a, for which iny > 0 the two relations hold: £* + Vx + f* = $y + Vy + f y, && + VxVv + fxfv = 0 with £z = (d/dx)£, . . . . Denote by £*(x, y), T)*(X, y), f *(x, y) conjugate harmonics of £(x, y), 17, f a»
|!*(«. h) - ?(a, h')\ < fj %(a, y)\dy< fht {£ + v? + tf)l/'dy < SkH£ + fi + &l/'dy < SJtfy + vl + W'dy. Now by Lemma 2, Jhm\S,(a, y)\dy < nm[Z{x,

0)] < i ( 0 ) .

Hence J"° \£y(a, y) \dy converges and is <3Z,(0), thereby insuring that £* tends to a limit as y —*• 0. Similar facts hold for rj*, f*. T h a t these boundary values of £*, 17*, f * are continuous follows easily from i r ( o . 0) - {*(&, 0) I < T"a [{*(*, 0)] < Km inf T06 [{*(*, y)]
< length of C from x = a to x = 6 which also shows that the total variation of the boundary values of £* is continuous and remains bounded. Hence Lemma 1' applies to £*, 17*, f* as well as to £, 77, f; Lemma 3 follows. LEMMA 3 ' : Under the hypotheses of Lemma 3, we have more generally Lb(y) = L*b(y) -+ Z»(0) = L*o\0). Proof: Because of semicontinuity the only other possibility could be Z,£(0) < lim inf Lba(y). But then L(0) = L"_-(0) + Lj(0) + L6-(0) < Hm inf L l „ ( y ) , which contradicts Lemma 1'. Analogous considerations exclude L*ba(0) < lim inf L*ba(y), as y —* 0 through an arbitrary sequence.

303 VOL. 37, 1951

MATHEMATICS:

H. LEWY

107

LEMMA 4: Denote by L(0, y) the length of the boundary curve of the surface with coordinates £ cos y + | * sin y, ij cos y + v* sin y, f cos y + f * w i 7r ^ » L * ( 0 ( T) = £*(0, 0) = Lj(0). Proof: An elementary computation shows Z,(y, 7) = L(y, 0), and the passage y —* 0 gives, in view of Lemma 3', the desired relation. LEMMA 5: Denote by s the length on C. Excepting a set of measure zero with respect to s, we have on C 0 = (dS/ds)(dS*/ds) + (dv/ds)(dv*/ds)

+ (dt/ds)(dt*/ds).

(4)

Proof: Since ia(0, 7) = Lj(0, 0) = f,hm

a

[(
we find by differentiation with respect to 7, setting 7 = 0, 0 = fxb- a m/dsKdp/ds)

+

...]ds

which is the integrated form of Lemma 5, from which it follows by differentiation with respect to 5 = s(b). THEOREM : Under the assumptions of Lemma 3 about S and its coordinates £» V, f(*> y)f suppose furthermore that the boundary curve contains an analytic arc'^4 which, without loss of generality, may be taken in the form V = 0, continuous in y ^ 0. If we take the integral J* \ d\ | + \dp\ + | dv \, extended over the rc-axis, a parallel to the .-c-axis, or over the y-axis, it remains by Lemmas 1, 2, 3 bounded above by 6L (0). We now define the continuous function a(x, y) = y Id\I + Idy.I -+- \dv\ by extending the integral from (0, 0) to (0, y) to (x, y) over straight segments. For sufficiently small h > 0, the square q of the corners — h, +h, h + 2ih, —h + 2ih has the property that within q, 0. Denote by p, M two positive numbers such that for the complex quantity E satisfying | E | = p, we have
<M\S\.

108

MA THEM A TICS: H. LEWY

PROC. N. A. S.

Take e so small that M e < 1/8 and 4 e < p. We determine a function E(z) as solution of the differential equation dZ = 2[1 + *>'2(E) + *'ȣ)]-*{<&. +

*>'(E)<*M

+
with E(0) = 0.* Successive approximations give in the usual manner E(z) by integrating over straight segments from 0 to iy to x + iy for all x + iy = z of q. To show this, we observe, calling E* the &th approximation, that Eo = 0, and \dZi\ < 2d(r, |Ei| < 2 0. scribed presently, we find

For a certain ilf*, to be de-

l^lHjt+2 - E*+i| | < |rf(Et+2 — E*+i)| < M*|E t+ i - E
G « = Gi(«V0O. G « = &(*)*'(*). Now set S =

lim E*. We have in q the integrated form of (5),

S = 2 / [ l + „"(E) + r*(3)]-i{d\

+
since the Stieltjes integral on the right hand is the limit as k —* °° of the analogous integral with E replaced by E*. That the construction yields an analytic function of z near all z = iy, y > 0, follows from the fact that (i) there is for arbitrary initial xo + iyo, yo > 0, a. solution of (5) which is analytic, according to the classical existence theorem and (ii) that there is no more than one solution of (5). For call 5S the difference of two solutions of (5). We have

VOL.

37, 1951

MATHEMATICS:

|5S| <

H. LEWY

109

Aff\5Z\d
with 5E vanishing initially. This inequality yields 5E = 0 in familiar manner. That the solution E is continuous in the whole of q follows from the same property of the S*. The latter fact follows from the convergence of the total variations of X, n, v over parallel segments which approach the x-axis normally. It is true that a priori (5) will hold on y = 0 only except for a set of measure zero relative to the variable s, but the integrated form holds everywhere. I say that E coincides with l-(x, 0) for y = 0. In fact, we have on y = 0 because ij = *>(£), f = ^(£), that almost everywhere in the sense mentioned, in view of (4),

2[i + w) + r\m-iUk/2 +
M(Z) = -J^S) + ^E(i)), *(z) = -Hi) + £(E(I)). First of all, these new definitions agree with the previous ones for n, v on y = 0, since there

- M * ) + tfS(*)) - -Of - «?*)/2 + *($) = (i, + *V)/2 = /•(«) and similarly

-Hx) + M£W) = »(*)• Secondly, the same argument as before yields the fact that for y < 0, fi and v are regular functions of z, thus becoming regular also on the x-axis.

110

MATHEMATICS:

PUTNAM

AND WINTNER

PROC N. A. S.

The Theorem is proved. The idea underlying the analytic extension, of forming a differential equation connecting the extension with the given function, is capable of application in many other situations. We have treated similarly the problem of the "free boundary surface" for minimal surfaces. Other problems concerning analytic boundary conditions come under the same heading. We intend to return to the subject elsewhere. 1

It suffices to demand assumption of boundary values upon normal approach to the boundary. * Evans, G. C , The Logarithmic Potential, Am. Math. Soc. Colloquium Publica-. tions VI, New York, 1927, pp. 35ff. 3 Courant, R., Dirichlet's Principle, Conformal Mapping and Minimal Surfaces, Interscience Publishers, 1950, esp. p. 118. 4 More precisely a solution of the integrated equation (6).

ANNALS OF MATHEMATICS

Vol. 66, No. 1, July, 1957 © 1957 The Johns Hopkins University Press. Reprinted with permission.

AN EXAMPLE OF A SMOOTH LINEAR PARTIAL DIFFERENTIAL EQUATION WITHOUT SOLUTION BY HANS

LEWY

(Received F e b r u a r y 10, 1957)

Introduction In dealing with the existence of solutions of partial differential equations it was customary during the nineteenth century and it still is today in many applications, to appeal to the theorem of Cauchy-Kowalewski which guarantees the existence of analytic solutions for analytic partial differential equations. On the other hand a deeper understanding of the nature of solutions requires the admission of non-analytic functions in equations and solutions. For large classes of equations this extension of the range of equation and solution has been carried out since the beginning of this century. In particular much attention has been given to linear partial differential equations and systems of such. Uniformly the experience of the investigated types has shown that—speaking of existence in the local senSe—there always were solutions, indeed, smooth solutions, provided the equations were smooth enough. It was therefore a matter of considerable surprise to this author, to discover that this inference is in general erroneous. More precisely, there exist linear partial differential equations with coefficients in C°° which possess not a single smooth solution in any neighborhood. The example to be presented in this paper is an equation of first order in three independent variables with complex-valued coefficients and unknown function, or, what amounts to the same, a system of two equations of first order for two functions of three variables, all real. A theorem We begin the discussion of this example by first deriving the following THEOREM. Let Xi, x>, f/i be independent real variables, u a dependent complexvalued variable and \p(yi) a real function of C . Consider the linear equation (1)

[-(d/dxi)

- iid/dxt) + 2i(xi + ixiXd/duMu

=

f'bii).

Assume that there exists a solution u(xi , x>, iji) of (1) in a neighborhood N = iV(0, 0, y\) of the point (0, 0, y\), with u in Cl. Then ^(//i) *s analytic at yx = y\. PROOF. Integrate (1) over a circle in N of equations xl + xi = const. = 1/2,

iji

= const.

On introducing the angle d by Xi -f- ix2 = yle'" we find (2)

d/dXi + id/dx2 = a(d/d log 7/2 + id/dd)

with a determined by applying (2) on log y\. Therefore a — (xt + ix2)y^1 and

308 156

HANS LEWY j.2r

/

Jo

.2ir

eieyV\{d/d log y\)u + i(d/d0)u] d6

(d/dxi + id/dx2)u de = /

Jo 2ir

/

e^ylHid/d log y\)u + u] d0

= (d/dyi)2 / Jo

e'ylu dB.

Setting (3)

U(yi,y,)

= i f

e"y\udB

Jo

we obtain from (1) (4)

dU/dyx + tdU/dy* = T^foi).

Note that (3) implies U(yi , 0 ) = 0. Furthermore, V(yi, 2/2) = V(y) = Ufa,

2/2) - WfoO

1

is in C and satisfies d P / t y i + idV/dyi

= 0

which states that V(y) is an analytic function of y whose domain of existence certainly includes all those points (2/1, 2/2) for which y2 > 0 is sufficiently small and 2/1 is such that {x\, x2,2/1) lies in N for xi + x2 ^ 2/2 • As V = — ir^ on 2/2 = 0 with ^ real by hypothesis, V can be continued across 2/2 = 0 as analytic function of y. Therefore V is analytic on 2/2 = 0. Hence ^(2/1) is analytic at and near 0

>A = 2/iWe apply the Theorem in a negative sense. Suppose we take an equation (1) in which ^(2/1) is real and in C°°, but not analytic at 2/1 = J/i, then the equation (1) can have no solution u which is in C in any .V = iV(0, 0, 2/1). It becomes desirable to remove the restriction to special neighborhoods which occurs in this example. Example of an equation without solution With the aid of a periodic real function ^(2/1.) of C°°, which is analytic in no 2/i-interval, we can construct a function F(x\, x 2 , iji) of C°° such that (5)

[-d/dxi

- id/dXi + 2i(xi + ix2)d/dyi}u = F(xi , x 2 , 2/1)

has no H -solution, no matter what open (x\, x 2 , >j\)-set is taken as domain of existence. A function is said to be in H* if its first partial derivatives satisfy a Holder condition with positive exponent, provided the distance of the points involved does not exceed 1. Choose a countable set of points Pi ,P*, • • • , which is dense in the (xi, x 2 , iji)space, and positive radii pi, p 2 , • • • with lim,,^ p„ = 0 and denote by N j the sphere of radius p, about P , . An arbitrary open set always contains some

309 A DIFFERENTIAL EQUATION WITHOUT SOLUTION

157

A^j . Designate by py and q, the x\ and .r2-coordinates of Pj and put Cj = m a x [j, | pj |, | gy |].

Consider the sequences e of real numbers £i, £2, • • • with (6)

l . U . b . j _ i . . . . . . I £j\

< CO

= £ r efi?W(yi

+ 2ffix, - 2p,x 2 ).

and set Ft(Xl, x2,

Vl)

F r is itself in C°°, for if we formally construct a j- t h derivative Z>" by termwise differentiation D'Fc(Xl,

xt, 2/0 = E " - i * * 7 V ' + u ( l / i + 2 ^ , -

2p J :r 2 )?; i (-p,r2' 1 + ' 2 > »"1 +

^2 = V

the series of the right member converges absolutely and uniformly as ip(y+>) is bounded on account of the periodicity of ^ and ^27=i CJC' < °° • Recall that the sequences f with norm (6) form a complete metric (Banach) space and consequently this space is not exhausted by a countable sum of nondense sets. This fact enables us to demonstrate the existence of a sequence £* in this space such that for F = Fc* there is no open set of points (x,, .T2 , 2/1) on which (5) has an H -solution. l PROOF. Denote by H nm the property of a function of having first partial derivatives satisfying a Holder condition of exponent \/n and constant m, where n, m = 1, 2,3, • • • . Evidently Hl = ^Z„,m F ^ a n d all functions which vanish at Pj and satisfy Hnm are compact so that the functions of H vanishing at Pj and existing in Nj are all contained in a countable sum of compact sets. Let Ejnm be the set of sequences E such that (5) with F = Fr has a solution existing in Nj and belonging there to Hlnm . Einn is closed for the following reason: we can always suppose that the solutions vanish at Pj and out of any sequence of solutions belonging to various e of Eilim tending to a limit sequence we can select a suitable subsequence of sequences e and corresponding solutions u of (5) with F = Fc such that the solutions converge in .V, together with their first partials. The limit function then satisfies the limit equation in A'; and lies h\H„,„ , proving that Ej„m is closed. But Ejnm is nondense. In view of the closure property of Ejlim it suffices to ascertain that every sequence of Ejlim is limit of sequences of the Banach space none of which is in Ej„m . Observe that if a and /3 arc elements of EjHm so is (a — /3)/2. Consider the particular sequence 5'' = (8\ , 52 , • • • ) where 5^ is the Kronecker symbol. It will be seen in the last § that (5) has no Cl-solution in Nj for F = Fit . Hence since (5) is linear, none of the sequences c + \8', X ^ 0, e in Ej,im , can be Ejnm . Taking j X j arbitrarily small, we obtain the sequence E of Ej„m as limit of sequences not in Ejnm . It follows that• ^ j , „ . m Ej„m does not exhaust the space of all sequences E

310 158

HANS LEWY

with (6). Let £* be a sequence not in X ^j»« • Then (5) with F = Fc. has no H -solution in any Nj, j = 1, 2, • • • or any other open set. The only thing remaining is the proof that Lu = [ — d/dxi — id/dXi + 2i(xi + ix^)d/dy^\u (7) = f'iy!

+ 2qjXi -

2pjX2)

has no C -solution in Nj. Consider the transformation Xi = xi - pj,

X2 = x2 — qj,

Fi = i/i -f- 2g,-a:1 — 2pyx2

whose inverse is xi = Xi + PJ ,

x2 = X 2 + qj,

?/i = Yx - 2qjXi + 2p,X 2

and for which d/dxi = (d/dXt) + 2qj(d/dY1),

d/dxi = (d/dX2) -

d/dyi =

2pj(d/dY1),

d/dYi.

We verify that (7) is transcribed into (8) Lu = [-d/dXi - id/dX, + 2i(X1 + iX,)d/dYAu = *'(Ki). If (7) had a C1 -solution in Nj, (8) would have a C l -solution in the neighborhood of the transform of the center of Nj, whose new coordinates are X, = X, = 0,

Fi = Y\ = 2 / ^ ) .

But here our Theorem applies that for the solution to exist i£( Fi) would have to be analytic at this F?, contrary to the hypothesis about \f/. U N I V E R S I T Y O F C A L I F O R N I A AT B E R K E L E Y

Wolf Prize in Mathematics, Vol. 2 (pp. 311-358) eds. S. S. Chern and F. Hirzebruch © 2001 World Scientific Publishing Co.

Curriculum Vitae

Born: March 9, 1948 in Budapest, Hungary Family. Married, 4 children Citizenship: Hungarian. Permanent resident in the US. Degrees: Dr.Rher.Nat., Eotvos Lorand University, Budapest, Hungary, 1971 Candidate of Math. Sci., Hungarian Academy of Sciences, Budapest, Hungary, 1970 Dr.Math.Sci., Hungarian Academy of Sciences, Budapest, Hungary, 1977 Elected memberships and positions in scientific societies: Hungarian Academy of Sciences, corresp. member, 1979; reg.member, 1985 European Academy of Sciences, Arts and Humanities, 1981 Academia Europaea, 1991 Nordrhein-Westfalische Akademie der Wissenschaften, corresponding member, 1993 Presiduum of the Hungarian Academy of Sciences, 1990-1993 Executive Committee of the International Mathematical Union, 1986-1994 Positions held: Research Associate, Eotvos Lorand University, Budapest, 1971-75 Docent, Jozsef Attila University, Szeged, 1975-78 Professor, Chair of Geometry, Jozsef Attila University, Szeged, 1978-82 Professor, Chair of Computer Science, Eotvos Lorand University, Budapest, 1983-93. Professor, Dept. of Computer Science, Yale University, 1993-99 Senior Researcher, Microsoft Research, 1999Visiting positions: Vanderbilt University, 1972/73 University of Waterloo, 1978/79 Universitat Bonn, 1984/85 University of Chicago, Spring 1985 Cornell University, Fall 1985 Mathematical Sciences Research Institute, Berkeley, Spring 1986 Princeton University, Fall 1987, Spring 1989, 1990/91, 1992/93

312 Honorary Degrees and Positions: Adjunct Professor, University of Waterloo, Waterloo, Ontario, Canada, 1980A.D.White Professor-at-Large, Cornell University, Ithaca, NY, 1982-1987 Honorary Professor, Universitat Bonn, 1984 John von Neumann Professor, Universitat Bonn, 1985 Honorary Professor, Academia Sinica, 1988 Doctor Honoris Causa, University of Waterloo, Ontario, Canada, 1992 Doctor Honoris Causa, University of Szeged, Hungary, 1999 Awards: Griinwald Geza Prize, Bolyai Society, 1970 George Polya Prize, Soc. Ind. Appl. Math., 1979 Best Information Theory Paper Award, IEEE, 1981 Ray D.Fulkerson Prize, Amer. Math. Soc.-Math. Prog. Soc, 1982 State Prize, Hungary, 1985 Tibor Szele Medal, Bolyai Society, 1992 Brouwer Medal, Dutch Matematical Society Royal Netherlands Academy of Sciences, 1993 National Order of Merit of Hungary, 1998 Bolzano Medal, Czech Mathematical Society, 1998 Wolf Prize, 1999 Knuth Prize, ACM SIGACT 1999 Editorial Boards: Combinatorica (Editor-in-Chief), Advances in Mathematics, J. Combinatorial Theory (B), Discrete Math., Discrete Applied Math., J. Graph Theory, Europ. J. Combinatorics, Discrete and Computational Geometry, Random Structures and Algorithms, Acta Mathematica Hungarica, Acta Cybernetica, Electronic Journal of Combinatorics Field of research: Combinatorial optimization, graph theory, theoretical computer science

313 List of Publications Books Kombinatorika (Pelikan J. es Vesztergombi K.) Tankonyvkiado, Budapest, 1977 (German translation: Teubner, 1977; Japanese translation: 1985). Algoritmusok (Gacs P.), Miiszaki Konyvkiado, Budapest, 1978; Tankonyvkiado, Budapest, 1987. Combinatorial Problems and Exercises, Akademiai Kiado - North-Holland, Budapest, 1979 (Japanese translation: Tokai Univ. Press, 1988; Hungarian translation: Typotech, 1999). Matching Theory (M. D. Plummer), Akademiai Kiado - North-Holland, Budapest, 1986 (Russian translation: Mir, 1998). An Algorithmic Theory of Numbers, Graphs, and Convexity, CBMS-NSF Regional Conference Series in Applied Mathematics 50, SIAM, 1986. Geometric Algorithms and Combinatorial Optimization (with M. Grotschel and A. Schrijver), Springer, 1988; Chinese edition: World Publishing Corp., Beijing, 1990. Greedoids (B. Korte and R. Schrader), Springer, 1991. Research papers 1965 Fiiggetlen koroket nem tartalmazo grafokrol (On graphs containing no independent circuits), Mat. Lapok 16, 289-299. 1966 On decomposition of graphs, Studia Math. Hung. 1, 237-238. 1967 On connected sets of points, Annates Univ. R. Eotvos 10, 203-204. Uber die starke Multiplication von geordneten Graphen, Acta Math. Hung. 18, 235-241. Operations with structures, Acta Math. Hung. 18, 321-328. 1968 Graphs and set-systems, in Beitrage zur Graphentheorie, Teubner, Leipzig, 99-106. On chromatic number of graphs and set-systems, Acta Math. Hung. 19, 59-67.

314 On covering of graphs, in Theory of Graphs (eds. P. Erdos and G. Katona), Akad. Kiado, Budapest, 231-236. 1969 Kapcsolatok polinomoknak es helyettesitesi ertekeiknek szamelmeleti tulajdonsagai kozott, Mat. Lapok 20, 129-132. 1970 Generalized factors of graphs, in Combinatorial Theory and its Applications, Coll. Math. Soc. J. Bolyai 4, 773-781. Subgraphs with prescribed valencies, J. Comb. Theory 8, 391-416. A generalization of Konig's theorem, Acta Math. Hung. 21, 443-446. A remark on Menger's theorem, Acta Math. Hung. 21, 365-368. The factorization of graphs, in Combinatorial Struc. AppL, Gordon and Breach, 243-246. Representation of integers by norm-forms II (with K. Gyory), Publ. Math. Debrecen 17, 173-181. 1971 On the cancellation law among finite relational structures, Periodica Math. Hung. 1, 145-156. On finite Dirichlet series, Acta Math. Hung. 22, 227-231. On the number of halving lines, Ann. Univ. Eotvos 14, 107-108. 1972 Normal hypergraphs and the perfect graph conjecture, Discrete Math. 2, 253-267; reprinted Ann. Discrete Math. 21 (1984) 29-42. On the structure of factorizable graphs, Acta Math. Hung. 23, 179-195. The factorization of graphs II, Acta Math. Hung. 23, 223-246. On the structure of factorizable graphs II, Acta Math. Hung. 23, 465-478. Direct product in locally finite categories, Acta Sci. Math. Szeged 23, 319-322. A characterization of perfect graphs, J. Comb. Theory 13, 95-98; reprinted in Classic Papers in Combinatorics (eds. I. Gessel and G. C. Rota), Birkhauser, 1987, 447-450. A note on the line reconstruction problem, J. Comb. Theory 13, 309-310; reprinted in Classic Papers in Combinatorics (ed. I. Gessel and G. C. Rota), Birkhauser, 1987, 451-452. A note on factor-critical graphs, Studia Sci. Math. 7, 279-280.

315 1973 On the eigenvalues of trees (with J. Pelikan), Periodica Math. Hung. 3, 175-182. Antifactors of graphs, Periodica Math. Hung. 4, 121-123. Dissection graphs of planar point sets (with P. Erdos, G. J. Simmons and E. G. Strauss), in A Survey of Comb. Theory (ed. S. Srivastava), Springer, 139-149. A note to a paper of Dudley (with P. Major), Studia Sci. Math. 8, 151-152. Permutation groups and almost regular graphs (with L. Babai), Studia Sci. Math. 8, 141-150. Finite homeomorphism groups of the 2-sphere (with L. Babai and W. Imrich), in Topics in Topology, Coll. Math. Soc. J. Bolyai 9, 61-75. Connectivity in digraphs, J. Comb. Theory 15, 174-177. Independent sets in critical chromatic graphs, Studia Sci. Math. 8, 165-168. On the sum of matroids (with A. Recski), Acta Math. Hung. 24, 329-333. Coverings and colorings of hypergraphs, in Proc. Ath Southeastern Conf. on Comb., Utilitas Math., 3-12. Factors of graphs, in Proc. Ath Southeastern Conf. on Comb., Utilitas Math., 13-22. 1974 Valencies of graphs with 1-factors, Periodica Math. Hung. 5, 149-151. Minimax theorems for hypergraphs, in Hypergraph Seminar (eds. C. Berge and D. K. Ray-Chaudhuri), Lecture Notes in Math. 411, Springer, 111-126. Every directed graph has a semi-kernel (with V. Chvatal), in Hypergraph Seminar (eds. C. Berge and D. K. Ray-Chaudhuri), Lecture Notes in Math. 411, Springer, 175. Applications of product coloring (with D. Greenwell), Acta Math. Hung. 25, 335-340. A family of planar bicritical graphs (with M. D. Plummer), in Combinatorics, London Math. Soc. Lecture Notes 13, 103-108. 1975 Problems and results on 3-chromatic hypergraphs and some related questions (with P. Erdos), in Infinite and Finite Sets, Coll. Math. Soc. J. Bolyai 11, 609-627. On bicritical graphs (with M. D. Plummer), in Infinite and Finite Sets, Coll. Math. Soc. J. Bolyai 11, 1051-1079. A characterization of cancellable fc-ary structures (with R. Appleson), Periodica Math. Hung. 6, 17-19. A family of planar bicritical graphs (with M. D. Plummer), Proc. London Math. Soc. 30, 160-176.

316

Three short proofs in graph theory, J. Comb. Theory 19, 269-271. Spectra of graphs with transitive groups, Periodica Math. Hung. 6, 191-195. 2-matchings and 2-covers of hypergraphs, Acta Math. Hung. 26, 433-444. On the ratio of optimal fractional and integral covers, Discrete Math. 13, 383-390. A kombinatorika minimax teteleirol (On the minimax theorems of combinatorics), Mat. Lapok 26, 209-264. 1976 The number of values of Boolean functions (with D. E. Daykin), J. London Math. Soc. 30, 160-176. On two minimax theorems in graph theory, J. Comb. Theory B21, 93-103. On graphs of Ramsey type (with S. A. Burr and P. Erdos), Ars Combinatoria 1, 167-190. On some connectivity properties of Eulerian graphs, Acta Math. Hung. 28, 129-138. Covers, packings and some heuristic algorithms, in Combinatorics, Proc. 5th British Comb. Conf. (eds. C. St. J. A. Nash-Williams and J. Sheehan), Utilitas Math., 417-429. On the number of complete subgraphs of a graph (with M. Simonovits), in Combinatorics, Proc. 5th British Comb. Conf. (eds. C. St. J. A. Nash-Williams and J. Sheehan), Utilitas Math., 439-441. A forbidden substructure characterization of Gauss codes (with M. Marx), Bull. Amer. Math. Soc. 82, 121-122. A forbidden substructure characterization of Gauss codes (with M. Marx), Acta. Sci. Math. Szeged 38, 115-119. Chromatic number of hypergraphs and linear algebra, Studia Sci. Math. 11, 113-114. 1977 Some remarks on generalized spectra (with P. Gacs), Zeitschr. Math. Logik Grundlagen Math. 23, 547-554. Certain duality principles in integer programming, Ann. Discrete Math. 1, 363-374. A homology theory for spanning trees of a graph, Acta Math. Hung. 30, 241-251. On minimal elementary bipartite graphs (with M. D. Plummer), J. Comb. Theory B23, 127-138. Flats in matroids and geometric graphs, in Combinatorial Surveys, Proc. 6th British Comb. Conf., Academic Press, 45-86. Polynomes de la matrice des distences d'un arbre (with R. L. Graham), in Problemes Combinatoires et Theorie de Graphes, CNRS, 189-190.

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321 On some combinatorial properties of algebraic matroids (with A. Dress), Combinatorica 7, 39-48. Matching structure and the matching lattice, J. Comb. Theory B43, 187-222. Pseudomodular lattices and continuous matroids (with A. Bjorner), Acta Sci. Math. Szeged 51, 295-308. 1988 Rubber bands, convex embeddings, and graph connectivity (with N. Linial and A. Wigderson), Combinatorica 8, 91-102. Geometry of numbers: An algorithmic view, in ICIAM '87: Proc. 1st Internatl. Conf. on Industr. Appl. Math. (eds. J. McKenna and R. Teman), SIAM, 144-152. Lattices, Mobius functions and communication complexity (with M. Saks), 29th IEEE Annual Symp. Found. Comp. Sci., 81-90. The chromatic number of the graph of large distances (with P. Erdos and K. Vesztergombi), in Combinatorics, Proc. Coll. Eger 1987, Coll. Math. Soc. J. Bolyai 52, North-Holland, 547-551. How to tidy up your set-system? (with C. A. J. Hurkens, A. Schrijver and Eva Tardos), in Combinatorics, Proc. Coll. Eger 1987, Coll. Math. Soc. J. Bolyai 52, North-Holland, 309-314. The intersection of matroids and antimatroids (with B. Korte), Discrete Math. 73, 143-157. Covering minima and lattice point free convex bodies (with R. Kannan), Ann. Math. 128, 577-602. 1989 Extremal problems for discrepancy (with K. Vesztergombi), in Irregularities of Partitions (eds. G. Halasz and V. T. Sos), Algorithms and Combinatorics 8, Springer, 107-113. Orthogonal representations and connectivity of graphs (with M. Saks and A. Schrijver), Linear Alg. Appl. 114/115, 439-454. Examples and algorithmic properties of greedoids (with B. Korte and O. Goecke), in Combinatorial Optimization, Como 1986 (ed. B. Simeone), Lecture Notes in Math. 1403, Springer, 113-161. Disks, balls and walls: The analysis of a combinatorial game (with R. Anderson, P. Shor, J. Spencer, E. Tardos and S. Winograd), Amer. Math. Monthly. An on-line graph coloring algorithm with sublinear performance ratio (with M. Saks and W. T. Trotter), Discrete Math. 75, 319-325. Faster algorithms for hard problems, Information Processing '89 (ed. G. X. Ritter), Elsevier, 135-141.

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Approximating clique is almost NP-complete (with U. Feige, S. Goldwasser, S. Safra and M. Szegedy), Proc. 32nd IEEE Annual Symp. on Found, of Comp. Sci., 2-12. Search problems in the decision tree model (with M. Naor, I. Newman and A. Wigderson), Proc. 32nd IEEE Annual Symp. on Found, of Comp. Sci., 576-585. 1992 A matching algorithm for regular bipartite graphs (with J. Csima), Discrete Appl. Math. 35, 197-203. How to compute the volume? Jber. d. Dt. Math.-Verein, Jubildumstagung 1990, B. G. Teubner, Stuttgart, 138-151. Linear decision trees, hyperplane arrangements, and Mobius functions (with A. Yao and A. Bjorner), in Proc. 24th ACM Symp. on Theory of Computing, 170-177. Two-prover one-round proof systems: their power and their problems (with U. Feige), in Proc. 24th ACM Symp. on Theory of Computing, 733-744. On integer points in polyhedra: a lower bound (with I. Barany and R. Howe), Combinatorica 12, 135-142. On the randomized complexity of volume and diameter (with M. Simonovits), Proc. 33rd IEEE Annual Symp. on Found, of Comp. Sci., 482-491. The generalized basis reduction algorithm (with H. Scarf), Math, of OR 17, 751-764. Chip-firing games on directed graphs (with A. Bjorner), J. Algebraic Combinatorics 1, 305-328. 1993 Dating to marriage (with J. Csima), Discrete Appl. Math. 41, 269-270. The cocycle lattice of binary matroids (with A. Seress), Euro. J. Combin. 14, 241-250. Random walks in a convex body and an improved volume algorithm (with M. Simonovits), Random Structures and Alg. 4, 359-412. Communication complexity and combinatorial lattice theory (with M. Saks), J. Comp. Sys. Sci. 47, 322-349. Note: on the last new vertex visited by a random walk (with P. Winkler), J. Graph Theory 17, 593-596. A Monte-Carlo algorithm for estimating the permanent (with N. Karmarkar, R. M. Karp, R. Lipton and M. Luby), SIAM J. Comput. 22, 284-293. 1994 Stable sets and polynomials, Discrete Math. 124, 137-153.

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Linear decision trees, subspace arrangements, and Mobius functions (with A. Bjorner), J. Amer. Math. Soc. 7, 677-706. Chessboard complexes and matching complexes (with A. Bjorner, S. Vrecica and R. Zivaljevic), J. London Math. Soc. 49, 25-39. 1995 Search problems in the decision tree model (with M. Naor, I. Newman and A. Wigderson), SI AM J. Disc. Math. 8, 119-132. Isoperimetric problems for convex bodies and a localization lemma (with R. Kannan and M. Simonovits), Disc. Comput. Geometry 13, 541-559. Randomized algorithms in combinatorial optimization, in Combinatorial Optimization, Papers from the DIMACS Special Year (eds. W. Cook, L. Lovasz and P. Seymour), DIMACS Series in Discrete Mathematics and Combinatorial Optimization 20, AMS, 153-179. Exact mixing in an unknown Markov chain (with P. Winkler), Electronic J. Combinatorics 2, paper R15, 1-14. Mixing of random walks and other diffusions on a graph (with P. Winkler), in Surveys in Combinatorics, 1995 (ed. P. Rowlinson), London Math. Soc. Lecture Notes Series 218, Cambridge Univ. Press, 119-154. Efficient stopping Rules for Markov Chains (with P. Winkler), Proceedings of the 1995 ACM Symposium on the Theory of Computing, 76-82. The cocycle lattice of binary matroids II (with A. Seress), Linear Algebra and its Applications 226-228, 553-566. On the invariance of Colin de Verdiere's graph parameter under clique sums, (with A. Schrijver and H. v. d. Hoist), Linear Algebra and its Applications 226-228, 509-518. 1996 The rank and size of graphs (with A. Kotlov), J. Graph Theory 23, 185-189. Random walks on graphs: A survey, in Combinatorics, Paul Erdos is Eighty, Vol. 2 (eds. D. Miklos, V. T. Sos and T. Szonyi), Janos Bolyai Mathematical Society, Budapest, 353-398. Approximating clique is almost NP-complete (with U. Feige, S. Goldwasser, S. Safra and M. Szegedy), J. ACM 43, 268-292. 1997 The membership problem in jump systems, J. Comb. Theory B70, 45-66. On Conway's thrackle conjecture (with J. Pach and M. Szegedy), Discrete Comput. Geom. 18, 369-376. Random walks and an 0*(n5) volume algorithm for convex bodies (with R. Kannan and M. Simonovits), Random Structures and Algorithms 11, 1-50.

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JOURNAL OF COMBINATORIAL THEORY, Series A 25. 319-324 (1978)

Note Kneser's Conjecture, Chromatic Number, and Homotopy L. LOVASZ Bolyai Institute, Jozsef A ttila University, H-6720 Szeged, Aradi vertanuk tere 1, Hungary Communicated by the Editors Received March 4, 1977

If the simplicial complex formed by the neighborhoods of points of a graph is (k — 2)-connected then the graph is not ^-colorable. As a corollary Kneser's conjecture is proved, asserting that if all n-subsets of a (2n — A:)-element set are divided into k + 1 classes, one of the classes contains two disjoint n-subsets.

1. INTRODUCTION

Kneser [6] formulated the following conjecture in 1955, whose proof is the main objective of this note. THEOREM 1. If we split the n-subsets of a (2n + k)-element set into k + I classes, one of the classes will contain two disjoint n-subsets.

It is easy to split the n-subsets into k + 2 classes so that the assertion does not remain valid. For let 1,..., 2n + k be the given elements and let Kt contain those subsets whose first element is i. Then Kx, K2,..., Kk+1, Kk+2. u • • • u Kk+n+1 is a partition of the n-subsets into k + 2 classes such that any two n-subsets in the same class intersect. Let us construct a graph KGn%k as follows. The vertices of KGnik are the n-subsets of {1,..., 2n + k) and two of them are joined by an edge iff they are disjoint. These graphs are often called Kneser's graphs. Note that KG2il is the well-known Petersen graph. Now Theorem 1 can be rephrased as follows: THEOREM

1'. The chromatic number of Kneser's graph KG„ik is k + 2.

This conjecture, or special cases of it, have turned out to play many roles in various fields of graph theory. In particular, the case n < 3 has been proved and applied by Garey and Johnson [4] and Stahl [5]. Here we mention the following: Kneser's graph has the property that each odd circuit of it has 319 Reprinted by permission of the publisher Copyright © 1978 by Academic Press, Inc. All rights of reproduction in any form reserved.

L. LOVASZ

320

lengths at least 2njk + 1. So if we know that it has high chromatic number, we see that Kneser's graph is an example of triangle-free high-chromatic graphs (and of even more). Erdos and Hajnal [2, 3] have constructed several other classes of graphs with similar properties, among others the following graph, which is often called Borsuk's graph. 1 Let the vertices of graph Bk be the points of the ^-sphere Sk, two of them being adjacent iff their distance is at least 2 — e for some e > 0 (i.e., iff they are almost antipodal). It is easy to see that if e is small, this graph contains no short odd circuits. The fact that its chromatic number is k + 2 is equivalent to the following well-known theorem of Borsuk: BORSUK'S THEOREM.

If

Sk = Fx u ••• u f

w

,

where F1,...,Fk+1

are

k

closed subsets ofS , then one of the sets F{ contains two antipodal points. We shall prove Theorem 1 by using some techniques of algebraic topology and Borsuk's theorem. In fact we shall derive a more general lower bound for the chromatic number of certain graphs. To formulate this result we need some preparation. Let G be a graph. Define the neighborhood complex Jf(G) as the simplicial complex whose vertices are the vertices of G and whose simplices are those subsets of V(G) which have a common neighbor. For any complex Jf, let Jf denote the polyhedron determined by Jf. A topological space T is called n-connected if each continuous mapping of the surface Sr of the (r + I) dimensional ball into T extends continuously to the whole ball, for r = 0, l,...,n. THEOREM

2. If ^(G)

COROLLARY.

is (k -\- 2)-connected then G is not k-colorable.

^V(.G) is never homotopically trivial.

In the case k = 2 we obtain: If the neighborhood complex of G is connected then G is not bipartite. This is trivial since the color-classes of any 2-coloration of G are components of Jf(G). For connected graphs the converse is also true if A: = 2: If G is not bipartite then any two vertices x, y can be connected by a walk x = x0, x1,..., x2j) = y of even length and then x 0 , x2,..., x2»-2 > r X23J i s a walk in ^V(G) connecting x to y, thus J (G) is connected. For k ^= 3 the condition of Theorem 2 is not necessary, which is shown by any graph with large chromatic number and girth. It seems to be an interesting question whether any topological property of ^(G) is equivalent to the A:-colorability of G. On the other hand, could Theorem 2 be strengthened 1

1 am indepted to Miklos Simonovits for pointing out the analogy between Kneser's and Borsuk's graphs, which is the underlying idea of this paper.

KNESER'S CONJECTURE

321

by considering homology instead of homotopy, or as follows ? If the (k — 2)dimensional homotopy group of J/~(G) is trivial, then the chromatic number of G differs from k. The fact that Kneser's graph satisfies the conditions of Theorem 2 is not quite obvious; we shall prove the following, slightly more general result: THEOREM 3. Let S be a finite set and n, k natural numbers. Consider the simplicial complex Jf whose vertices are the n-subsets ofS and whose simplices are those sets A0 ,..., Am of n-subsets for which

IM

< n + k.

Then J f is (k — \)-connected. Since for | S | = 2n + k the complex X above is the neighborhood complex of KG„ik, Theorems 2 and 3 together imply Theorem 1.

2. PROOF OF THEOREM 2

Let J^G) denote the barycentric subdivision of JV{G). The vertices of Jf-^G) are those sets XC V(G) whose elements have a common neighbor, and some of them span a simplex iff they form a chain with respect to inclusion. It is trivial that ^(G) and ^\(G) are homeomorphic. Let X C V(G) and denote by v(X) the set of common neighbors of X. Then v maps the set of vertices of ^V[(G) into itself, and since X C Y implies v{X) 2 v{Y), it is simplicial, i.e., maps the vertices of any simplex onto vertices of a simplex of •^l(G). Let us extend it simplicially to a continuous mapping of -^[(G) into itself. We denote this extended mapping by v. Note that v3 = v

and

v3 — v.

(1)

We define mappings 9r:

ST -> A{G)

(r = 0, 1,..., k - 1)

by induction on r such that
(2)

for all x e Sr (here — x is the point antipodal to x). First let r = 0 and v an arbitrary point of A(G). Set
322

L. LOVASZ

Second, let r ^ 1 and assume that cpr-^. S r _ 1 ->• *A^(G) is defined so that (2) holds. Denote by 5+ and S~ the upper and lower hemisphere of Sr, so that S+ n S~ = S r _ 1 . Let us extend <pr_! to a continuous mapping ifi: S+ —*• *^(G). This is possible by the assumption that *^(G) is ^-connected. Define now
if x e 5+,

,„

On 5 r _ 1 = S+ n S" the two definitions coincide, and in fact both yield
=

t\<pr-i(x))

= v(9r-l(

— x)) =

<pr-l(x),

since (2) is valid for r — 1. Thus (3) defines a continuous mapping of 5 r into ^ ( G ) . Moreover, if x e 5+ then

= ?2(«A(-x)) = v( 9r (x)).

So (2) is inherited and the definition of <pr is complete for all r < k — 1. Suppose now that G admits a ^-coloration. Let ^ denote the subcomplex of ~W(G) formed by those simplices whose vertices have a common neighbor of color i (1 < / < £). Then trivially

jr(G) = J[\J ••• u Jfk. Moreover, ^

n v(^) =

0.

Assume indirectly that x e JV{ and v(x) e JTt. Then x belongs to the simplex. of Jf(G) spanned by the neighborhood of a vertex v e V(G) of color /. In the barycentric subdivision ~^(G), x belongs to the interior of a unique simplex spanned by vertices of ^[(G), i.e., subsets of V(G), say X1,..., Xm; and we have X j , X2,..., Xm C v(v). Then y, = v(A'?) a r and v(x) is contained in the interior of the simplex of ^ ( G ) spanned by some of Y1, Y2,..., Ym . Since v(x) eJfi,\X follows similarly that there is a vertex u e V{G) of color / such that some F,- C v(w). But then u, v are adjacent vertices of G of color /, which is a contradiction. Now let Fj = cp^xi^K). Then Ft is a closed subset of S1"1 and clearly fj u ••• u f t = 5,fc_1. Also Ft contains no two antipodal points; in fact, if x eFj and —xeFt then 9k-i(X>

6

^

323

KNESER'S CONJECTURE

and
3. PROOF OF THEOREM 3

Let A = {A0,..., Am} be a simplex in Jf. Put m

U(A)

=[]Ai, i=0

and denote by M(A) the simplex spanned by all ^-subsets of U(A). We call A crowded if | U(A)\ < n + k. The proof goes by induction on \S\. For \S\ < n + k the assertion is obvious, since J f is a simplex. So we may assume that \ S\ > n + k. Let Jf' denote the closed subcomplex of c€ whose simplices are the crowded simplices of Jf, and let Cf0 be the subcomplex whose simplices are the simplices of dimension < k — 1 (the (k — l)-dimensional skeleton of df). First we show that Cf0 can be deformed into JT' in ct. We do so by defining a continuous mapping >p: X~0 —• Jf' such that (*)

for each simplex A of df0,

I/J(A)

lies in M(A).

This condition clearly implies that tfi is homotopic in Jf to the injection of cta into jf. We define tp(A) by induction on the dimension of A. Tf dim A = 0 then 4 is automatically crowded and we may set >p(A) = A. Assume now that dim A > 0 and that I/J is defined on the boundary A of A such that (*) is fulfilled. Consider the subcomplex jf'A of Jf' induced by the vertices of M{A). By the induction hypothesis, I/J(A) lies in Cfc'A and hence, by the other induction hypothesis on | S |, from which we know that c£"A is (r — l)-connected, we know that ifi can be extended over the interior of A. This completes the definition of
ifueXbutv£X,

Then cpuv is simplicial, i.e., if A = {A0,..., Am} is a simplex in Jf' then so is
324

L. LOVASZ

Am then U(tpuv(A)) C U(A); if w e A0 u ••• u Am and r £ ^ 0 u ••• u 4TO then. U(
I U(
k-l.

Thus 9oUK can be considered as a continuous mapping of X' into itself. Also observe that (**)

9uv(A) u A is contained in the simplex of j f spanned by the n-subsets of U(A) u {v}.

Therefore
This maps each w-subset of S on {u±,..., un}. Since it is, by the remark above, homotopic to the injection of X' into X, it follows that X' can be contracted in X to a single point. Since we have shown that X0 can be deformed into X', it follows that X0 can be contracted in X to a single point. This completes the proof.

REFERENCES 1. K. BORSUK, Drei Satze iiber die H-dimensionale euklidische sphare, Fund. Math. 20 (1933), 177-190. 2. P. ERDOS AND A. HAJNAL, On chromatic number of graphs and set-systems, Acta Math. Acad. Sci. Hungar. 17 (1966), 61-99. 3. P. ERDOS AND A. HAJNAL, Kromatikus grafokrol, Mat. Lapok 18 (1967), 1-4. 4. M. GAREY AND D . S. JOHNSON, The complexity of near-optimal graph coloring, J. Assoc. Comput. Much. 23 (1976), 43-49. 5. S. STAHL, n-tuple colorings and associated graphs, / . Combinatorial Theory Ser. B 20 (1976), 185-203. 6. M. KNESER, Aufgabe 300, Jber. Deutsch. Math.-Verein. 58 (1955).

333 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-25, NO. 1, JANUARY 1979

On the Shannon Capacity of a Graph LASZLO LOVASZ

Abstract—It is proved that the Shannon zero-error capacity of the pentagon is V5 . The method is then generalized to obtain upper bounds on the capacity of an arbitrary graph. A well-characterized, and in a sense easily computable, function is introduced which bounds the capacity from above and equals the capacity in a large number of cases. Several results are obtained on the capacity of special graphs; for example, the Petersen graph has capacity four and a self-complementary graph with n points and with a vertex-rransitive automorphism group has capacity Vn .

I.

INTRODUCTION

L

ET THERE BE a graph G, whose vertices are letters in an alphabet and in which adjacency means that the letters can be confused. Then the maximum number of one-letter messages which can be sent without danger of confusion is clearly a(G), the maximum number of independent points in the graph G. Denote by ot(Gk) the maximum number of ^-letter messages which can be sent without danger of confusion (two ^-letter words are confoundable if for each 1 < / < k, their rth letters are confoundable or equal). It is clear that there are at least a(Gf such words (formed from a maximum set of nonconfoundable letters), but one may be able to do better. For example, if C5 is a pentagon, then a(C|) = 5. In fact, if i;,,' • • ,c 5 are the vertices of the pentagon (in this cyclic order), then the words 0,0,, o2°3> v3v5, t)4u2, and t)5o4 are nonconfoundable. It is easily seen that 0(G)=sup^ct(G*) = lim yja(Gk) . k

*—*

A general upper bound on 0(G) was also given in [6] (this bound was discussed in detail by Rosenfeld [5]). We assign nonnegative weights w(x) to the vertices x of G such that 2

w(x)
for every complete subgraph C in G; such an assignment is called a fractional vertex packing. The maximum of l£xw(x), taken over all fractional vertex packings, is denoted by a*(G). It follows easily from the duality theorem of linear programming that a*(G) can be defined dually as follows: we assign nonnegative weights q(C) to the cliques C of G such that

2 9(C) > i CBx

for each point x of G and minimize ~Zcq(C). With this notation Shannon's theorem states 6(G)
This number was introduced by Shannon [6] and is called the Shannon capacity of the graph G. The previous considII. THE CAPACITY OF THE PENTAGON eration shows that e(G)>a(G) and that, in general, equality does not hold. Let G be a finite undirected graph without loops. We The determination of the Shannon capacity is a very say that two vertices of G are adjacent if they are either difficult problem even for very simple small graphs. connected by an edge or are equal. Shannon proved that a(G)-8(G) for those graphs which The set of points of the graph G is denoted by V(G). can be covered by a(G) cliques (the best known such The complementary graph of G is defined as the graph G graphs are the so-called perfect graphs; see [1]). However, with V(G)= V(G) and in which two points are connected even for the simplest graph not covered by this result— by an edge iff they are not connected in G. A k-coloration •he pentagon—the Shannon capacity was previously un- of G is a partition of V(G) into k sets independent in G. known. Note that this corresponds to a covering of the points of the complementary graph by k cliques. The least k for which G admits a A-coloration is called its chromatic number. lanuscript received February 23, 1978; revised June 20, 1978. A permutation of V(G) is an automorphism if it prehe author is with the Department of Combinatorics, University of serves adjacency of the points. The automorphisms of G erloo, Walerloo, ON, Canada, on leave from the Bolyai Institute, ef Attila University, H-6720 Szeged, Aradi vertaniik t. 1, Hungary.

01979 IEEE

334 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-25, NO. 1, JANUARY 1979

form a permutation group called the automorphism group hence of G. If for each pair of points JC, y G V(G) there exists an automorphism mapping x onto_y, then the automorphism 9(G°H)< max group is called vertex transitive. Edge transitivity is defined in an analog manner. A graph is called regular of = 9(G)9(H). degree d if each point is incident with d edges. Note that graphs whose automorphism groups are vertex transitive Remark: We shall see later that equality holds in are regular. This does not necessarily hold for edge transi- Lemma 2. tivity (as, for example, in the case of a star). Lemma 3: a(G)<9(G). If G and H are two graphs, then their strong product Proof: Let («„••-,«„) be an optimal orthonormal G-H is defined as the graph with V{G-H)= V(G)X V{H), in which (x,y) is adjacent to (x',y') iff x is adjacent to x' representation of G with handle c. Let {1.••-,&}, for k in G a n d / is adjacent t o y in H. If we denote by G the example, be a maximum independent set in G. Then strong product of k copies of G, then a(Gk) is indeed the a,,- • • ,uk are pairwise orthogonal, and so maximum number of independent points in Gk. We shall use linear algebra extensively. For various I = <:-'> 2 {cTu,)2>a(G)/9(G). properties of (mostly semidefinite) matrices, see, for exami= \ ple, [4]. All vectors will be column vectors. We shall denote by / the identity matrix, by / the square matrix all Theorem 1: 9(G) <9(G). of whose entries are ones, and by j the vector whose Proof: By Lemmas 1 and 2, a(Gk)<9(Gk)<9(G)k. entries are ones (the dimension of these matrices and vectors will be clear from the context). Theorem 2: 0 ( C 5 ) = V 5 . Besides the inner product of vectors v, w (denoted by Proof: Consider an umbrella whose handle and five r t; w, where T denotes transpose), we shall use the tensor ribs have unit length. Open the umbrella to the point product, defined as follows. If v = (v],- • • ,vn) and w = where the maximum angle between the ribs is TT/2. Let u„ (M/|, • • • , wm), then we denote by v ° w the vector u2, «3. «4, "5 be the ribs and c be the handle, as vectors (OIMV* • • ,viwm,v2wl,- • • ,vnwm)T of length nm. A simple oriented away from their common point. Then «,,- • • ,a 5 computation shows that the two kinds of vector multi- is an orthonormal representation of C5. Moreover, it is plication are connected by easy to compute from the spherical cosine theorem that c r«1, = 5 ~' / 4 , and hence (x°y)T(u°w) = (xrv)(yTw). (1) 1 0(C 5 )„) of unit vectors in a Euclidean follows. space such that if i a n d / are nonadjacent vertices, then t), and Vj are orthogonal. Clearly, every graph has an orthonormal representation, for example, by pairwise orthogoIII. FORMULAS FOR 9(G) nal vectors. To be able to apply Theorem 1 to estimate or calculate Lemma 1: Let {«,,-••,«„) and (U],••-,»„) be orthonormal representations of .G and //, respectively. Then the the Shannon capacity of other graphs we must investigate vectors u, ° »• form an orthonormal representation of GH. the number 9(G) in greater detail. The proof is immediate from (1). Theorem 3: Let G be a graph on vertices {1, • • - , « } . Define the value of an orthonormal representation Then 9(G) is the minimum of the largest eigenvalue of ( a , , - • • ,«„) to be any symmetric matrix (a^)"^ [ such that 1 mm max a:y = l, if i = / o r if / a n d / a r e nonadjacent. (2) c "'<"(cr«,)2 Proof: where c ranges over all unit vectors. The vector c yielding 1) Let («,,••• ,«„) be an optimal orthonormal repthe minimum is called the handle of the representation. Let 9(G) denote the minimum value over all representa- resentation of G with handle c. Define tions of G. It is easy to see that this minimum is attained. Call a representation optimal if it achieves this minimum value. Lemma 2: 9(G-H) <9(G)»(H). Proof: Let («!,,••• .«„) and (v„-• • ,v„) be optimal and orthonormal representations of G and H, with handles c and d, respectively. Then c°d is a unit vector by (1), and

a„=l-

('Wl)'

",v=l. A

=(«iXJ-r

'*j.

335 LOVASZ: SHANNON CAPACITY OF A G R A P H

Then (2) is satisfied. Moreover,

matrix satisfying (3) and (4). Then using (2) and (3), Tr BJ = 2

>*j,

2 btJ= 2

i-ij-i

2

"ijb^TrAB,

,-ij-i

and so

and

#(C)-Trfl/ = Tr(#(G)/-^)£. + #(G)--

&(G)-a,

Here both d(G)I-A and B are positive semidefinite. Let ' - ( - * ) • «,,-•• ,«„ be a set of mutually orthogonal eigenvectors of These equations imply that d(G)l — A is positive semidefiB, with corresponding eigenvalues X„- • • , \ , > 0. Then nite, and hence the largest eigenvalue of A is at most 9(G). T r ( # ( G ) / - / 0 f l = 2 e/WG)/-/!)&>,. 2) Conversely, let A = (a,-,) be any matrix satisfying (2), and let X be its largest eigenvalue. Then XI —A is positive semidefinite, and hence there exist vectors = 2\-*,r(#(G)/-/Oe,>0. i= i xv- • • ,xn such that 2) We have to construct a matrix B which satisfies the previous inequality with equality. For this purpose let Let c be a unit vector perpendicular to xt,- • • ,xn, and set ('n/i)."'' '('mJm)('k<Jk) be the edges of G. Consider the (m + l)-dimensional vectors 1

W

VX «?=^(l + x?)-l,

(h,hJy,hlmh^(2hl)2)T

h=

Then

where h = (hl,- • • ,hn) ranges through all unit vectors and

i = l, • • • , « ,

z = (0,0,--,0,#(G))r and for nonadjacent / and_/',

Claim: z is in the convex hull of the vectors A. Suppose this is not the case. Since the vectors A form a compact set, there exists a hyperplane separating z from So (i^,-• • ,i/n) is an orthonormal representation of G. all the A, i.e., there exists a vector a and a real number a such that aTha. Moreover, Set 1 = \, i = l,---,n, a = (al,- • • ,am,y) . (c\f ^ . - i ( l + ^ ) - 0 .

and hence &(G)<\. theorem.

This completes the proof of the

Then in particular aTh <,a, for A = (1,0,-• • ,0); whence y» implies #(G).v>0. Note that it also follows that among the optimal repre- Hencey >0, and a > 0 . We may suppose t h a t y ~ \ , and so a<&(G). sentations there is one such that Now define »(G) = ! =••• = ! . 2 a * + 1 ' if {'V} = {W*} 11. otherwise; then aTh
The next theorem gives a good characterization of the value »(G). Theorem 4: Let G be a graph on the set of vertices 0 ' " • " . " } , and let B — (biJ)1J_, range over all positive semidefinite symmetric matrices such that *, = 0

(3) (4)

Then t?(G)=max B

Since the largest eigenvalue of -4 = (fly) is equal to max [xTAx: \x\ = 1),

for every pair (ij) of distinct adjacent vertices and T r £ = l.

2 2 "„AA<«-

i-ij-i

TTBJ.

Note that Tr BJ is the sum of the entries in B. Proof; 1) Let A =(aj).)JJ._, be a matrix satisfying (2) with largest eigenvalue #(G\ and let B be any symmetric

this implies that the largest eigenvalue of (a ) is at most a. Since (atJ) satisfies (2), this implies &(G)*ia, a contradiction. This proves the claim. By the claim, there exist a finite number of unit vectors A,,- • • ,AW and nonnegative reals a,,- • • .a^ such that ", + ••• +
= ;.

(5) (6)

336 *

IEE

Set

TRANSACTIONS ON INFORMATION THEORY, VOL. I T - 2 5 . NO. 1, JANUARY 1 9 7 9

Set v = w

i

N b

a

i/\w\

d-\ 2

w

i\/\ 2

w

i •

h

ij= 2 pK> pj P-\

B'(bu). The matrix B is clearly symmetric and positive semidefinite. Further. (6) implies b,, =0,

Then the vectors vt form an orthonormal representation of G by (7) and (3). Moreover, using the Cauchy-Schwarz inequality we get

k=l,--,m

>(tH(dTv,)J=(±dTW)j2

and Tr A / - » ( G ) while (5) implies

= (f2>,) = (.£ •".•) =*(G)-

TrB=l. This completes the proof.

This completes the proof. Lemma 4: Let («,,•-•.«„) be an orthonormal repreNote that since we have equality in the Cauchysentation of G and (a,,- • • ,vn) be an orthonormal repre- Schwarz inequality, it also follows that sentation of the complementary graph G. Moreover, let c and d be any vectors. Then ( 2 = # ( G ) 6 „ , (8) («/i0 2 ( O , 7 d) 2
£ i-1

/Yoo/; By (1). the vectors «,•"<;,• satisfy ( V !*)(., • » , ) - ( n r « , ) ( t > r » > « , . Thus they form an orthonormal system, and we have (c>df>

£((c"
which is just the inequality in Lemma 4. Corollary // If (C|,•••,«„) is an orthonormal representation of G and d is any unit vector, then

Theorem 6: Let A range over all matrices such that a,j = 0 if ij are adjacent in G, and let \,(A)> • • • >X„(A) denote the eigenvalues of A. Then

Proof: 1) Let A be any matrix such that a, ; =0 if i and 7 are adjacent. L e t / = ( / „ • • • ,/„) r be an eigenvector belonging to A,(/() such t h a t / 2 = - \/\n(A) (note that since Tr A = 0, the least eigenvalue of A is negative). Consider the matrices F=diag (/,,-•• ,/„) and B = F(A-\„(A)I)F.

9(G)> 2 (t^rf)2. Corollary 2: »(G)9(G)>n. We give now another minimax formula for the value 9(G), which shows a very surprising duality between G and its complementary graph G. Theorem 5: Let (i?,,- • • ,t>m) range over all orthonormal representations of G and d over all unit vectors. Then #(G) = max 2

T

(d v:f.

Obviously B is positive semidefinite. Moreover, bfJ = 0 if / and j are distinct adjacent points, and Tr£=-A„(/4)TrF2=l. So by Theorem 4,

#(G)>TrB/= £ £ V^-A„(,0 £ / 2 '" - 1 y - 1

2

1-1

2

= 2{A l (^)/ -A^)/ } =

l - ^ .

Proof: By Corollary 1 we already know that the 2) The fact that equality is attained here follows by inequality > holds. We construct now a representation of a more or less straightforward inversion of this argument G and a unit vector • • • > \, be the (4) such that Tr BJ = ft(G). Since B is positive semidefieigenvalues of the adjacency matrix of a graph G. Then nite, we have vectors »,,••• , H>„ such that the chromatic number of G is at least Note that

i> 2 =i.

(iU)2=»(G).

Proof: The chromatic number of G is least #(G). In fact, if («,,• • • ,«„) is an orthonormal representation of G,

337 LOVAS2: SHANNON CAPACITY OF A GRAPH

c is any unit vector, and ./,,- • • ,Jk are the color classes in Then trivially, B also satisfies (3), and any A-coloration of G, then TTB=\

2(<-r«,f= 2 2 UT«,f< 2 i-* /-l

m - l IEJ„

»-l

from which the assertion follows by Theorem 5. Now the adjacency matrix of G satisfies the condition in the theorem (with G instead of G), which implies the inequality in the corollary. IV.

TrBJ =

9(G)

(using PJ — JP = J). Also trivially, Bis symmetric and positive semidefinite and satisfies P^'BP^B, for all PS T. Since T is transitive on the vertices, this implies btl — l / « , for all ;'. Constructing the orthonormal representation (f|,- • • ,vn) and the unit vector d as in the proof of Theorem 5, we have

SOME FURTHER PROPERTIES OF 9(G)

The results in the previous section make the value 9(G) by (8). So from the definition of 9(G), quite easy to handle. Let us derive some consequences. Theorem 7: »(G-H) =

9(G)9(H).

9(G) < max '<•'•''(jTv,f

Proof: We already know that

= -r-^-, 9 G ( )

and hence »(GH)i9(G)9(H).

9(G)9(G) X2 > • • • >X„ be the eigenvalues of its adjacency matrix A. T 2 2(v, c) = »(G) f,(^df = 9(H). Then I- I

/= 1

Then u( ° H>- is an orthonormal representation of GH (this follows since it is an orthonormal representation of GH and GND G-H). Moreover, c°d is a unit vector. So

9(GH)> 2 £((«>, °M,)V
= 2 2 W'fWf

Equality holds if the automorphism group of G is transitive on the edges. Corollary 5: For odd n. n cos (ir/n) »(C„) = 1 +cos {ir/n) '

i-ij-i

Proof: Consider the matrix J — xA, where x will be chosen later. This satisfies condition (2) in Theorem 3, and hence its largest eigenvalue is at least &(G). Let t> Theorem 8: If G has a vertex-transitive automorphism denote the eigenvector of A belonging to \ . Then since A is regular, vi =/', and therefore, y, «2, - • • ,vh are also eigengroup, then vectors of J. So the eigenvalues of J — xA are n — 9(G)9(G) = n. JCX], — xX2,- • •, — x \ , . The largest of these is either the first or the last, and the optimal choice of JC is x = n/(ki — Corollary 4: If G has a vertex-transitive automorphism X„) when they are both equal to - n\n/(\l — Xn). This group, then proves the first assertion. e(G)@(G)n having largest eigenvalue &(G). As in the proof of Theo(for example, a star). rem 8, consider Proof: Let T be the automorphism group of G. We •nay consider the elements of F as nXn permutation 'CP. matrices. Let B = (b0) be a matrix satisfying (3) and (4) such that Tr BJ = 9(G). Consider Then C also satisfies (2), and moreover, its largest eigen-

-2(«,r*)22(»/<02-#(G)*(/o. , - i

S

,--i

U

-&H(&'-' '}

value is at most &(G). By Theorem 3, it is equal to &(G). Moreover, C is clearly of the form J — xA. Hence the second assertion follows.

338 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-25, NO. 1, JANUARY 1979 V.

COMPARISON WITH OTHER BOUNDS ON

If G is self-complementary, then

CAPACITY

0(G-G) = ©(G 2 ) = 0(G) 2 . This proves the theorem. The proof also shows that in Proof: We use Theorem 4. Let («,) be an orthonor- these cases & = #. mal representation of G and c be a unit vector such that Theorem 13: Let n>2r, and let the graph K(n,r) be defined as the graph whose vertices are the r-subsets of an ^-element set S, two subsets being adjacent iff they are disjoint. Then Let C be any clique in G. Then { « , : / £ C} is an orthonormal set of vectors, and hence Q(K(n,r))-(»2\). Theorem 10: &(G)
Corollary 6: The Petersen graph, which is isomorphic with A"(5,2), has capacity four. Hence the weights (c */,) form a fractional vertex packCorollary 7: (See Erdos, Ko, and Rado [2].) ing, and so a(K(n,r)) = ("-_\). /EC 7

2

Note that

1= 1

A very simple upper bound on @(G) is the dimension of an orthonormal representation of G.

«*W«,r))-(?)/[i]

Theorem 11: Assume that G admits an orthonormal which is larger than " unless r is a divisor of n. representation in dimension d. Then Proof of Theorem 13: The r subsets containing a ${G)a(K(n,r))>("rZ\]j. «;,- • • ,un °un) is another orthonormal representation of G. Let {el}- • • ,ed] be an orthonormal basis and On the other hand, we calculate &(K(n,r)). Since the automorphism group of K(n,r) is clearly transitive on the b=—-(el°e]+e2°e2+--+ ed°ed). vertices and edges, we may use Theorem 9. So let us Vd calculate the eigenvalues of K(n,r). Clearly j is an eigen2 vector with eigenvalue {nZr)Then b = 1, and Let 1 < ( < r. For each 7" c S such that | T\ = t, let xT be a real number such that for every UcS with \U\ = t— 1. (U,=U,)r*=^ f (ek°ek)T(u,°u,) Vd * - i

(9)

2 *r = °1

V , 7- \2

1

UczT

There are ( 1 — I _ , ) linearly independent vectors (xT) of this type. For each such vector, define

Therefore 9(G)
*A = VI.

We can use our methods to calculate the Shannon capacity of graphs other than the pentagon. We of course deal only with graphs G such that a(G)a(G-G)>\V(G)\. On the other hand, we have by Theorems 1, 6, and 7 that e ( G - G ) < » ( G - G ) - » ( G ) * ( G ) = |K(G)|.

2

*T

TCA

APPLICATIONS

m-r for every AcS, \A\ = r. It is not difficult to see, and actually well-known, that the numbers xT can be calculated from the numbers xA, whence there are I

I—I . I linearly independent vectors of type (xA). Claim: Every (xA) is an eigenvector of the adja-

cency matrix of K(n,r) with eigenvalue (— l)'!

I-

In fact, for any A0c S such that \A0\ = r, we have /fn/<„-0

TnA„-2>\

r

To determine this value we set

A=

2 \TnA0\-

l I

\

r

i I

339 7

LOVASZ: SHANNON CAPACITY OF A GRAPH

Then summing (9) for every UcS and \U nA0\-i, we get

such that \U\ = t~\

(,+ l)/3, + l + ( / - 0 / 3 , = 0. This may be considered as a recurrence relation for the fj: and yields

A=(-iy(;.)/3 0 whence

A,=(-i)'/3, = (-i)'^„ which proves the claim. By this construction we have found

' + ,?,(CH,-, ))=(") linearly independent eigenvectors (there is no problem with the eigenvectors belonging to different values of 1 since they belong to different eigenvalues). Therefore, we have all eigenvectors, and it follows that the eigenvalues of K(n,r) are the numbers

(-!)'(«;:;'),

I, respectively, and Theorem 9 yields !n-r-\\ln\

VII.

0(G)0(G) = 0(GG)>a(GG)>|K(G)|. This, in turn, would imply an affirmative answer to the last question of Problem 1: n<0(C„)0(C„)<#(C„)O(C„) = «; hence 0(C„) = t)(C„) and 0(C„) = i»(C„)). Corollary 7 shows an example where the calculation of &(G) helps to determine a(G) in a nontrivial way. Are there any further examples? ACKNOWLEDGMENT

,=o,i,..,,

So the largest and smallest eigenvalues are ( " 7 r ) and - 1 " — r~

Various properties of ir(G) established in this paper suggest further problems which would be solved by an affirmative answer to Problem 1. Problem 2: Is 0(G- H) = 0(G)0(W)? (Note that @(GH)>@(G)@(H) is obvious.) Problem 3: Is it true that 0 ( G ) - 0 ( G ) > | K(G)|? Note that an affirmative answer to Problem 2 would imply an affirmative answer to Problem 3:

My sincere thanks are due to K. Vesztergombi and M. Rosenfeld for numerous conversations on the topic of this paper—also to T. Nemetz, A. Schrijver, and the referees of the paper for pointing out several errors in, and suggesting many improvements to, the original text. It is a pleasure to acknowledge that A. J. Hoffman and M. Rosenfeld have also found extensions of the original idea (in Section II) to other graphs. Among others Hoffman found Theorem 9. and Rosenfeld found #(C„).

CONCLUDING REMARKS

The purpose of introducing 9(G) has been to estimate 0(G). So the obvious question is as follows. Problem 1: Is # = 0? More modestly, find further graphs with #(G) = 0(G). In particular, do odd circuits satisfy #(G) = 0(G)? This last question pinpoints a difficulty which seems to be crucial. In all cases known to the author where 0 ( G ) is precisely determined, there is some k (k = \ or 2, in fact) such that a(Gk) = Q{G)k. But if 0(G) = t»(G) for the seven-circuit, for example, then no such k can exist, since no power of #(C 7 ) is an integer.

REFERENCES [1] C. Berge, Graphs and Hypergraphs. Amsterdam and London North-Holland; New York: American Elsevier, 1973. [2] P. Erdos, C. Ko, and R. Rado, "Intersection theorems for systems of finite sets," Quart. J. Malk. Oxford, vol. 12, pp. 313-320, 1961. [3] A. J. Hoffman, "On eigenvalues and colorings of graphs," in B. Harris, Ed., Graph Theory and its Applications. New York and London: Academic, \910,pp. 79-91. [4] P. Lancaster, Theory of Matrices. New York and London: Academic, 1969. [5] M. Rosenfeld, "On a problem of Shannon," Proc. Amer. Math. Soc, vol 18, pp. 315-319, 1967. [6] C. E. Shannon, "The zero-error capacity of a noisy channel," IRE Trans. Inform. Thoery, vol. IT-2, no. 3, pp. 8-19. Sept. 1956.

340 August 1988

Algorithmic mathematics: an old aspect with a new emphasis Ldszlo Lovdsz

0. Introduction The development of computers is perhaps the single most significant technological breakthrough in this century. It is natural that it has not left untouched closely related branches of science like mathematics and its education. It is also natural that whichever fields have come into contact with it, heated debates have started and very different views, extreme and moderate, progressive and conservative, have been put forth. Is algorithmic mathematics of higher value than classical, structureoriented, theorem-proof mathematics, or does it just hide the essence of things by making them more complicated then necessary? Does teaching of an algorithm lead to a better understanding of the underlying structure, or is it a more abstract, more elegant setting that does so? Is the algorithmic way of life best (Maurer 1985), or is applied mathematics just bad mathematics (Halmos 1981)? Should computers be introduced in elementary/secondary/college education of mathematics? I want to start with a disclaimer: I will not attempt to give an answer to all these questions. The point I will try to make is that algorithmic mathematics (put into focus by computers, but existent and important way before their development!) is not an antithesis of the "theorem-proof" type classical mathematics. Rather, it enriches several classical branches of mathematics with new insight, new kinds of problems, and new approaches to solve these. So: not algorithmic or structural mathematics, but algorithmic and structural mathematics! The interplay between the algorithmic and structural sides of mathematics is manyfold; I will only mention the two most important lines. The design and analysis of algorithms and the study of algorithmic solvability uses deeper and deeper tools from classical structural mathematics on the one hand; and an algorithmic perspective has a more and more profound effect on the whole framework of many fields of classical mathematics on the other hand. Let me examplify. Perhaps the first notion of an "algorithm" that was defined clearly enough so that the question of "algorithmic solvability" could be raised was the notion of a geometric construction by ruler and compass, formulated by the Greek geometers. The mathematical interest of this notion is that there are both solvable and unsolvable construction problems. The design of construction algo-

341 rithms has been stimulating in geometry for a long time, and has contributed to the development of important tools (very useful also independently of construction problems) like inversion or the golden ratio. But the proof of unsolvability of basic construction problems (trisecting an angle, squaring a circle, doubling a cube, constructing a regular heptagon etc.) illustrates this effect more dramatically. Such negative results were inaccessible for the Greek mathematics and for quite a while later on; it required the notion of real numbers and a substantial part of modern algebra to prove them, or even to formulate them with an exactness that made them accessible to mathematical methods. In fact, modern algebra was inspired by the desire to prove such negative results (besides the non-constructibility of certain configurations, the non-solvability of equations of degree at least 5 was of the same nature). As another example, let us consider the notion of primes. These numbers were also studied by the ancient Greeks, and they proved several basic properties of them. The beautiful but very hard theory of prime numbers has been a major branch of mathematics (and a source of inspiration for many other branches) throughout the history of modern mathematics too. But only the development of computers, and even more the establishment of computational complexity theory, raised the fundamental algorithmic problem: how to test whether a given number is prime? how to find the prime factorization if it is not? (Mathematicians in the 18-th and 19-th century, in particular Gauss, did make extensive computations regarding primes, and developed ingenious tricks to help their work. But appearantly they did not consider their algorithms as mathematical results.) These questions have profound applications in computing, and the simple elementary procedures to solve them are by far not satisfactory (practice requires the consideration of numbers up to several hundred digits). Over the last 10 years or so, more and more advanced methods from number theory have been applied to design more and more sophisticated and efficient algorithms to answer these questions. Note that the interaction of mathematics and algorithms in these two examples is different: in the first, we have a certain notion of an "algorithm" (a construction procedure), and we want to prove that it does not suffice to solve certain problems. Such questions may be notoriously hard, and, as our example shows, the proof may require deep mathematics or even the development of entirely new fields. The theory of computing today is full of unsolved problems of a similar nature; there are very few methods to prove negative results about the algorithmic solvability of problems, in particular if limits are imposed on the time (or other resources) that the algorithm can use. In such cases, computer science is an "external" user of

342 mathematics: it supplies hard problems, which need be modelled and solved just like hard problems in mechanics or astronomy. In the second example, computer science penetrates classical mathematics, putting old questions in a new perspective. Often, this is done by requiring constructions where "pure" existence proofs have been supplied by the classical theories. In other cases, one requires efficient procedures where "theoretically finite" case-analyses have been at hand (like in prime testing). In some branches of mathematics, e.g. in graph theory, this process has provided a whole new framework for the field (cf. Lovasz 1986). In my talk, I would like to focus on this second development. This new perspective on several issues in classical mathematics is, of course, a challenge to mathematics education. The introduction of computers (at whatever level) is only a very partial answer. I will give some remarks about how I think this challenge can be met; but I believe that it will take much more work — experimental and theoretical — before the contours of the answer will be clear.

1. Some old results from a new perspective The approximation of irrational numbers by rational ones has a long history. Suppose that we have a real number a, that we want to approximate by a rational number with a small denominator. The first, trivial approach is to round: take any positive integer q, and round the product qa to the nearest integer p. Then p/q is a reasonably good rational approximation of a; in fact, \a — E\ < j - . Can we do better, with some positive integer Q in place of the 2 here? More precisely, can we find, for a given real number a and positive integer Q, a rational number p/q such that \a — 2| < -X-l It is clear that in this case q cannot be chosen arbitrarily any more; but how large does it have to be? A theorem of Dirichlet states that for every given real number a and positive integer Q, we can find integers p and q such that 0 < q < Q and \a — 2| < ^ - . In other words, \qa — p\ < i . There are two basic proofs of this fundamental result, and I would like to discuss them here to illustrate the difference between the algorithmic and nonalgorithmic approaches. First proof: Consider the Q + 1 numbers Qa - [OaJ,

la - [la\,

,... ,

Qa - [Qa\

These numbers all lie in the interval [0,1), so some two of them, say ka — [ka\ and la — [la\ (0 < k < I < Q) are closer to each other than 1/Q. Let q = I — k and

343 p = [la\ — [ka\. Then q < Q and \qa -p\ = \{l - k)a - {{la\ - \ka\)\ = \(ka - [ka\) - {la - \la\)\ < - . So the rational number p/q proves Dirichlet's Theorem. Second proof (sketch): It is well known that every real number a can be expressed as a continued fraction 1

a = CLQ -\

.

ai +

1 02 +

a3 + ...

This expansion may be finite (if a is rational) or infinite (if a is irrational). For example, we have the expansions 11

,

3

,

1 2+

,

1 2

3

T

+ 1 +

2

and v/2 = l + ( v / 2 - l ) = l + ^ — = 1 + ~ - = 1+ v/2 + 1 2+ (V2-l)

2 +

_

1 1

2+ ^ Now if we stop in a continued fraction expansion in the k-th step, we get a rational number Pk , 1

— qk = a0 +

1

1

a2+ ... + —

a-k

There are a number of basic facts that can be proved about this rational number, called the A;-th convergent of a. What we need here is the fact that Pk/qk is a very good approximation of a: \a

< qk

qkqk+i

Hence, if we let k be the largest subscript for which qk < Q (it is known that qk tends to infinity, in fact exponentially fast), then

344

i.e. the rational number Pk/qk proves Dirichlet's Theorem. Which of these proofs is "better" ? There is little doubt that the first one is much simpler: not only is it shorter but it is self-contained, while the second uses quite a bit from the theory of continued fractions. But suppose that we also want to find the rational number in question. Before going into a discussion of this, we have to clarify one thing: how is a given? Since we want to have an algorithm now, we better assume that a is represented in some finite explicit form; let us assume that it is rational, say a = a/b. Of course, we have to assume then that Q < b, else the approximation problem is trivial. Which algorithm can we derive from the first proof? It tells us that we should form all the numbers hot — [ka\, then find two, which are close, then ... But if we want to form all these numbers anyway, we can check with the same amount of work which of them is less that 1/Q. So the first proof of the theorem does not provide us with any non-trivial algorithm to find the rational number whose existence it certifies; it is a "pure existence proof" in this sense. (It could be argued that the structural insight gained from the first proof can be developed further to obtain an efficient algorithm. This, however, takes further work and deeper insight.) Is the second proof better from this point of view? How much work does it take to compute the continued fraction expansion? We show that not only is it easy to obtain this expansion, but also that there is nothing mystical about its use in an approximation problem. Suppose that we want to find a good rational approximations of a number a. As a first approach, we could use the rational number ao/1, where ao = [^JTo obtain a better approximation, we have to approximate the error, i.e., x\ = a — ao. Since this number is less than 1, it is a natural idea to replace it with its reciprocal, and approximate this by its integer part a\ = [ l / ^ i j - Now this approximation again has an error X2 = l/x\ — a%. Take the reciprocal again etc. The positive integers a0,ai,... obtained this way are just the coefficients in the continued fraction expansion of a. If a = a/b then we obtain ao by dividing a by b with remainder; then ao is the quotient and if r is the remainder then the error x\ = r/b. So l / z i is the rational number b/r and we repeat the procedure with this in place of a/b. It is clear that this is a finite procedure, and so the continued fraction expansion of a can be obtained. The rest of the proof gives a simple recipe to obtain the approximating rational number. (Some of my readers may have observed at this point that to obtain the continued fraction expansion of a rational number a/b, we carry out exactly the same

345 arithmetic operations as in the euclidean algorithm used to compute the greatest common divisor of a and b.) So the second proof is just the analysis of a very natural iterative algorithm to find better and better approximations of a number. But is this algorithm any better than the trivial one derived from the first proof? The answer is: much better! The number of steps in the continued fraction expansion of a/b (equivalently, the number of steps in the euclidean algorithm to compute g.c.d.(a, b)), is proportional to the number of digits of a and b\ the number of steps in the first trivial algorithm is proportional to Q, which may be as large as b itself. If a, b and Q are 100 digit numbers, then the first algorithm takes 10 100 steps while the second, less than 500. The usual way to measure the running time of an algorithm is to compare the number of bit-operations with the number of bits necessary to write down the input. So a fc-bit integer (having k bits in its base 2 expansion) contributes k to the input; in the diophantine approximation problem, the size of the input is the number of bits in a, b and Q, which is essentially log2 a + log2 b + log2 Q. In this model of computation, the length of the numbers also influences the time spent on a single arithmetic operation. For example, multiplication of two fc-bit integers by the method taught at school takes about k2 bit-operations, so such an operation contributes k2 to the running time etc. An important distinction to make is whether the running time grows as a polynomial of the input length or faster. Polynomial algorithms tend to be mathematically interesting and usually — though not always — practically feasible. Brute force case-distinctions often lead to exponentially many cases to distinguish (all subsets of a set, etc.) and thereby to exponentially growing running times. The algorithm derived from the first proof needs exponential time; the algorithm derived from the second is polynomial. So we see that: - The first proof is an existence proof; it is elegant, short, but does not give an algorithm to find the approximation. It takes more work and further ideas to develop it into an efficient algorithm. - The second proof is just an analysis of an elegant, natural and efficient algorithm to construct good approximations. It takes work to analyse how good these approximations are. The first proof also generalizes easily to the problem of simultaneously approximating several numbers by rationals with a common denominator. The algorithmic proof (continued fractions) also has some generalizations to this case, but those are

346

substantially less elegant and do not yield as good approximations as the generalization of the first proof. To find a polynomial time algorithm for this simultaneous diophantine approximation problem, which would find the approximating rational numbers whose existence is guaranteed by the almost straightforward extension of the first proof is unsettled!

2. A glimpse of complexity theory Most of us have met problems sounding like "Characterize those sets (numbers,...) with the property that ***.", and also the student who gives, provocatively, the answer "A set has property *** if and only it has property ***." What is wrong with this answer? And what happens if the student hides the triviality by slightly re-phrasing the property * * *? When does the answer begin to be non-trivial and thereby acceptable? Or should we simply ban this kind of problems as meaningless? One of the great successes of the theory of computing is that it is able to define in a mathematically exact way which characterizations are "good" and which are more-or-less just rephasing the question, at least for a large class of structures and properties. There is no room in this talk, of course, to develop this theory. But I try to illustrate the idea on an important example. Consider a system of inequalities anxi + auX2 + • • • + ainxn a2\X\

< b\,

+ a22X2 + • • • + CL2nXn < b2, (1)

: a-mi^i + am2X2 H

h amnxn

< bm.

Since we are interested in algorithmic aspects, we assume that the inequalities have rational coefficients. When does (1) have a solution? Let us investigate the following two answers to this question: Theorem A. The system of inequalities (1) has a solution if and only if the system o n ( « i - vi) + ai2(u2 -V2)-\

h ain(un

- vn) < &i,

fl2l(«l - ^ l ) + 022("2 - v2) H

\- a2n(un ~ Vn) < b2, (2)

o m i(ui - vi) + am2(u2 -v2)-\ has a non-negative solution.

1- amn(un

- vn) < bm

347 T h e o r e m B . The system of inequalities (1) has a solution if and only if the system «n2/i + «2i^2 ~i

h amiym

= 0,

ai22/i + «222/2 -i

\- am2ym = 0, ;

O-lnVl + a2nV2 H hyi

+ b2V2 ^

(3)

h a-mnVm = 0, 1- bmym

= -1

has no non-negative solution. (Note that a solution of (3) can be viewed as a linear combination of the inequalities in (1) with non-negative multipliers — the j/j — that yields the trivially non-solvable inequality Ozi + . . . + 0xn < —1.) Both theorems give necessary and sufficient conditions for the solvability of (1). But Theorem A is an essentially trivial trick to show that solvability is easily transformed into non-negative solvability, while Theorem B is an important result (called Farkas' Lemma). What makes the difference? Assume that I want to use a concrete case of (1) in this talk as an illustration of a system of solvable linear inequalities. How can I convince you that it is indeed solvable? Easy: I just show you a solution. Suppose now that I want to use another concrete case of (1) to illustrate an unsolvable system. How can I convince you that it is indeed unsolvable? To try all possible values for the variables? There is no easy way at hand. Do the necessary and sufficient conditions formulated in the two theorems above help? If we apply Theorem A, I have to exhibit that (2) does not have a non-negative solution — this is not any easier than the original task. But if we apply Theorem B, it suffices to exhibit that (3) does have a non-negative solution — and this I can do by showing you one. So the condition in Theorem B does have an entirely different logical structure from the original property and from the condition given in Theorem A. It turns out that a large part of graph theory, optimization, number theory etc. has an analogous structure. Important properties of graphs, numbers etc. have the feature that if they are present, there is an easy way to exhibit this (e.g. composite numbers, 4-colorable graphs etc.) Basically, these properties are defined in terms of the existence of a certain object (a number is composite if it has a proper divisor; a graph is 4-colorable if it has a proper coloring with 4 colors etc.) If I want to convince you that the number 617532176011725742711064069114397791170668109866281

348 is composite, all I have to do is to show you the divisor 7858321551080267055879091 (Of course you still have to verify that'this is a divisor; but, as we know, this takes only polynomial time.) Such properties are called NP-properties (named after a technical definition involving Non-deterministic Polynomial-time Turing-machines, whose details I don't have to give here). Sometimes (but not always!) the negation of NP-properties is again an NP-property. Theorems establishing such equivalences are often among the most important results in the field (like the Farkas Lemma above). They are sometimes called good characterizations. This classification of properties is also closely related to algorithms. Most properties for which we would like to find a polynomial-time algorithm to decide them belong to NP in a natural way. If a property can be decided by a polynomialtime algorithm, then both the property and its negation are NP-properties, i.e., it has a good characterization. The converse may not be true: it is not known whether every well-characterized property can be decided in polynomial time (probably not, but as I remarked before, we do not have the means to prove such negative results in this area). But, usually, to find a good characterization is an important step towards the construction of a polynomial time algorithm. The example of the Farkas Lemma is very illustrative: almost a century after the proof of the lemma, a polynomial time algorithm to decide if (1) has a solution was given by Khachiyan in 1978 (the famous Ellipsoid Method). Of course, this article cannot discuss the theory of algorithms in any reasonable detail; we only sketched what was necessary to support our arguments on a possible new framework for various fields in mathematics. (For an introduction to the theory of algorithms, see e.g. Sedgewick 1983.)

3. What does this imply in math education? Whatever it implies, should be regarded with utmost caution and moderation. I feel that math education should follow what happens in math research, at least to a certain extent, in particular those (rare) developments there that fundamentally change the whole framework of the subject. Algorithmic mathematics is one of these. However, the range of the penetration of an algorithmic perspective in classical mathematics is not yet clear at all, and varies very much from subject to subject (as well as from lecturer to lecturer). Graph theory and optimization, for example, have been thoroughly re-worked from a computational complexity point

349 of view; number theory and parts of algebra are studied from such an aspect, but many basic questions are unresolved; in analysis and differential equations, such an approach may or may not be a great success; set theory does not appear to have much to do with algorithms at all. Our experience with "New Math", the adaptation of the set-theoretic foundations of mathematics in lower level mathematics education, warns us that drastic changes may be disastrous even if the new framework is well established in research and college mathematics. So let me just raise some ideas on the teaching of algorithms on various levels, emphasizing that they must be carefully discussed and tried out before any large scale implementation is attempted. Basically, some algorithms and their analysis could be taught about the same time when theorems and their proofs first occur, perhaps around the age of 14. Of course, certain algorithms (for multiplication and division etc.) occur quite early in the curriculum. But these are more recipes than algorithms; no correctness proofs are given (naturally), and the efficiency is not analyzed. The children have to learn (and practice) how to carry out these simple algorithms. This is like teaching theorems (axioms) without proofs, or teaching empirical facts in the sciences without experiments: necessary but not leading to really deep understanding. What I would consider as the beginning of learning "algorithmics" is to learn to design, rather than execute, algorithms. (For an elaboration of this idea, see e.g. Maurer 1984). The euclidean algorithm, for example, is one that can be "discovered" by students in class. In time, a collection of "algorithm design problems" will arise (just as there are large collections of problems and exercises in algebraic identities, geometric constructions or elementary proofs in geometry). Along with these concrete algorithms, the students should get familiar with basic notions of the theory of algorithms: input-output, correctness and its proof, analysis of running time and space, good characterizations read off from algorithms, algorithms motivated by good characterizations etc. Some possible types of algorithm-design problems, suitable probably already on the high school level: enumeration problems where no closed formula exists; elementary optimization problems in graph theory (e.g. maximum independent sets in trees, shortest paths, listing of cliques, circuits etc.); sorting and searching; simple (though inefficient) methods for primality testing, factorization, and many other problems in number theory; Gaussian elimination and other manipulations in linear algebra; convex hull and other elementary plane geometry constructions. In college, the shift to a more algorithmic presentation of the material should, and will, be easier and faster. Already now, some subjects like graph theory are taught in many colleges quite algorithmically: shortest spanning tree, maximum

350 flow and maximum matching algorithms are standard topics in most graph theory courses. This is quite natural since, as I have remarked, computational complexity theory provides a unifying framework for many of the basic graph-theoretic results. In other fields this is not quite so at the moment; but some topics like primality testing or cryptographic protocols provide nice applications for a large part of classical number theory.

4. Computers and algorithms At this point, I have to comment on the use of computers in the teaching of these topics. One should distinguish between an algorithm and its implementation as a computer program. The algorithm itself is a mathematical object; the program depends on the machine and/or on the programming language. It is of course necessary that the students see how an algorithm leads to a program that runs on a computer; but it is not necessary that every algorithm they learn about or they design be implemented. The situation is (again) analogous to that of geometric constructions with ruler and compass: some constructions have to be carried out on paper, but for some more, it may be enough to give the mathematical solution (since the point is not to learn to draw but to provide a field of applications for a variety of geometric notions and results). Let me insert a warning about the shortcomings of algorithmic language. There is no generally accepted form of presenting an algorithm, even in the research literature (and as far as I see, computer science text books for secondary schools are even less standardized and often even more extravagant in handling this problem.) The practice ranges from an entirely informal description to programs in specific programming languages. There are good arguments in favor of both solutions; I am leaning towards informality, since I feel that implementation details often cover up the mathematical essence. Let me illustrate this by two examples. An algorithm may contain a step "Select any element of set 5". In an implementation, we have to specify which element to choose, so this step necessarily becomes something like "Select the first element of set S". But there may be another algorithm, where it is important the we select the first element; turning both into programs hides this important detail. (Also, it may turn out that there is some advantage in selecting the last element of S. Giving an informal description leaves this option open, while turning the algorithm into a program forbids it.) To give my second example, recall that the Fibonacci numbers are defined by the recurrence -Pfc+i = -Pfe + jFfc-i

351 (and by F0 = 0 and Fi = 1). This recurrence provides a trivial algorithm to compute these numbers. Turning this recurrence into a program, we would get something like this: if F is the current Fibonacci number, and G is the previous, then (with an auxiliary variable H) H:=F,

F:=G

+ F,

G := H.

We see that the program contains a number of details that do not belong mathematically to the procedure of computing Fibonacci numbers: it stores Fk+i in the same register where Fk used to sit, but to do so, it has to "salvage" Fk since its value will be needed in the next step, and to do so, we need an "auxiliary variable" etc. 1 To show the other side of the coin, the main problem with the informal presentation of algorithms is that the "running time" or "number of steps" is difficult to define — as we have experienced above. Unfortunately, this depends on the details of implementation (down to a level below the programming language; it depends on the data representation and data structures used). Sometimes there is a way out by specifying which steps are counted (e.g. comparisons in a sorting algorithm, or arithmetic operations in an algebraic algorithm); but this is "cheating" in a sense since we disregard the time needed ' o handle the data, which may be as much as, or even more than, the time used by the "mathematical essential" steps. It should be mentioned that the polynomiality of an algorithm is "robust", i.e., it does not depend on implementation (although different implementations may have different polynomials in the bound on their running time). The route from the mathematical idea of an algorithm to a computer program is long. It takes the careful design of the algorithm; analysis and improvements of running time and space requirements; selection of (sometimes mathematically very involved) data structures; and programming. In college, to follow this route is very instructive for the students. But even in secondary school mathematics, at least the mathematics and implementation of an algorithm should be distinguished. An important task for mathematics educators of the near future (both in college and high school) is to develop a smooth and unified style of describing and analyzing algorithms. A style that shows the mathematical ideas behind the design; that facilitates analysis; that is concise and elegant would also be of great help in overcoming the contempt against algorithms that is felt nowadays both from the side of the teacher and of the student. X

I am grateful to Jack Edmonds for these examples and arguments — and regret not to have given a more detailed exposition of them.

352 References P. R. Halmos (1981), Applied mathematics is bad mathematics, in Mathematics Tomorrow (ed. L. A. Steen), Springer, 9-20. L. Lovasz (1986), An algorithmic Theory of Numbers, Graphs, and Convexity, CBMS-NSF Reg. Conf. Series in Appl. Math. 50; SIAM. S. Maurer (1984), Two meanings of algorithmic mathematics, Mathematics Teacher 430-435. S. Maurer (1985), The algorithmic way of life is best; reflexions by R. G. Douglas, B. Korte, P. Hilton, P. Renz, C. Smorynski, J. M. Hammersley and P. R. Halmos; College Math. Journal 16, 2-18. R. Sedgewick (1983): Algorithms, Addison-Wesley.

,ASZL6 LOVASZ

One Mathematics "here i s no n a t u r a l way ;o divide mathematics vlathematics is torn by many divi;ion lines. The most prominent of hese runs between Pure and applied Mathematics. The conroversy around Bourbaki focuses in Abstract vs. Concrete. The listinction between Structural vlathematics (whose main results ire theorems and proofs) and Ugorithmic Mathematics (whose •esults are algorithms and their inalysis) can be traced back to incient times.Thcre is a deep iivision (or at least so it appears) >etween Continuous and Discrete Vlathematics. ioinc of these divisions are consequences of different working environments: For example, applied nathematicians work under very lifferent financing conditions and ;uccess criteria from pure mathenaticians. Some others are cul:ural: Various branches of mathenatics have their own system of inferences, journals, prizes; a lifferent set of mathematical con:epts, basics and perhaps values hat are assumed in conversation. ^jid of course the appreciation of :he abstract varies very much with personality and individual taste. Should we accept this partition as i fact of life: "I am a bad pure abstract algorithmic discrete mathematician"? Some say that this is rood enough, and one should ac:ept the splitting of mathematics nto smaller and smaller independint branches as a fact of life. . feel that accepting this would •eally lead to tragic consequences,

ind that there is a deep unity of our sci;nce that gives it its strength and vitality. will argue that recent trends in our icience make these division lines more ;omplex than they appear; that we have to io our best to bridge these gaps; and that :hese same new trends may provide means :o do so.

Three new trends changing the tforld of mathematics The size of the community. It is a commonplace that the number of mathematical jublications (along with publications in 'ther sciences) has increased exponenially in the last 50 years. Mathematics has jutgrown the small and close-knit comnunity that it used to be. And with inreasing size, the profession is becoming nore diverse, more structured and more ;omplex. Vlathematicians are conservative people; '. don't mean that we are right-wing (as far ts I can see, we span the political spec:rurn just like every other profession), but ve don't push for changes: "We are reluc.ant to spend time on anything other than rying to prove that p * NP (or the Rienann Hypothesis, or whatever problem ceeps us infatuated at the moment). So pretend that mathematical research is ts it used to be. We believe that we find ill the information that might be relevant jy browsing through the new periodicals •n the table in the library, and that if we publish a paper in an established journal, hen it will reach all the people whose •esearch might utilize our results. V larger structure is never just a scaled-up version of the smaller. In larger and nore complex animals an increasingly arge fraction of the body is devoted to

Reprinted with permission. The Berlin Inteligencer, International Congress of Vlathematics, Berlin, August 1998, pp. 10-] 5. © Springer-Verlag.

354

overhead": The transportation ofmatediscrete: simple basic questions ial and the coordination of the function ike finding matching patterns, ot if various parts. In larger and more :racing consequences of flipping :omplex socieries an increasingly large aver substrings, sound more raction of the resources is devoted to ramiliar to the graph theorist than ion-productive activities like transporta- :o the researcher of differential :ion and information processing. We have equations. A question about the o realize and accept that a larger and nformation content, redundancy, larger part of our mathematical activity ir stability of the code may sound should, and will, be devoted to communi- :oo vague to a classical mathemacation. This is easy to observe: The numtician but a theoretical computer ler of professional visits, conferences, icientisc will immediately see at workshops, research institutes is increaseast some tools to formalize it ng fast, e-mail is used more and more. 'even if to find the answer may be The percentage of papers with multiple :oo difficult at the moment). luthors has also jumped. But probably we Even physics has its encounters will reach the point soon where mutual with unusual discrete mathematipersonal contact does not provide suffi;al structures: elementary parcient information flow. ades, quarks and the like are very There is another consequence of the Combinatorial; understanding increase in mass: the inevitable formation basic models in statistical mechanof smaller communities, one might say ics requires graph theoty and subcultures. These seem to arise on a probability. random basis, but then they persist and Economics is a heavy user of determine research directions for quite a mathematics - a n d much of its long time. One such subculture is Disneed is not part of the traditional crete Mathematics —Theory of Compuapplied mathematics toolbox. ting - Operations Research. The success of linear programming in economics and operations research depends on conditions I DON T SEE ANY REASON, OTHER THAN of convexity and unlimited divisiC U L T U R A L , WHY COMPUTATIONAL C O M P L E X I bility; taking indivisibilities into TY THEORY SHOULD BE EMBRACED BY THE account (for example, logical DESIGNERS OF DISCRETE A L G O R I T H M S , BUT decisions, or individuals) leads to VIEWED: WITH GRAVE S U S P I C I O N BY THE integer programming and other M A J O R I T Y OF DESIGNERS OF NUMERICAL combinatorial optimization ALGORITHMS . models, which are much more 'New areas of application. The traditional (difficult to handle. finally, there is a completely new area of application of mathematics is area of applied mathematics: comphysics; and no doubt this area involves J:he deepest mathematics and the greatest puter science. The development of success stories. The branch of mathemat- electronic computation provides a vast array of well-formulated, ics used in these applications is analysis, difficult, and important mathematthe real hatd core of mathematics. ical problems, raised by the study But in the boom of scientific research in 0f algorithms, data bases, format the second half of this century, many languages, ctyptography and comother sciences have come to the point puter security, VLSI layout, and whete they need serious mathematical much more. Most of these have to tools. Quite often the traditional tools of do with discrete mathematics, fotanalysis are not adequate. Tial logic, and ptobability. For example, biology tries to understand Which branches of mathematics :he genetic code: a gigantic task, which ,vill be applicable in the near future s the key to understanding life and, ultis utterly unpredictable. Just 25 Tiately, ourselves. The genetic code is

vears ago questions in number theory like j low many primes there are between j ; x 10 200 and 4 x io 2 0 0 seemed to belong ! :o the purest, most classical and comI iletely inapplicable mathematics; now j 'elated questions belong to the core of J piathematical cryptology and computer \ security. ! \t would seem that this diversity of appli- ! cations is anothet centrifugal force; but I :hink that, to the contrary, it should itrengthen the flow of information across . division lines. j

NO FIELD CAN R E T R E A T I N T O ITS I V O R Y

I

TOWER AND C L O S E ITS D O O R S TO APPL

j

C A T I O N S ; NOR CAN ANY FIELD CLAIf)l TO

I

BE THE A P P L I E D M A T H E M A T I C S .

New tools: computers. Computers, of course, are not only sources of interesting and novel mathematical problems. They also provide new tools for doing and organizing our research. There is obviously a large variation in the relationship between mathematicians and computers. Some avoid computers altogether; others are glued to theirs. I use them for e-mail and word processing like inost of us; less regularly, I use them for experimentation, and for getting information through the web. I have become iddicted to searching in the Mathematical Reviews database, and I •ind it mote and more convenient to get nformation by browsing through electronic journals and, perhaps even tiore significantly, through home pages if other mathematicians. Are these uses of computers just toys or at lest matters of convenience? I think not, md that each of these is going to have a profound impact on our science. Electronic journals and databases, home pages, and e-mail provide new ways of dissemination of results and ideas. In a sense, they reinforce the increase in the volume of research: not only are there increasingly more people doing research, but an increasingly large fraction of this information is available at our fingertips [and often increasingly loudly and aggressively: the etiquette of e-mail is far from feolid). But we can also use them as ways

if coping with the information explosion. At first sight, word processing just looks like a convenient way of writing papers. JThe final output of mathematical research [s still a printed paper, read by others in a journal or perhaps more and more from a manuscript printed out in their office. But slowly many features of electronic versions become available that are superior to the usual printed papers: Hyperlinks, colored figures and illustrations, animations and the like. A mathematical paper is almost never read in a strictly linear fashion: one jumps back to refresh a definition, jumps ahead to see how a certain lemma is applied, skips proofs at first reading, returns repeatedly to check how the arguments work on a certain example-this is more reminescent of browsing the internet than of reading a novel. And if a document is not read in a linear way, why wrirc it in a linear way? I will not discuss here the opportuniries fand traps) provided by these features of felectronic publication; but it is quite probable that they will gradually transorm the way we write papers; and jossibly through rhis also the way wc do *esearch. New forms of mathematical activity The traditional 2500 year old paradigm pf mathematical research is defining horions, stating theorems and proving them. Perhaps less recognized, but almost this old, is algorithm design (rhink of rhe Euclidean Algorithm or Newton's Method). While different, these two ways of doing mathematics are strongly interconnected. It is also obvious that computers have increased the visibility and respectability of algorithm design substantially.

Surveys. The most serious threat p the unity of mathematics is the sheer size of mathematical research. No one can read even a tiny fraction of new research papers. One solution to this problem is the creation of an activity that deals with the secondary processing of research results. For lack of (1 better word, I'll call rhis expository writing, although I'd like to consider it more as a form of mathematical research than as a

F

orm of writing: finding the ramiScations of a result, its connections with results in other fields, explaining, perhaps translaring it for people coming from a different subculture.

The community has invented this activity already: There is more and more demand for expositions, surveys, mini courses, handbooks and encyclopedias. Many conferences are mostly or exclusively devoted to survey-type talks; publishers much prefer volumes of survey articles to volumes of research papers. Wc organize the International Congress every 4 years (and many other regional meerings of the

Same kind). While some matheI inaticians feel that the Congress is worthless (and it is if you consider ir as a big research conference), people in other fields envy us for r. It is a great asset if used for naintaining the unity of our field, is a forum forgiving surveys, expositions of the most important new results and new areas and methods of applications. Yet we all feel reluctance towards accepting expository and survey However, as a consequence of the increase writing as scientific achievement. jn the size of the research community, There is often a reservation abour this paradigm must be enriched by new somebody's writing an exposition forms of scienrific achievemenr. These of somebody else's new result may include writing good expositions and (I personally feel that this activity surveys, formulating problems and conjecphould be encouraged instead). tures, compiling examples, experimenting If, as suggested, writing expositions ind reporting the results. Let me comshould become a highly regarded nent on the first two of these. research activity, one has to find

356

conjecture is g o o d , o n e expects that its

Whole euclidean plane). It is a

into o u r p i c t u r e of achievements, i n c l u d i n g jobs,

resolution should advance o u r k n o w l e d g e

patural a n d powerful m e t h o d to

P r o m o t i o n s , grants?

iubstantially. M a n y m a t h e m a t i c i a n s feel

s t u d y discrete structures by

{We k n o w little a b o u t the criteria for a good

:hat this is the case w h e n we can clearly

' e m b e d d i n g " t h e m in the c o n t i n -

m a t h e m a t i c a l survey. We don't have a good for-

;ee the place of the conjecture, a n d its

uous world.

hial criterion m a r k i n g a g o o d t h e o r e m , either.

?robable solution, in the b u i l d i n g of

T h e leading r h e m e in c o m b i n a t o -

However, t h e r e are s o m e reasonably exact neces-

Tiathematics; b u t there are conjectures so

al o p t i m i z a t i o n in the 6o's a n d

;ary c o n d i t i o n s (the t h e o r e m should be new, n o n -

iurprising, so utterly inaccessible by

o's was the application of tech-

:riviai, a n d correct), a n d the c o m m u n i t y tends

c u r r e n t m e t h o d s , t h a t their resolution

niques o f linear p r o g r a m m i n g to

:o agree o n o t h e r criteria t h a t are m o r e difficult

m u s t b r i n g s o m e t h i n g n e w - w e just don't

c o m b i n a t o r i c s . It is q u i t e easy to

to formalize like interest a n d significance.

k n o w where.

f o r m u l a t e the m o s t i m p o r t a n t

ays of evaluating it. H o w s h o u l d surveys fit

Let m e propose a radical idea: Let us evaluate Surveys in the way h u m a n i t i e s evaluate their Achievements. W e t e n d t o look d o w n u p o n these ;ireas as "soft", a n d believe t h a t (in c o n t r a s t to o u r own "hard a n d exact" science) success is a •natter of luck or, worse, good abilities in self-pro-

t

combinatorial optimization prob-

i s c r e t e and c o n t i n u o u s

ems as linear p r o g r a m s w i t h i n t e -

he m o s t intrinsic a m o n g the division

grality c o n d i t i o n s , and it is quite

lines is discrete vs. c o n t i n u o u s , because it

easy to solve these, if we disregard

nvolves basic structures a n d m e t h o d s of ur science. In this lasr section, which

the integrality c o n d i t i o n s ; the g a m e is to find ways to wrire u p

m o t i o n . Clearly this feeling is far from the t r u t h ,

necessarily gets a bit m o r e technical, I p u t

these linear p r o g r a m s in such a

ind. h u m a n i t i e s have their own ways of recog-

forth s o m e examples d e m o n s t r a t i n g h o w

way t h a t disregarding integrality

l i z i n g excellence in intellectual achievements.

m u c h we c o u l d lose if we let this c h a s m

c o n d i t i o n s is justified.

We c o u l d o n l y gain b y l e a r n i n g h o w t o d o this

g r o w wider, a n d h o w m u c h we can gain

\The power of tools from elsewhere,

without o u r m o r e direct criteria.

by b u i l d i n g bridges over it.

To s u p p o r t m y plea for the u n i t y of

fnftnite

Tiathematics, let me discuss o n e

OUR S C I E N C E C O U L D O N L Y C A I N 8V HtJRE OF THE M E T H O D S

FROM THE

T R E A S U R Y OF H U M A N T H O U G H T

ADAPTING

VAST

I N T O OUR

OWN

P U R S U I T OF K N O W L E D G E .

Problems and conjectures. In a small c o m m u n i t y , everybody knows w h a t the m a i n p r o b l e m s are. p u t in a c o m m u n i t y o f 1 0 0 , 0 0 0 people, p r o b l e m s have to be identified a n d stated in a precise way. 'oorly stated p r o b l e m s lead to b o r i n g , irrelevant ults. T h i s elevates the f o r m u l a t i o n of conjecures t o t h e r a n k of research results. C o n j e c t u r i n g

i

iecame an art in the h a n d s of the late Paul Erdos, / h o f o r m u l a t e d m o r e conjectures t h a n p e r h a p s

£ll m a t h e m a t i c i a n s before h i m p u t together. H e Considered his conjecrures as p a r t of his m a t h e matical oeuvre as m u c h as his t h e o r e m s . O n e of U y m o s t prized m e m o r i e s is the following c o m m e n t from h i m : "I never envied a t h e o r e m from Enybody; b u t I envy you for this conjecture." )f course, w i t h conjectures we run into the same ifficulty as with surveys: It is difficult to f o r m u late w h a t makes a good conjecture. A n d indeed, there is a lot of controversy a r o u n d the style of fcrdos conjectures. It is easy to agree that if a

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to finite.

It is perhaps unnecessary

:o argue t h a t discrete a n d c o n t i n u o u s

recent d e v e l o p m e n t in the t h e o r y

Tiathematics c o m p l e m e n t each o t h e r

of algorithms. M y starting example

m d that each utilizes m e t h o d s a n d tools

(s a simple a l g o r i t h m i c p r o b l e m in

from the other. W e use the finite to

g r a p h t h e o r y : Given a (finite)

a p p r o x i m a t e rhe infinite. T o discretize a

g r a p h , find a p a r t i t i o n of its n o d e

c o m p l i c a t e d c o n t i n u o u s s t r u c t u r e has

pet into two classes so t h a t the

ilways been a basic m e t h o d - from the

l u m b e r of edges c o n n e c t i n g the

definition o f the R i e m a n n integral

w o classes is as large as possible.

:hrough t r i a n g u l a t i n g a manifold in (say)

In spite of its simplicity, this is a

l o m o l o g y t h e o r y to numerically solving a

rather i m p o r t a n t p r o b l e m ; see the

martial differential e q u a t i o n o n a grid.

t n o n o g r a p h o f Deza and Laurent

'.n spite o f this, I feel t h a t t h e status o f

for a d e s c r i p t i o n o f its far-reaching

amplications of discrete m a t h e m a t i c a l

;onnections.)

m e t h o d s in c o n t i n u o u s m a t h e m a t i c s s less than satisfactory. Perhaps c o m b i n a rorics has nor yet reached the d e p t h a n d lower of analysis or algebra. Perhaps p a r t )f the reasons is cultural: A discrete n a t h e m a t i d a n is m o r e likely to have studied Galois T h e o r y or the B o r s u k - U l a m JTheorem than a "classical" m a t h e m a t i c i a n , say, R a m s e y T h e o r y or the M a x - F l o w Vlin-Cut T h e o r e m . Finite to infinite.

It is a slightly m o r e

mbtle t h o u g h t t h a t the infinite is often 'or p e r h a p s always?) a n a p p r o x i m a t i o n o f h e large finite. C o n t i n u o u s structures ire often cleaner, m o r e s y m m e t r i c , a n d icher t h a n their discrete c o u n t e r p a r t s Tor example, a p l a n a r grid has a m u c h mailer degree of s y m m e t r y t h a n the

Unfortunately, this p r o b l e m is N P iard. If you are n o t familiar w i t h :his basic n o t i o n of complexity :heory, it means roughly t h a t there js n o efficient (polynomial time) ilvay no find t h e best p a r t i t i o n (at least subject to the hypothesis t h a t p). W e m u s t settle for less: ay, to find an a p p r o x i m a t e l y o p t i m a l p a r t i t i o n . It is very easy to find a p a r t i t i o n w h e r e at least half Lf the edges go across; this was first observed by E r d o s in t h e 6o's in a different c o n t e x t . Since no p a r t i t i o n can pick u p m o r e t h a n all edges, this gives an a p p r o x i m a t e s o l u t i o n that achieves at feast 50% of the o p t i m u m .

Can we do better? This really innocent question remained unanswered until fairly recently, almost Simultaneously, two important results were obtained: Goemans and Williamson gave an efficient approximation algorithm that gets within 13% of the optimum; and building on a series of weaker results, Hastad proved that no efficient (polynomial time) approximation algorithm whatever can do better than 6%. This is a surprisingly small gap for this kind of problem, but from our point of view, it is more important that both results depend on tools that come from quite unexpected places. The negative result is one of several lower bounds an approximability of various optima, proved by iimilar means.The first of these proofs was an application of a result in the theory of interactive proof systems, a very interesting, but at first sight, quite specialized area in complexity theory. Later improvements revealed that the most important mathematical construction in the b roof is an error-correcting code obtained by algebraic methods.

Deeper u n i t y Probability theory is only one illustration of the unity in mathematics that goes deepet than just using tools from other branches. Many of the basic questions are not a priori discrete or continuous in nature - they can be modelled as discrete problems, or continous problems. In the last years, I have worked on sampling algorithms (algorithms generating a uniformly distributed random element From a set that is large and often only implicitly described). This question leads to estimating the mixing time of Markov chains (the number of steps before the chain becomes essentially stationary). From the point of view of this application, |t is natural to consider finite Markov chains - a computation in a computer is necessarily finite. But in the analysis, it depends on the particular application whether one prefers to use a finite, or a general measurable, state space. All the essential (and very interesting) connec(The algorithm itself depends on another line of tions that have been discovered hold in previous results based on connections between both models. In fact, the genetal mathedistant areas. The key step is the use of semimatical issue is dispersion: We might definite optimization, which is an extension of be intetested in dispersion of heat in linear programming, building heavily on the a material, or dispersion of probability spectral theory of symmetric matrices. Again, during a random walk, or many other this is not an isolated result; the use of semirelated questions. There is always a Lapladefinite optimization (combined with randomcian operator that describes one step of ized algorithms) has been very successful in the the dispersion. The speed of the disperdesign of approximation algorithms. sion is governed by the spectral gap of Probability. This brings me to a topic that seems the Laplacian; but if infotmation about to bridge most of the division lines in mathematthe spectral gap is not available, one can ics. The importance of probabilistic methods also relate the dispersion speed to isopein combinatorics, graph theory, and the theory of rimetric inequalities in the state space. {llgorithms is exploding. Beside their traditional To establish isoperimetric inequalities, use in Monte-Carlo methods in integration and pne most often constructs (explicitly or simulation, randomized algorithms are used for implicitly) muiticommodity flows. counting, exact and approximate optimization, My second example is more vague. I start primality testing, and one could go on. With probably the most important series In non-algorithmic graph theory, the probabiliof results in graph theory in the last stic method was first introduced in the 50's decade or two, the Graph Minor Theory by Erdos. As a method for showing the existence developed mainly by Robertson and bf objects (graphs, or colorings of a given graph, Seymour. Recall Kutatowski's classical etc.), it is now basic and extremely powerful. theorem: A graph can be embedded in the Probability enters proofs of theorems whose plane if and only if it does not contain statement has nothing to do with probability. pvo specific gtaphs. The notion of conThe role of probability is certainly not restricted tainment can be defined here in several to combinatorics and graph theory: Just let me different but equivalent ways; let us settle mention sieve methods in prime number theory, on "containment as a minor", which Or the explanation of tutbulence in terms of means the following: H is a minor oft; if it statistical mechanics.

can be obtained from G by deleting edges and nodes, and contracting some edges to single nodes. The class of planar graphs (just like the class of graphs embeddable in any other fixed surface) is closed under taking of minors. It is clear therefore that this class can be characterized by excluded minors (just list alt minor-minimal nonplanar graphs). The point in Kuratowski's Theorem is to show that this set of excluded minors consists of two graphs only. Wagner formulated the daring conjecture in the 30's that every class of graphs that is closed under taking of minors can be pharacterized by a finite list of excluded minors. The central result in rhe Robertfcon-Seymour theory is the proof of this conjectute. However, what 1 would like to comment on is an "auxiliary" result, Which describes, in a sense, large graphs not containing a fixed graph H as a minor. Informally, it says that every such graph can be constructed in the following way: Take an arbitrarily large graph embedded in a surface with bounded genus; add edges connecting nodes on the same face at bounded distance; add abounded

number of further nodes; and glue together such graphs along bounded sets of nodes in a tree-like fashion. Here,"bounded"means a bound depending on the graph G but nothing else. We can see the beginnings of a "global" theory of graphs emerging here: What does a huge graph look like? What hidden structures can be identified in this seemingly unstructured universe? Probably there is a more general theory, identifying 3-dimensional, 4-dimensional etc. structures in large graphs. But the formidable difficulties in the Robertson-Seymour theory (stretching over 19 papers now) warn us that such a theory ivill not be easy to establish. Recently I learned about the work of David and Semmes, and I could not help noticing some analogy with the Robertson-Seymour theory. They give a decomposition of a "reasonable"metric space into pieces of different dimension on different scales. Is there more to this analogy? put speaking of "global" graph theory brings other important results to mind. The Regularity Lemma of Szemeredi states that every huge graph can be "decomposed" into a bounded number of pieces that look "random" (the number of pieces depends on the error in approximating randomness; an exact statement would take too much

preparation again). Recently, a flow of exciting applications of this basic lemma emerged. Does Szemeredi's Lemma have a more general setting? An indication may be the recent work of Frieze and Kannan, showing a connection between Szemeredi's Lemma and low-rank approximation of matrices. There is no natural way to divide mathematics, but serious communication gaps can arise unless we realize that we have to pay for avoiding them: Pay not only with organizational effort but also with research time devoted to expository Writing and reading those expositions, to popularizing mathematics and to listening to mathematical problems from various areas of applications. Acknowledge!

am indebted to Tom Zaslavsky for

reulingrimu

refully and for suggesting many

aszld Lovasz is Profe 1 fJomputer Sc ence at Yale University, (view Haven.

Wolf Prize in Mathematics, Vol. 2 (pp. 359^15) eds. S. S. Chern and F. Hirzebruch © 2001 World Scientific Publishing Co.

Curriculum Vitae

Born Orange, New Jersey, USA, February 20, 1931 Positions Held 1954-57 Assistant Professor, Princeton University 1957-60 Associate Professor, Princeton University 1960-67 Professor, Princeton University 7/1/1963-7/1/73 Member, Institute for Advanced Study 1967-69 Professor, University of California, Los Angeles 1969-70 Professor, Massachusetts Institute of Technology 7/1/1970-7/88 Professor, Institute for Advanced Study 8/88- Professor and Director, Institute for Mathematical Sciences, SUNY at Stony Brook Honors and Societies American Mathematical Society Fields Medal, 1962 National Academy of Sciences, USA, 1963American Philosophical Society, 1965National Medal of Science, 1967 James Madison Medal, Princeton University, 1977 Wolf Prize, 1989

360 Topological Methods in Modern Mathematics

PUBLICATIONS OF J O H N MILNOR —THROUGH 1992

1.

On the total curvature of knots, Ann. of Math. (2) 52 (1950), 248-257.

2.

An axiomatic approach to measurable utility, (with I. N. Herstein), Econometrica 21 (1953), pp. 291-297.

3.

The characteristics of a vector field on the two-sphere, Ann. of Math. (2) 58 (1953), 253-257.

4.

On total curvatures of closed space curves, Math. Scand. 1 (1953), 2 8 9 296.

5.

Sums of positional games, in "Contributions to the Theory of Games, Vol. II," (H. W. Kuhn and A. W. Tucker, eds.), Ann. of Math. Stud. No. 28, Princeton University Press, Princeton, N. J., 1953, pp. 2 9 1 301.

6.

Games against nature, in "Decision Processes," (R. M. Thrall, C. H. Coombs and R. L. Davis, eds.), Wiley, New York; Chapman and Hall Ltd., London, 1954, p p . 49-59. (Reprinted in "Game Theory and Related Approaches to Social Behavior," (M. Shubik, ed.), Wiley, New York, 1964.)

7.

Some experimental n-person games, (with G. Kalish, J. Nash, and E. D. Nering), in "Decision Processes," (seeprevious reference), pp. 301-327.

8.

Link groups, Ann. of Math. (2) 59 (1954), 177-295.

9.

Construction of universal bundles, I, Ann. of Math. (2) 63 (1956), 2 7 2 284.

xv

PUBLICATIONS OF JOHN MILNOR

10.

Construction of universal bundles, II, Ann. of Math. (2) 63 (1956), 430436.

11.

On the immersion of n-manifolds in n + l-space, Comment. Math. Helv. 30 (1956), 275-284.

12.

On manifolds homeomorphic to the 1-sphere, Ann. of Math. (2) 64 (1956), 399-405.

13.

The geometric realization of a semi-simplicial complex, Ann. of Math. (2) 65 (1957), 357-362.

14.

Groups which act on Sn without fixed points, Amer. J. Math. 79 (1957), 623-630.

15.

Isotopy of links, in "Algebraic Geometry and Topology. A Symposium in Honor of S. Lefschetz," (R. H. Fox, D. C. Spencer and A. W. Tucker, eds.), Princeton University Press, Princeton, N.J., 1957, pp. 280-306.

16.

On games ofsurvival, (with L. S. Shapley), in "Contributions to the Theory of Games, Vol. Ill," (M. Dresher, A. W. Tucker and P. Wolfe, eds.), Ann. of Math. Stud. No. 39, Princeton University Press, Princeton, N. J., 1957, p p . 15-45.

17.

On the existence of a connection with curvature zero, Comment. Math. Helv. 32 (1958), 215-223.

18.

Some consequences of a theorem of Bott, Ann. of Math. (2) 68 (1958), 444-449.

19.

On the parallelizability of the spheres, (with R. Bott), Bull. Amer. Math. Soc. (N.S.) 64 (1958), 87-89. .

20.

The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), 150-171.

21.

On simply connected A-manifolds, in "Symposium Internacional de Topologia Algebraica," Univ. Nac. Autonoma de Mexico and UNESCO, Mexico City, 1958, pp. 122-128.

22.

On the Whitehead homomorphism J, Bull. Amer. Math. Soc. (N.S.) 64 (1958), 79-82.

23.

Differentiable structures on spheres, Amer. J. Math. 81 (1959), 962-972.

24.

Sommes de varietes differentiables et structures differentiables des spheres, Bull. Soc. Math. France 87 (1959), 439-444.

25.

On spaces having the homotopy type of a CW-complex, Trans. Amer. Math. Soc. 90 (1959), 272-280.

26.

Bernoulli numbers, homotopy groups, and a theorem of Rohlin, (with M. A. Kervaire), in "Proc. Internat. Congress Math. 1958," Cambridge University Press, 1960, pp. 454-458.

362 PUBLICATIONS OF JOHN MILNOR

27.

Two remarks on fibre homotopy type, (with E. Spanier), Pacific J. Math. 10 (1960), 585-590.

28.

On the cobordism ring Q* and a complex analogue, I, Amer. J. Math. 82 (1960), 505-521.

29.

A procedure for killing homotopy groups of differentiable manifolds, in "Differential Geometry. Proc. Symposia Pure Math. Vol. Ill," Amer. Math. S o c , Providence, R. I., 1961, pp. 39-55.

30.

Two complexes which are homeomorphic but combinatorially distinct, Ann. of Math. (2) 74 (1961), 575-590.

31.

On 2-spheres in 4-manifolds, (with M. A. Kervaire), Proc. Nat. Acad. Sci. U.S.A. 4 7 ( 1 9 6 1 ) , 1651-1657.

32.

Variedades lisas confrontera, An. Inst. Mat. Univ. Nac. Autonoma Mexico 1 (1961), 82-116.

33.

A unique decomposition theoremfor 3-manifolds, Amer. J. Math. 84 (1962), 1-7.

34.

A duality theorem for Reidemeister torsion, Ann. of Math. (2) 76 (1962), 137-147.

35.

A survey of cobordism theory, Enseign. Math. (2) 8 (1962), pp. 16-23.

36.

On axiomatic homology theory, Pacific J. Math 12 (1962), 337-341. (Reprinted in "Algebraic Topology: A Student's Guide," by J. F. Adams, Cambridge University Press, 1972.)

37.

An example of anomalous singular homology, (with M. G. Barratt), Proc. Amer. Math. Soc. 13 (1962), 293-297.

38.

The work of J. H. C. Whitehead, in "The Mathemadcal Works of J. H. C. Whitehead, Vol. 1," Pergamon Press, Oxford, New York, 1962, pp. xxi-xxxiii.

39.

Topological manifolds and smooth manifolds, in "Proc. Internat. Congress Math. Stockholm 1962," Inst. Mittag-Leffler, Djursholm, 1963, pp. 132-138.

40.

Spin structures on manifolds, Enseign. Math. (2) 9 (1963), 198-203.

41.

Groups of homotopy spheres, (with M. A. Kervaire), Ann. of Math. (2) 77 (1963), 504-537.

42.

"Morse Theory. Based on lecture notes by M. Spivak and R. Wells," Ann. of Math. Stud. No. 51, Princeton University Press, Princeton, N. J., 1963. (Russian translation 1965, Japanese translation 1968.)

43.

On the Betti numbers of real varieties, Proc. Amer. Math. Soc. 15 (1964), 275-280.

xvii

PUBLICATIONS OF JOHN MILNOR

44.

Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 542.

45.

Microbundles, I, Topology 3 (1964) Suppl. 1, 53-80.

46.

Most knots are xvild, Fund. Math. 54 (1964), 335-338.

47.

Differential topology, in "Lectures on Modern Mathematics, Vol. II," T L. Saaty, ed., Wiley, New York, 1964, pp. 165-183. (Russian translation: Uspehi Mat. Nauk 20 (1965), pp. 41-54.)

48.

Some curious involutions of spheres, (with M. W. Hirsch), Bull. Amer. Math. Soc. (N.S.) 70 (1964), 372-377.

49.

Some free actions of cyclic groups on spheres, in "Differential Analysis. Bombay Colloquium," Oxford University Press, London, 1964, pp. 3 7 42.

50.

On the structure of Hopf algebras, (with J. C. Moore), Ann. of Math. (2) 81 (1965), 211-264.

51.

On the Stiefel-Whitney numbers of complex manifolds and of spin manifolds, Topology 3 (1965), 223-230.

52.

Remarks concerning spin manifolds, in "Differential and Combinatorial Topology. A Symposium in Honor ofMarston Morse,'" Princeton University Press, Princeton, N. J., 1965, pp. 55-62.

53.

"Lectures on the /i-Cobordism Theorem. Notes by L. Siebenmann and J. Sondow," Princeton University Press, Princeton, N.J., 1965. (Russian translation 1969.)

54.

"Topology from the Differential Viewpoint," University Press of Virginia, 1965. (Polish translation 1969.)

55.

Whitehead torsion, Bull. Amer. Math. Soc. (N.S.) 72 (1966), 358-426.

56.

Singularities of 2-spheres in 4-space and cobordism of knots, (with R. H. Fox), Osaka J. Math. 3 (1966), 257-267.

57.

Solution of the congruence subgroup problem for SL„(/i > 3) and Sp2„(fl > 2), (widi H. Bass and J. P. Serre), IHES Publ. Math. No. 33 (1967), 59-137.

58.

On characteristic classes for spherical fiber spaces, Comment. Math. Helv. 43 (1968), 51-77.

59.

A note on curvature and fundamental (1968), 1-7.

60.

"Singular Points of Complex Hypersurfaces," Ann. of Math. Stud. No. 6 1 , Princeton University Press, Princeton, N. J., 1968.

group, J. Differential Geom. 2

364 PUBLICATIONS OF JOHN MILNOR

61.

Infinite cyclic covering, in "Conference on the Topology of Manifolds," (J. G. Hocking, ed.), Prindle, Weber and Schmidt, Boston, 1968, pp.115-113.

62.

Growth offinitelygenerated solvable groups,]. Differential Geom. 2 (1968), 447-449. On isometries of inner product spaces, Invent. Math. 8 (1969), 83-97. Semi-characteristics and cobordism, (with G. Lusztig and F. P. Peterson), Topology 8 (1969), 357-359. A problem of cartography, Amer. Math. Monthly 76 (1969), 1102-1112. Torsion et type simple d'homotopie, (with O. Burlet), in "Essays on Topology and Related Topics. Memoires dedies a George de Rham," SpringerVerlag, Berlin, New York, 1970, pp. 12-17. Algebraic K -theory and quadratic forms, Invent. Math. 9 (1970), 318-344.

63. 64. 65. 66.

67. 68.

Isolated singularities defined by weighted homogeneous polynomials, (with P. Orlik), Topology 9 (1970), 385-393. 69. Symmetric inner product spaces in characteristic 2, in "Prospects in Mathematics," Ann. of Math. Stud. No. 70, Princeton University Press, Princeton, N. J., 1971, pp. 59-75. 70. "Introduction to Algebraic A'-Theory," Ann. of Math. Stud. No. 72, Princeton University Press, Princeton, N. J., 1971. (Russian translation 1974.) 71. On the construction FK, in "Algebraic Topology: A Student's Guide," by J. F. Adams, Cambridge University Press, 1972, pp. 119-136. 72. "Symmetric Bilinear Forms," (with D. Husemoller), Springer-Verlag, Berlin, New York, 1973. 73. 74.

75.

76.

"Characteristic Classes," (with J. Stasheff), Ann. of Math. Stud. No. 76, Princeton University Press, Princeton, N. J., 1974. Isolated critical points of complex functions, in "Differential Geometry. Proc. Symposia Pure Math. Vol. XXVII, Part I," Amer. Math. S o c , Providence, R. I., 1975, pp. 381-382. On the 3-dimensional Brieskorn manifolds M(p, q, r), in "Knots, Groups and 3-manifolds. Papers dedicated to the memory of R. H. Fox," (L. P. Neuwirth, ed.), Ann. of Math. Stud. No. 84, Princeton University Press, Princeton, N.J., 1975, p p . 175-225. Hilbert's problem 18: on crystallographic groups, fundamental domains, and on sphere packing, in "Mathematical Developments Arising from Hilbert Problems. Proc. Symposia Pure Math. Vol. XXVIII," Amer. Math. S o c , Providence, R. I., 1976, pp. 491-506.

xix

PUBLICATIONS OF JOHN MILNOR

77.

Differential Geometry (problem list), ibid., 54-57.

78.

Curvatures of left invariant metrics on Lie groups, Adv. in Math. 21 (1976), 293-329.

79.

On deciding whether a surface is parabolic or hyperbolic, Amer. Math. Monthly 84 (1977), 43-46.

80.

On fundamental groups of complete affinely flat manifolds, Adv. in Math. 25 (1977), 178-187.

81.

Characteristic numbers of 3-manifolds, (with W. Thurston), Enseign. Math. (2) 23 (1977), 249-254.

82.

Analytic proofs of the "hairy ball theorem " and the Brouwerfixed point theorem, Amer. Math. Monthly 85 (1978), 521-524.

83.

Hyperbolic geometry: the first 150 years, Bull. Amer. Math. Soc. (N.S.) 6 (1982), 9-24.

84.

On the homology of Lie groups made discrete, Comment. Math. Helv. 58 (1983), 72-85.

85.

On the geometry of the Kepler problem, Amer. Math. Monthly 90 (1983), 353-365.

86.

Remarks on infinite-dimensional Lie groups, in "Relativity, Groups and Topology II," Les Houches Session XL, 1983, B. S. DeWitt and R. Stora, eds., Elsevier Science Publishers B.V., North Holland, 1986, p p . 1007-1058.

87.

Values of large games II: oceanic games, (with L. S. Shapley, written in 1961), Math. Oper. Res. 3 (1978), 290-307.

88.

On polylogarithms, Hurwitz zeta functions, and the Kubert identities, Enseign. Math. (2) 29 (1983), 281-322.

89.

On the concept ofattractor, Comm. Math. Phys. 99 (1985), 177-195; 102 (1985), 517-519.

90.

Directional entropies of cellular automaton-maps, in "Disordered Systems and Biological Organization," (Bienenstock et al., eds.), SpringerVerlag, Berlin, New York, 1986, pp. 113-115.

91.

The work of M. H. Freedman, in "Proc. Internat. Congress Math. Berkeley, 1986," Amer. Math. S o c , Providence, R. I., 1987, 13-15.

92.

Non-expansive Henon maps, Adv. in Math. 69 (1988), 109-114.

93.

On the entropy geometry of cellular automata, Complex Systems 2 (1988), 357-386.

PUBLICATIONS OF JOHN MILNOR

94.

On iterated maps of the interval, (with W. Thurston, preprint 1977), in "Dynamical Systems" (J. C. Alexander, ed.), Lect. Notes Math. 1342, Springer-Verlag, Berlin, New York, 1988, pp. 465-563. 95. Dynamical properties of plane polynomial automorphisms, (with S. Friedland), Ergodic Theory Dynamical Systems 9 (1989), 667-699. 96. Self-similarity and hairiness in the Mandelbrot set, in "Computers in Geometry and Topologoy," (M. C. Tangora, ed.), Marcel Dekker, New York, Basel, 1989, pp. 211-257. 97. Dynamics in one complex variable, Stony Brook Institute for Mathematical Sciences Preprint # 1 9 9 0 / 5 . 98. Remarks on iterated cubic maps, Stony Brook Institute for Mathematical Sciences Preprint #1990/6. Experiment. Math. 1 (1992), 5-24. 99. Fixed points of polynomial maps, II. Fixed point portraits, (with L. R. Goldberg), Stony Brook Institute for Mathematical Sciences Preprint #1990/14. To appear in Ann. Sci. Ecole Norm. Sup. 100. The Fibonacci unimodal map, (with M. Lyubich), Stony Brook Institute for Mathematical Sciences Preprint #1991/15. To appear in J. Amer. Math. Soc. 101. Hyperbolic components in spaces of polynomial maps, Stony Brook Institute for Mathematical Sciences Preprint # 1 9 9 2 / 3 . 102. Local connectivity of Julia sets: expository lectures, Stony Brook Institute for Mathematical Sciences Preprint # 1 9 9 2 / 1 1 . 103. Remarks on quadratic rational maps, Stony Brook Institute for Mathematical Sciences Preprint #1992/14. 104. On cubic polynomials with periodic critical point, in preparation.

xxi

367 ANNALS OF MATHEMATICS

Vol. 64, No. 2, September, 1956 © 1956 The Johns Hopkins University Press. Reprinted with permission.

ON MANIFOLDS HOMEOMORPHIC TO THE 7-SPHERE BY JOHN MILNOR1

(Received June 14, 1956)

The object of this note will be to show that the 7-sphere possesses several distinct differentiable structures. In §1 an invariant X is constructed for oriented, differentiable 7-manifolds M satisfying the hypothesis (*) H3(M7) = H*(M7) = 0. (Integer coefficients are to be understood.) In §2 a general criterion is given for proving that an n-manifold is homeomorphic to the sphere S". Some examples of 7-manifolds are studied in §3 (namely 3-sphere bundles over the 4-sphere). The results of the preceding two sections are used to show that certain of these manifolds are topological 7-spheres, but not differentiable 7-spheres. Several related problems are studied in §4. All manifolds considered, with or without boundary, are to be differentiable, orientable and compact. The word differentiable will mean differentiable of class C. A closed manifold M" is oriented if one generator n tHn(Mn) is distinguished. §1. The invariant X(Af7) For every closed, oriented 7-manifold satisfying (*) we will define a residue class X(Af7) modulo 7. According to Thorn [5] every closed 7-manifold M7 is the boundary of an 8-manifold B*. The invariant X(M') will be defined as a function of the index T and the Pontrjagin class pi of B*. An orientation v t Hs(Bg, M7) is determined by the relation 6V = ft. Define a quadratic form over the group H*(Bg, M 7 )/(torsion) by the formula a —> (v, a 2 ). Let T(B ) be the index of this form (the number of positive terms minus the number of negative terms, when the form is diagonalized over the real numbers). Let pi tH (Bg) be the first Pontrjagin class of the tangent bundle of B*. (For the definition of Pontrjagin classes see [2] or [6].) The hypothesis (*) implies that the inclusion homomorphism i:H\B\

M7) -»H*(B g )

is an isomorphism. Therefore we can define a "Pontrjagin number"

q(B*) = <,, (f W ) THEOREM 1. The residue class of 2q(B*) — r(B*) modulo 7 does not depend on the choice of the manifold B8. Define \(M ) as this residue class.2 As an immediate consequence we have: 7 COROLLARY 1. If \(M ) j£ 0 then M is not the boundary of any 8-manifold having fourth Betti number zero. 1

The author holds an Alfred P. Sloan fellowship. 399

400

JOHN MILNOR

Let B\, B\ be two manifolds with boundary M7. (We may assume they are disjoint.) Then C8 = B\ U B\ is a closed 8-manifold which possesses a differentiable structure compatible with that of Bi and B\. Choose that orientation v for C8 which is consistent with the orientation in of B\ (and therefore consistent with — v2). Let q(Cs) denote the Pontrjagin number {v, p\((?)). According to Thom [5] or Hirzebruch [2] we have r(C8) = < y , A ( 7 p 2 ( C 8 ) - p ? ( C 8 ) ) ; and therefore 45r(C 8 ) + q(C8) = 7(v, p2(C*)) = 0

(mod 7).

2q((?) - r(C 8 ) = 0

(mod 7).

This implies (1) LEMMA

1. Under the above conditions we have

(2)

r(Cf) = r(B\) -

r(B\)


q{B\).

and (3)

Formulas 1, 2, 3 clearly imply that 2q{B\) - r{B\) = 2q(Bl) - r{B%)

(mod 7);

which is just the assertion of Theorem 1. PROOF OF LEMMA 1. Consider the diagram

# n ( B i , M) © Hn(B2, M) —

IT(C, M)

t'i © it

Hn(Bx) © Hn(B2)

k

Hn(C)

£Jote that for n = 4, these homomorphisms are all isomorphisms. If a = jh~\ai © «2) tH\C), then (4)

(v, a) = (v;jh~\al

© at)) = (Vl © ( - * ) , a\ © a\) = (vu a\) - <m , a\).

Thus the quadratic form of C8 is the "direct sum" of the quadratic form of By and the negative of the quadratic form of B 2 . This clearly implies formula (2). Define ax = i7lpi(-Bi) and a2 = iT'piC^). Then the relation *(pi(C)) = pi(Bi) © pi(B»)

implies that 4

Similarly for n = 4fc — 1 a residue class X(M") modulo snn(Lt) could be defined. (See [2] page 14.) For k = 1, 2, 3, 4 we have stti(Lk) = 1,7, 62, 381 respectively.

369 MANIFOLDS HOMEOMORPHIC TO THE 7-SPHERE

jh~\ax

-101

0 a2) = pi(C).

The computation (4) now shows that (y, pl(0)

= <*, , al) - (* , a\),

which is just formula (3). This completes the proof of Theorem 1. The following property of the invariant X is clear. LEMMA 2. If the orientation of M is reversed then X(M ) is multiplied by — 1. As a consequence we have 7 1 COROLLARY 2. If \(M ) ^ 0 then M possesses no orientation reversing diffeo3 morphism onto itself. §2. A partial characterization of the n-sphere Consider the following hypothesis concerning a closed manifold M" (where R denotes the real numbers). (H) There exists a differentiable function f:M" —> R having only two critical points Xo, Xi. Furthermore these critical points are non-degenerate. (That is if Wi, • • • , un are local coordinates in a neighborhood of Xo (or xi) then the matrix (d^f/dUidUj) is non-singular at xo (or xi).) n THEOREM 2. If M satisfies the hypothesis (H) then these exists a homeomorphism n of M onto S" which is a diffeomorphism except possibly at a single point. Added in proof. This result is essentially due to Reeb [7]. The proof will be based on the orthogonal trajectories of the manifolds / = constant. Normalize the function / so that /(xo) = 0, f(xi) = 1. According to Morse ([3] Lemma 4) there exist local coordinates Vi, • • • , vn in a neighborhood V of xt so that /(x) = v\ + • • • + i)!„ for i e F. (Morse assumes that / is of class C3, and constructs coordinates of class C1; but the same proof works in the CT case.) The expression ds — dv\ + • • • + dv2n defines a Riemannian metric in the neighborhood V. Choose a differentiable Riemannian metric for Mn which coincides with this one in some neighborhood4 V of x 0 . Now the gradien of / can be considered as a contravariant vector field. Following Morse we consider the differential equation J=grad//||grad/||2In the neighborhood V this equation has solutions

(»i(0, • • • , ».(0) = (oi(0*, • • • , aM) for 0 ±S t < E, where a = (ai, • • • , o„) is any n-tuple with ^fa\ = 1. These can be extended uniquely to solutions xa(t) ior 0 ^ t ^ 1. Note that these solutions satisfy the identity 3

A diffeomorphism /is a homeomorphism onto, such that both/ and/ -1 are differentiable. * This is possible by [4] 6.7 and 12.2.

370 402

JOHN MILNOR

/(*«(<)) = IMap the interior of the unit sphere of Rn into Mn by the map MO*,'

• • • , an(ty) -» x„(t).

It is easily verified that this defines a diffeomorphism of the open n-cell onto Mn — (xi). The assertion of Theorem 2 now follows. Given any diffeomorphism g'.S"^ —> S"~x, an n-manifold can be obtained as follows. n CONSTRUCTION (C). Let M (g) be the manifold obtained from two copies of R" by matching the subsets R" — (0) under the diffeomorphism 1 u —+ v ~

T.

/ n

u

\

q I ,,——- I.

II « II VII « 1 1 / (Such a manifold is clearly homeomorphic to <Sn. If g is the identity map then n M (g) is diffeomorphic to Sn.) n COROLLARY 3. A manifold M can be obtained by the construction (C) if and only if it satisfies the hypothesis (H). PROOF. If M"(g) is obtained by the construction (C) then the function

/(x) = || w j|7(i+'IMI 2 ) = i / ( i + || Ml2) will satisfy the hypothesis (H). The converse can be established by a slight modification of the proof of Theorem 2. §3. Examples of 7-manifolds Consider 3-sphere bundles over the 4-sphere with the rotation group SO(4) as structural group. The equivalence classes of such bundles are in one-one correspondence6 with elements of the group ir3(SO(4)) ^ Z + Z. A specific isomorphism between these groups is obtained as follows. For each (h, j) e Z + Z letfhj:B3 — SO(4) be defined by fkj(u) • v = u vu', for v « R . Quaternion multiplication is understood on the right. Let i be the standard generator for ^(S4). Let &,• denote the sphere bundle corresponding to (ft,) t ir3(SO(4)). LEMMA 3. The Pontrjagin class pi(£»y) equals ± 2{h — j)t. (The proof will be given later. One can show that the characteristic class c(hj) (see [4]) is equal to (h + j)t.) For each odd integer k let Mk be the total space of the bundle &, where h and j are determined by the equations h + j ' = 1, h — j = k. This manifold M\ has a natural differentiable structure and orientation, which will be described later. LEMMA 4. The invariant \(Mk) is the residue class modulo 7 of k — 1. LEMMA 5. The manifold Ml satisfies the hypothesis (H). Combining these we have: 8

See [4] §18.

371 MANIFOLDS HOMEOMORPHIC TO THE 7-SPHERE

403

THEOREM 3. For k ^ 1 mod 7 the manifold Mk is homeomorphic to S but not diffeomorphic to S . (For k = ± 1 the manifold Ml is diffeomorphic to S 7 ; but it is not known whether this is true for any other k.) Clearly any differentiable structure on S7 can be extended through R* — (0). However: COROLLARY 4. There exists a differentiable structure on S which cannot be extended throughout R . This follows immediately from the preceding assertions, together with Corollary 1. PROOF OF LEMMA 3. It is clear that the Pontrjagin class pife.,) is a linear function of h and.?. Furthermore it is known that it is independent of the orientation of the fibre. But if the orientation of S is reversed, then &y is replaced by £_;•_/• . This shows that Pi(^y) is given by an expression of the form c(h — j)i. Here c is a constant which will be evaluated later. A PROOF OF LEMMA 4. Associated with each 3-sphere bundle Mk —> S there is a 4-eell bundle Pk:Bl -* S\ The total space Bk of this bundle is a differentiable manifold with boundary Ml . The cohomology group H (Bk) is generated by the element a = pk (i). Choose orientations n, v for Mk and Bk so that

{v, or1*)2) = +i. Then the index r(Bl) will be + 1 . The tangent bundle of Bk is the "Whitney sum" of (1) the bundle of vectors tangent to the fibre, and (2) the bundle of vectors normal to the fibre. The first bundle (1) is induced (under pk) from the bundle £hj, and therefore has Pontrjagin class pi = pk {c(h — j)i) = cka. The second is induced from the tangent bundle of S4, and therefore has first Pontrjagin class zero. Now by the Whitney product theorem ([2] or [6]) Pi(Bt) = cka + 0. For the special case A: = 1 it is easily verified that B\ is the quaternion projective plane Pz{K) with an 8-cell removed. But the Pontrjagin class pi(P2(K)) is known to be twice a generator of H (P^K)). (See Hirzebruch [1].) Therefore the constant c must be ± 2 , which completes the proof of Lemma 3. Now q(B\) = <*, (r 1 (±2fca)) : ) = 4fc2; and 2q - r = 8fc2 - 1 = k2 - 1 (mod 7). This completes the proof of Lemma 4. 4 PROOF OF LEMMA 5. As coordinate neighborhoods in the base space <S take the complement of the north pole, and the complement of the south pole. These can be identified with euclidean space R under stereographic projection. Then a point which corresponds to u t R under one projection will correspond to u' = u/\\ u ||2 under the other. The total space Mk can now be obtained as follows. Take two copies of R X S3 and identify the subsets (724 — (0)) X S under the diffeomorphism (u, v) -

«i/)

=

(M/||U||2,MW/||M||)

372 404

JOHN MILNOR

(using quaternion multiplication). This makes the differentiable structure of Ml precise. Replace the coordinates («', v') by («", v') where u" = w'(t/) _1 . Consider the function f:MI —* R defined by /(*) = »(p)/(l + || « ||*)* = « ( « » ) / ( l + || u" ||V; where 9?(t>) denotes the real part of the quaternion v. I t is easily verified that / has only two critical points (namely (u, v) = (0, ± 1 ) ) and that these are nondegenerate. This completes the proof. §4. Miscellaneous results THEOREM 4. Either (a) there exists a closed topological 8-manifold which does not possess any differentiable structure; or (b) the Pontrjagin class pi of an open 8-manifold is not a topological invariant. (The author has no idea which alternative holds.) PROOF. Let Xk be the topological 8-manifold obtained from Bl by collapsing its boundary (a topological 7-sphere) to a point x0. Let or « H*(Xl) correspond to the generator a e H*(Bk). Suppose that Xl, possesses a differentiable structure, and that pi(Xk — (x0)) is a topological invariant. Then pi(Xl) must equal ±2fca, hence

2q(Xl) - r(Xl) = 8A;2 - 1 = k2 - 1

(mod 7).

2

But for fc ^k 1 (mod 7) this is impossible. Two diffeomorphisms /, g'-Mi —> Mt will be called differentiably isotopic if there exists a diffeomorphism M" X R —* M% X R of the form (x, t) —» (h(x, I), t) such that f{x)

{t

0)

HX t) = ! ^ h{x l) ' \g(x) (t i> l ) . LEMMA 6. If the diffeomorphisms f, g'.S""1 —* S"~l are differentiably isotopic, then the manifolds Mn(j), Mn(g) obtained by the construction (C) are diffeomorphic. The proof is straightforward. 6 THEOREM 5. There exists a diffeomorphism f:S* —* <S of degree +1 which is not differentiably isotopic to the identity. Proof. By Lemma 5 and Corollary 3 the manifold Ml is diffeomorphic to M7(f) for some / . If / were differentiably isotopic to the identity then Lemma 6 would imply that M\ was diffeomorphic to <S7. But this is false by Lemma 4. PRINCETON UNIVERSITY REFERENCES

1. F. HIRZEBRTJCH, UeberdiequaternionalenprojektivenR&ume, S.-Ber. math.- naturw. Kl. Bayer. Akad. Wiss. Munchen (1953), pp. 301-312. 2. , Neue topologische Methoden in der algebraischen Geometrie, Berlin, 1956.

MANIFOLDS

HO.MEO-MOHPHIC TO T H E

/-SPHERE

405

3. M. M O R S E , Relations between the numbers of critical points of a real function of n independent variables, T r a n s . Amcr. M a t h . S o c , 27 (1925), p p . 345-396. 4. X. STEENHOD, T h e topology of fibre bundles, Princeton, 1951. 5. R. T H O M , Qitelgues proprieles qlohale des variUCs dijferentiables, Comment. Math. Helv., 28 (1954), p p . 17-80. 6. Wc W E X - T S U N , Sur les classes caractcristiques des structures fibrics sphcriques, Actual. sci. industr. 1183, Paris, 1952, pp. 5-89. 7. O. K E E B , Sur certain proprieles topoloiiiques des varictes feuilletecs. Actual, sci. industr. 1183, Paris, 1952, p p . 91-154.

374 ANNALS OF MATHEMATICS

Vol. 68, No. 2, September, 1958 © 1958 The Johns Hopkins University Press. Reprinted with permission.

SOME CONSEQUENCES OF A THEOREM OF BOTT BY JOHN MILNOR1

(Received February 11, 1958)

It will be shown that the following theorem, due to R. Bott [3], can be used to solve several well known problems ; including the problem of the existence of division algebras, and the parallelizability of spheres2. (Independent solutions of these problems, also based on Bott's work, have been given by Kervaire and Hirzebruch.) THEOREM OF BOTT. For any Om-bundle ? over the sphere S4k, the Pontrjagin class py(£) e Hik(Sik ; Z) is divisible by (2k — 1)!. (This result was conjectured, and proved up to powers of 2, by Borel and Hirzebruch [I])The following result, which follows from Wu Wen-Tsiin [19], will also be needed. Since Wu's paper is in Chinese a proof is included in the appendix. The epimorphism Z -> Z4 induces a homomorphism H*(K; Z) -> H*(K; Zt) which will be denoted by a -*• (a)t. Let i : Zz-^ Zt denote the inclusion homomorphism. For any Om-bundle f over a complex K, the class (Pk(£))i 6 H (K\ Z4) is determined by the Stiefel-Whitney classes w,(6) e Hl(K; Zz). In particular if the Stiefel-Whitney classes w^), •• • , wik-i(E) are zero then (pk(£))i = i*wilc(£). Combining these two results, the following is obtained. THEOREM OF W U . ik

1. There exists an Om-bundle £ over the sphere Sn with w„(?) ^t 0 only for n = 1, 2, 4 or 8. (Examples of such bundles can be given as follows : for n — 1 the 2fold covering of the circle, and for n = 2, 4 or 8 the On-bundle over Sn associated with the Hopf fibering S2""1 ->• S*.) THEOREM

PROOF. According to Wu [16] such a bundle can exists only if n is a power of 2. Hence it is certainly sufficient to consider the case n = 4k, k > 2. The identity

(ft(O)* = **«>„(£) e ff"0S"; zt) = zt is valid, since the lower Stiefel-Whitney classes must be zero. In other words the class w4Jt(?) is zero if and only if pk(£) is divisible by 4. But 1 J

The author holds a Sloan fellowship. A preliminary account of this work has been given in [20]. 444

375 A THEOREM OF BOTT

445

pt(f) is known to be divisible by (2k — 1)!. For k > 2 this proves that w«(f) = 0. r THEOREM 2. The sphere S is parallelizable only for r = 1,3, 7. (Compare Steenrod and Whitehead [10].) / PROOF. The fibering SOr > SOr+1 > Sr associated with the tangent r bundle of S has the following homotopy sequence : » 7Tr(SOr +l) - ^ nr(Sn —

TTr-ASO,)

>

ffr.,(SOrtl)

>0.

r

The group 7Tr(S ) will be identified with the integers. Then 9(1) e nr~i(SOr) is the element which corresponds to the tangent bundle of Sr. (See Steenrod [9, § 18]). For each X e 7rr(SOr+l) let £ denote the corresponding SO r+r bundle over Sr+1, and let X(£) denote its Euler class ( = top Stief el-Whitney class with integer coefficients). Let ^ be the standard generator of Hr+1(Sr+1 ; Z). Then f*(X) is equal to the negative of the " Euler number " <X(f), fi). [Proof. Let o(f) e Hr+i(Sr+'; 7tr(SOr+l)) denote the obstruction to the existence of a cross-section of $. Then X($), the obstruction to the existence of a cross-section in the associated sphere bundle, is equal to /*(o(f)). According to Steenrod [9, p. 180] the identity = - / * ( ; ) ] . Now if Sr is parallelizable then 9(1) = 0, hence there exists k e 7zy(SOr+1) with f*(X) = 1. For the corresponding bundle $, the class X($) generates Hr+1(Sr+1; Z); hence the class wr+1(£) = (X(£))2 is non-zero. Together with Theorem 1 this complete the proof. It follows immediately that the real projective space PT is parallelizable only for r = 1, 3, 7. (For consequences concerning the immersion of Pr in euclidean space see Milnor, Comm. Math. Helv. 30 (1956), p. 284). COROLLARY 1. There exists a division algebra of rank n over the real numbers only for n = 1, 2, 4, 8. PROOF. The existence of a bilinear product operation without zerodivisors in the vector space Rn implies that the projective space P71'1 is parallelizable. (See Stiefel [11, p. 216]). [ALTERNATIVE PROOF. Suppose that such a product operation in Rn is given. Then the correspondence Sn~l —• GLn defined by x -*• (left multiplication by x) gives rise to a GLnbundle f over S". It is not hard to verify that w„(£) =£ 0]. COROLLARY 2. For r ^ 8 the groups nr^1(SOr) are as follows : r modulo 8: K,-i(SO r ):

0 Z+Z

1 %%%!£

f

2

3

4

5

6

7

Z + Zi

Z%

Z +Z

Z2

Z

Z-2

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JOHN MILNOR

PROOF. This follows from Bott's computation [2] of the stable groups ffr_1(SOr+1), together with the exact homotopy sequence used to prove Theorem 2.

3. Let Min be a simply-connected differentidble manifold such that the cohomology group H\M!n ; Z) is infinite cyclic for i — 0, n, 2n, and zero otherwise. Then n must be 2, 4, or 8. (Examples are provided by the complex, quaternion, and Cayley projective planes. It will be shown in a later paper [7] that the condition of simple-connectivity can be eliminated. This will give an answer to Problem 5 of [5]). THEOREM

If a generates H"(Min ; Z), then the Poincare duality theorem implies that a w a generates Hin(Mln ; Z). Hence PROOF.

Sq" : Hn(M2n ; Z2)

> Hin{Mln ; Z,)

in non-zero. The formulas of Wu [15] now imply that the Stiefel-Whitney class w„ of the tangent bundle 6 is non-zero. Choose a map g : Sn ->• Min which, under the Hurewicz homomorphism, corresponds to a generator of H„(MM ; Z). Then the bundle 0' over Sn induced from 6 by g will satisfy w„(^') =£ 0. Therefore n must be 1, 2, 4 or 8. Since the case n = 1 is easily excluded, this completes the proof. Bott's theorem is related to the question of the existence of maps with Hopf invariant 1 as follows. Let J : nn^SOJ) -*• n-m+n_1(Sm) be the homomorphism of G. W. Whitehead [13], and let rn : 7Tm+n.1{Sm) -> Z3 be the generalized Hopf invariant of Steenrod [8], which is defined using the functional Sqn operation. For each odd prime q let Tq.i '• ^m + 2i(,-l)-l('S' m )

> Zt

denote the corresponding homomorphism based on the reduced gth power ^'. THEOREM 4a. The image J^n^SOn), m^n, contains an element JX with generalized Hopf invariant Tn(Jfy different from zero only if n equals 2, 4, or 8. THEOREM 4b. The image «7^24(,_1)_1(iSOro), m 2; 2i(q — 1), contains an element JX with r 9 ,«(^) different from zero only ifi = 1. PROOF OF 4b. Let $ be the SOm-bundle over S" associated with J, where n = 2i(q — 1). Let E be the total space of the associated bundle having the unit ball Bm as fibre, so that the boundary E is the total space of the associated sphere bundle. According to [7, Theorem 3, Corollary 1],

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447

the collapsed space E\E can be obtained from the sphere Sm by attaching an (m + w)-cell, using on attaching map in the homotopy class JX. Thus the generalized Hopf invariant rg,4(«//l) is non-zero if and only if the homomorphism ^ " : Hm(E, E ; Zq)

> Hm+n(E, E; Zq)

is non-zero. Let 0 : H3{Sn ; Zq) - • H1+m(E, E ; Z„) denote the isomorphism of Thorn [12]. According to Wu [18, § IV] the class tfr^W) e H"(Sn ; Zt) can be expressed as a polynomial in the Pontrjagin classes of £, reduced modulo q. But these Pontrjagin classes are zero, except for p((,-i)/2(f) which is divisible by (i(q — 1) — 1)!. For i > 1, since the number (i(q — 1) — 1)! is divisible by q, it follows that the operation ^ " must be zero. Theorem 4a is proved in a similar way, using Theorem 1 together with Thorn's definition of the Stiefel-Whitney classes. (See [12]). Appendix Following Hirzebruch [6] define the Pontrjagin class pk of an Om-bundle as (— 1)* times the Chern class c2t of the Z7m-bundle induced by the inclusion Om-*Um. This is slightly different from the Pontrjagin class as defined by Pontrjagin and Wu. (Compare [17, Theorem 4]). Consider the exact sequence of cohomology group corresponding to the coefficient sequence PROOF OF THE THEOREM OF W U .

0

> Z2 —^-+ Z, - ^ Z2

•0;

as well as the Pontrjagin squaring operation 5P : H*(K; Zt)

> H4*(K; Z<) .

(See for example Whitehead [14]). LEMMA 1. The Pontrjagin class pk of any Om-bundle is related to the Stiefel-Whitney classes wu ••• , wik by an identity

(P*)t — ¥>(wiK) + i+Mw^, ••• , wik) where fk is a polynomial with coefficients in Z2. It is clearly sufficient to consider the case of the universal bundle over the Grassmann space Gm(R), with m large. The identity j^(w) = w^jw holds for any cohomology class w. Comparing this with the relation3 PROOF.

~3~See Wu [Tvfheorem 3}.

448

JOHN MILNOR 3*{(Pk\)

= (P*)i =

W2k^lV.lk

it follows that (p*)4 - ^w.ik e (kernel j j = i*Hlk{Gn{R); Z2) . Since the cohomology ring H*{Gm{R); J?,) is generated by the StiefelWhitney classes, this proves Lemma 1. To prove the theorem it is only necessary to show that the coefficient of wik in fk is non-zero. Let T denote the universal E7m-bundle over the complex Grassmann space GJC). Recall that the cohomology ring H*(Gm(C) ; Z) is a polynomial ring4 generated by the Chern classes of T. The inclusion Um -> 02m induces an 02m-bundle over Gm(C) which will be denoted by TB. Applying Lemma 1 to this bundle TJJ, the relations5 P*(T*) = ck(ry - 2c*_1(T)ct+1(T) +

± 2c0(T)c«(T)

and «w(Tfl) = 0 ,

w.2r(rR) = (cr(T))2

show that the polynomial fk must satisfy fk(0, w2, 0,wt,---,

wik) = w2k-2w.ik+z + w2k.iW2k+i + • • • + wawlk .

Therefore/^(O, 0, • • • , 0, ivik) — wik ; which completes the proof. PRINCETON

UNIVERSITY REFERENCES

1. A. BOREL and F . HlRZEBRUCH, Characteristic classes and homogeneous spaces, to appear (Amer. J. Math.). 2. R. BOTT, On the stable homotopy of the classical groups, Proc. Nat. Acad. Sci. U.S.A. 43 (1957), 933-935. 3. , The space of loops on a Lie-group, to appear (Mich. J. Math.). 4. S. S. CHERN, Characteristic classes of Hermitian manifolds, Ann. of Math. 47 (1946), 85-121. 5. F . HlRZEBRUCH, Some problems on differentiate and complex manifolds, Ann. of Math. 60 (1954), 213-236. 6. , Neue topologische Methoden in der algebraischen Geometrie, Springer, 1956. 7. J. MILNOR, On spaces with a gap in cohomology, to appear. 8. N. E. STEENROD, Coholomogy invariants of mappings, Ann. of Math. 50(1949), 954-988. 9. , T h e topology of fibre bundles, Princeton, 1951. 10. and J.H.C. WHITEHEAD, Vector fields on the n-sphere, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 58-63. 11. E. STIEFEL, Uber Richtungsfelder in den projektiven Rmimen und, einen Satz aus der reelen Algebra, Comm. Math. Helv. 13 (1940), 201-218. 12. R. THOM, Espaces fibres en spheres et carres de Steenrod, Ann. sci. Ecole norm, sup 69 (1952), 109-182. * Chern [4]. 5 See Hirzebruch [6, p. 68] and Steenrod [9, p. 212].

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13. G. W. WHITEHEAD, On the homotopy groups of spheres and rotation groups, Ann. of Math. 43 (1942), 634-640. 14. J.H.C. WHITEHEAD, On simply connected 4-dimensional polyhedra, Comm. Math. Helv. 22 (1949), 48-92. 15. WU WEN-TSUN, Classes caracteristigues et i-carres d'une variete, C. R. Acad. Sci. Paris 230 (1950), 508-511. 16. , Les i-carres dans une variete grassmannienne, C. R. Acad. Sci. Paris 230 (1950), 918-920. 17. , On Pontrjagin classes I, Scientia Sinica 3 (1954), 353-367. 18. , On Pontrjagin classes II, Scientia Sinica 4 (1955), 455-490. 19. , On Pontrjagin classes III, Acta Math. Sinica 4 (1954), 323-346. 20. R. BOTT and J. MILNOK, On the parallelizability of the spheres, Bull. Amer. Math. Soc. 64 (1958), 87-89.

Inventiones math. 9, 318-344 (1970)

Algebraic X-Theory and Quadratic Forms JOHN MILNOR (Cambridge,

Massachusetts)

The first section of this paper defines and studies a graded ring K^F associated to any field F. By definition, KnF is the target group of the universal n-linear function from F* x • • • x F' to an additive group, satisfying the condition that ax x • • • x a„ should map to zero whenever a i + a i+i = l- Here F' denotes the multiplicative group F - 0 ^ Section2 constructs a homomorphism d: KnF—>.£„_! F associated with a discrete valuation on F with residue class field F. These homomorphisms 8 are used to compute the ring K^Fff) of a rational function field, using a technique due to John Tate. Section 3 relates K^ F to the theory of quadratic modules by defining certain " Stiefel-Whitney invariants " of a quadratic module over a field F of characteristic 4=2. The definition is closely related to Delzant [5]. Let W be the Witt ring of anisotropic quadratic modules over F, and let IcW be the maximal ideal, consisting of modules of even rank. Section 4 studies the conjecture that the associated graded ring {W/I,I/I2,I2/I\...) is canonically isomorphic to K^F/IK^F. Section 5 computes the Witt ring of a field F(t) of rational functions. Section 6 describes the conjecture that K^F/IK^F is canonically isomorphic to the cohomology ring H*(GF;Z/2Z); where GF denotes the Galois group of the separable closure of F. An appendix, due to Tate, computes K^F/HC^F for a global field. Throughout the exposition I have made free use of unpublished theorems and ideas due to Bass and Tate. I want particularly to thank Tate for his generous help. §1. The Ring K^F To any field F we associate a graded ring K.F^KoF.J^F^F,...) as follows. By definition, KXF is just the multiplicative group F* written additively. To keep notation straight, we introduce the canonical isomorphism

Algebraic K-Theory and Quadratic Forms

319

where l(ab)=l(a) + l(b). Then K^F is defined to be the quotient of the tensor algebra (Z, KtF, KlF®KlF, KlF®K1F®K1F,...) by the ideal generated by all I {a}® I (I — a), with a 4=0,1. In other words each KnF,n^.2,is the quotient of the w-fold tensor product Kx F® • • • ® Kx F by the subgroup generated by all l(at)®•••®l{a„) such that a ; + a i + i = l for some i. In terms of generators and relations, K^ F can be described as the associative ring with unit which is generated by symbols 1(a), aeF', subject only to the defining relations l(a b)=1(a) + 1(b) and 1(a) 1(1 — a)=0. Explanation. This definition of the group K2 F is motivated by work of R. Steinberg, C. Moore, and H. Matsumoto on algebraic groups; and has already been the object of much study. (Compare references [2 — 4, 7 — 9, 17].) For n ^ 3 , the definition is purely ad hoc. Quite different definition of K„ for n2:3 have been proposed by Swan [18] and by Nobile and Villamayor [11]; but no relationship between the various definitions is known. First let us describe some fundamental properties of the ring K^ F. (Examples will be given in §§1.5 — 1.8.) Lemma 1.1. For every £eKmF and every neKnF, the identity

r,i = (-ir^n is valid in Km+nF. Proof (following Steinberg). Clearly it suffices to consider the case m = n = l. Since — a = (l — a)/(l — a - 1 ) for a4=1, we have

l(a)l(-a)=l(a)l(l-a)-l(a)l(l-a-1) = l(a)l(l-a) + l(a~l)l(l-a-1)

= 0.

Hence the sum 1(a) 1(b)+1(b) 1(a) is equal to 1(a) l(-a) + 1(a) 1(b) + 1(b) 1(a) + 1(b) l(-b) =

l(a)l(-ab)+l(b)l(-ab)

= l(ab)l(-ab)

= 0;

which completes the proof. Here are two further consequences of this argument: Lemma 1.2. The identity l(a)2 = l(a) l( — 1) is valid for every 2

l(a)eK1F.

For the equation 1(a) l( — a) = 0 implies that 1(a) = 1(a) (/(—1) + /(— a)) must be equal to 1(a) l( — l).

320

J. Milnor:

Lemma 1.3. / / the sum a1 + ---+a„of non-zero field elements is equal to either 0 or 1, then l(a^) ... l(a„) = 0. Proof by Induction on n. The statement is certainly true for n = 1,2; so we may assume that nS:3. If a1 + a2-0, then the product l{a^l{a2) is already zero. But if a t + a2 4= 0, then the equation «i/(«i + a2) + a 2 /(ai + a2) = l implies that

{lia^-li^

+ a^^ia^-lia.

+

a^Q.

Multiplying by l(a3)... l(a„), and using 1.1 and the inductive hypothesis l(a1 + a2)l(ai)...l(a„)

= 0,

the conclusion follows. Here is an application. Theorem 1.4. The element —I is a sum of squares in F if and only if every positive dimensional element of K^ F is nilpotent. Proof. If — 1 is not a sum of squares, then F can be embedded in a real closed field, and hence can be ordered. Choosing some fixed ordering, define an n-linear mapping from Kt F x • • • x Kj F to the integers modulo 2 by the correspondence it w ~n \^ l - s g n f o ) l-sgn(a„) /(a 1 )x---x/(a„)i-^ ^ • Evidently the right hand side is zero whenever a ; + aj + 1 = l. Hence this correspondence induces a homomorphism K„F—Z/2Z; which carries /( —1)" to 1. This proves that the element /( — 1) is not nilpotent. Conversely, if say — l = aj-\ \-af, then it follows from 1.3 that /(-a 1 2 ).../(-a r 2 ) = 0; hence

/(-l)r = 0mod2XrF.

Since 2J(-1) = 0, it follows immediately that i ( - l ) r + 1 = 0 . For any generator y = l(ax)... l(an) of the group K„F, it follows from 1.2 that ys is equal to a multiple of f(—1)"<S-1). Hence y s =0 whenever n ( s - l ) > r . Similarly, for any sum y1H \-yk of generators, the power {y1 H h yk)s can be expressed as a linear combination of monomials y'i •••y'k w i t n h H 1- k = s- Choosing s > k, note that each such monomial is a multiple of/(-l)" ( s - f c ) . If s>k + r/n, it follows that (V! + --- + yfc)s = 0; which completes the proof.

Algebraic K-Theory and Quadratic Forms

321

To conclude this section, the ring K^F will be described in four interesting special cases. Example 1.5 (Steinberg). If the field is finite, then iC 2 F=0. In fact K^ F is cyclic, say of order q — 1; so § 1.1 implies that K2 F is either trivial or of order ^ 2, according as q is even or odd. But, if q is odd, then an easy counting argument shows that 1 is the sum of two quadratic nonresidues in F; from which it follows that X 2 F = 0. This implies, of course, thatK„F=0forn>2also. Example 1.6. Let R be the field of real numbers. Then every KnR, n ^ l , splits as the direct sum of a cyclic group of order 2 generated by 1{~Y)", and a divisible group generated by all products 1(0^... l{a„) with at,..., a„ > 0. This is easily proved by induction on n, using the argument of §1.4 to show that /( —1)" is not divisible. Example 1.7. Let F be a local field (i.e. complete under a discrete valuation with finite residue class field), and let m be the number of roots of unity in F. Calvin Moore [10] proves that K2 F is the direct sum of a cyclic group of order m and a divisible group. We will show that K„F is divisible for n ^ 3 . Consider the algebra K^F/pK^F over Z/pZ; where p is a fixed prime. If p does not divide m, then Moore's theorem clearly implies that K2 F/p K2F = 0. Suppose that p does divide m. We claim then that: (1) the vector space K1F/pK1F has dimension 2^2 over Z/pZ; (2) the vector space K2F/pK2F has dimension 1; and (3) for each a=t=0 in K^/pK^F there exists /? in K1F/pK1F so that

a 0*0.

In fact (1) is clear; (2) follows from Moore's theorem; and (3) is an immediate consequence of the classical theorem which asserts that, for each aeF' which is not a p-th power, there exists b so that the p-th power norm residue symbol (a, b)Fis non-trivial. (See for example [20, p. 260].) The correspondence l(a)l(b)^(a,b)F clearly extends to a homomorphism from K2 F to the group of p-th roots of unity. So, taking a=1(a), /? = /(&), the conclusion (3) follows. Proof that every generator a P y of K3F/pK3F is zero. Given a, /?, y one can first choose /?'=(= 0 so that oe/?' = 0 (using (1) and (2)), and then choose y' so that /?' y' = /? y (using (2) and (3)). The required equation

Thus K3F/pK3F = 0 for every prime p; which proves that K3F is divisible.

322

J. Milnor:

Example 1.8. Let F be a global field (that is a finite extension of the field Q of rational numbers, or of the field of rational functions in one indeterminate over a finite field). Let Fv range over all local or real completions of F. The complex completions (if any) can be ignored for our purposes. The inclusions F —> Fv induce a homomorphism K2 F —> ©„ K2 F„/(max. divis. subgr.), where each summand on the right isfinitecyclic by 1.6 and 1.7. Bass and Tate [3] have shown that the kernel of this homomorphism is finitely generated, but the precise structure of the kernel is not known. Moore has shown that the cokerael is isomorphic to the group of roots of unity inf. The structure of KnF is not known for n ^ 3 , but Tate has proved the following partial result: The quotient K„F/2K„F maps isomorphically to the direct sum, over all real completions Fv, of KmFJ2K„Fv*Z/2Z. Thus the dimension of K„F/2K„F as a mod 2 vector space is equal to the number of real completions. Tate's proof of this result is presented in the Appendix. It may be conjectured that the subgroup 2K„F is actually zero for w^3, so that K„F itself is a vector space over Z/2Z. As an example, for the field Q of rational numbers the isomorphism KnQ^Z/2Z for ng;3 can be established by methods similar to those of §2.3. § 2. Discrete Valuations and the Computation of K*F(i) Suppose_that a field F has a discrete valuation v with residue class field F (=FV). The group of units (elements u with ord„u=0) will be denoted by U, and the natural homomorphism U—>Fm by ui->u. An element n of F' is prime if ord„ n — 1. Lemma 2.1. There exists one and only one homomorphism d = dv from KnF to K„_iF which carries the product l(n)l(u2)...l(u„) to l(u2)...l(iin) for every prime element n and for all units u2, ...,«„. This homomorphism d annihilates every product of the form /(u t )... l(u„). (For n = 1 the defining property is to be that dl{n) = 1.) Remarks. Evidently d is always surjective. For n = l the homomorphism d can essentially be identified with the homomorphism ord„: F'-^Z;

Algebraic K-Theory and Quadratic Forms

323

and for n = 2 it is closely related to the classical "tame symbol" n'u1,niu2h->(—l)iJ M'2/U{ which is utilized for example in [3]. To begin the proof, note that any unit ut can be expressed as the quotient % ujn of two prime elements. So the property

follows immediately from the defining equation. Proof of Uniqueness. Choose a prime element n. Since F' is generated by 7t and U, it follows that K„ F is generated by products of the form l(n)rl(ur+1)...l(un). If r = l , then the image of any such product under d has been specified; and if r > l then using the identity l{n)r = l(n) /( — l) r _ 1 it is also specified. But if r=0, then any such product maps to zero. This proves that d is unique, if it exists. Proofof Existence1. It will be convenient to introduce an indeterminate symbol x which is to anticommute with all elements of Kx F. Given any n-tuple of elements l(nilu1),...,l(ni''un)eKlF, construct a sequence of elements cpjeKjF by the formula (xi1 + l(u1))...(xi„ + l(un)) = xn(p0 + xn-1(p1 + --- + (p„. Evidently each
Thus (peK„_1F, and evidently