No title

The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.

Prediction and the Spiteful Computer 1 A deterministic system consists of two computers, Laplace and Baby Dostoevsky. Laplace is programmed to say at time T1 what Baby Dostoevsky will do at some later time T2. Baby Dostoevsky is programmed to do at time T2 the opposite of what Laplace has said at time T 1. Baby Dostoevsky's method is the obvious one. Laplace's method is to calculate the state of the system at time T2 given the initial state; this should be possible since the system is deterministic. I suggest resolving this paradox as follows. Laplace's program includes a description of the initial state of the s~ystem. On the other hand, Laplace's program is part of the initial state of the system. Therefore, Laplace's program has to include a description of itself. There is no reason to suppose that the constraints this requirement imposes are consistent, and this resolves the paradox. Going further, one can say that, because the supposed initial state (or program) leads to a contradiction, in fact there is no such initial state. This is more or less the same as some of the solutions suggested in Akin's article, but it is perhaps expressed in a more mathematical and less physical way. Richard Steiner Department of Mathematics University of Glasgow University Gardens Glasgow, G12 8QW Scotland

This is not a physics paradox. The physical assumptions, Newtonian determinism, and uniform continuity of phase flow, as well as the requirement that predictions be secured through detailed microphysical computation, are all unnecessary scaffolding. The crux of the paradox is that the megacomputer L. is allegedly unable to make a certain prediction, which from other considerations, it obviously should be able to make. The draconian computational protocol, coupled with an assumption of determinism, is presumably intended to secure this predictability. This is computational overkill, as can be seen from the fact that the 1 See MathematicalIntelligencer. vol. 14, no. 2, 45-47

predictions L. has to make are trivially easy. L. needs to predict its own output and Baby D.'s kneejerk response. The catch-up problem is irrelevant to the paradox's resolution. This problem arises from the profligate stipulation that predictions are to be secured through a detailed calculation based on an exact microphysical theory. We can communicate the full force of the paradox without this extra baggage, in fact, without significant physical assumptions: Call the realistic computer in this version, R. R. sticks to the essentials; it predicts only its own output and Baby D.'s inevitable negation. This is quite easy, so R. can be a small device that is not afflicted with a catch-up problem. Both are incapable of making the prediction in the required form, owing to Baby D.'s simple-minded spoiler tactics. In short, R.'s version communicates the essential paradox, and is not tied to any particular physical theory; any possible world with enough stability for the construction of simple machines would suffice. Not a physics paradox, this is a logical paradox, furthermore a semantic one because it springs from too permissive a stance on the issue of when to permit one thing to be "about" another thing. (The clearest example is Epimenides's paradox: "This sentence is false.") In Akin's paradox, L.'s output is a prediction about Baby D.'s response. Under the terms of the problem, specification of Baby D.'s response is tantamount to specification of L.'s prediction. Thus, L.'s prediction makes an indirect statement about itself. This dooms the prediction to be false under the specified semantic assignment. It is a lesson of modern logic that whereas rigorous use of self-application can be a wellspring, unbridled use is sure to generate paradox and selfcontradiction. L. can make the needed prediction and, say, store it in memory or relay it through a channel that Baby D. does not monitor. There is no failure of predictability, and, hence no conflict with determinism. But L. has a semantic difficulty; it cannot, without falling into error, have a certain one of its outputs mean (or represent or be about) its prediction. L. is not precluded from making a valid prediction; it is precluded from expressing its prediction in a certain

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2 9 1993 Springer-Ver|ag New York 3

manner. If L. were a human whose predictive utterances were monitored by Baby D., we might say that L. can know the truth (on the matter of Baby D.'s response), but cannot speak it. William Eckhardt 250 South Wacker Drive #650 Chicago, IL 60606 USA

I think the self-reference in the input, in spite of being an obstacle, is not a valid objection, as J. von Neumann s h o w e d w h e n h e g a v e d e s c r i p t i o n s of selfreproductive automata. My opinion is that the solution is in the catch-up problem. For every T I 0, let f(T) be the time needed by Laplace to predict the output of Baby Dostoevsky at time T + 1. The paradox of the spiteful computer is just a simple proof of fiT) > T for every T I> 0. It looks like a diagonalization argument against the existence of a "universal future predictor." This seems convincing to me. Miguel A. Lerma Facultad de Informdtica Universidad Politdcnica de Madrid 28660 Boadilla del Monte Madrid Spain

From my own "classical physicist's" point of view, storing numerical data incurs a cost which increases, logarithmically, with desired accuracy. (Storing D digits of information requires space, time, and expense of order D.) With chaotic dynamics, this cost increases further, logarithmically with time. A classical computer can neither contain an accurate description of its state nor predict its own future. This mechanistic pic-

ture avoids the paradox of Ethan Akin's spiteful computer. Feedback differs from prediction in influencing the future rather than foretelling it. Thus, there is no predictive paradox in Lee Lorenz's wonderful portrayal of "Self-Awareness," from the 25 May 1992 New Yorker. Predicting the future is impossibly hard, while influencing it is easy. Useful to keep in mind in an election year! William G. Hoover Department of Applied Science University of California at Davis P.O. Box 808, L-794 Livermore, CA 94550 USA

.Akin R e p l i e s These letters confirm my experiences discussing this puzzle. Everyone says that the problem is simple, but the proposed answers display considerable variety. Certainly, this is a conceptual puzzle and not a physics paradox. Contra Lerma, I have reluctantly concluded that the catch-up problem is not the answer. This does not mean that for a computer to predict its own output is the trivial task that Eckhardt suggests. Such self-prediction is the heart of the paradox. The separation of the system into Laplace and Dostoevsky is just a convenient portrayal. My residual fondness for the catch-up problem comes from its suggestion that relative size is the binding constraint against successful prediction. Such a result would free me from the Walden Two nightmare: My fear that something of roughly m y size and complexity, for example, B. F. Skinner, could predict, and so

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4 THE MATHEMATICALINTELLIGENCERVOL. 15, NO. 2, 1993

(working, CO says Ans, C 1 says Co says Ans . . . . ), whereas at time T + 1 the output is (Ans, Co says Ans, C1 says CO says Ans . . . . ).

Drawing by Lorenz; 9 1992 The New Yorker Magazine, Inc. control, my behavior. I am less bothered by predictability by something vastly greater than myself, for example, an angel. Notice that as long as Skinner does not inform me of his predictions, his control of me does not appear to raise any more logical contradiction than does his training of any other pigeon. He merely adjusts, unknown to me, parameters whose effects on m y behavior he can, by assumption, predict. The catch-up result, or Hoover's storage-size variant, would suggest that the paradox reveals a limitation upon Skinner which would deny the possibility of even such nonparadoxical control. The trouble is that the paradox can be reconstructed with gadgets which clearly do admit a kind of selfdescription. Although I originally described it using a finite array of particles, the puzzle remains in force even if the computers are infinite. For infinite computers certain kinds of self-description and even selfprediction are possible. Let {Ci: i = 0, 1. . . . } be a sequence of finite computers increasing in size so that computer C i can predict by time T (fixed throughout) the results of any T + 1 computation by any hookup of the earlier Cj's. The infinite computer is the union of the Ci's with each receiving inputs only from the programmer and the previous ones in line. Give Co a problem, C 1 the problem of predicting Co, C2 the problem of predicting C1, etc. After completing its task, each component just keeps printing the same output. At time T the output is the sequence:

Thus, the subsystem, {C1, C2. . . . }, does predict the outcome of the entire system but only provided the feedback necessary to exploit the prediction does not exist. We are left with the issue of self-reference which, I think, holds the key. However, I disagree with Eckhardt's semantic analysis. Questions about the meaning and reference of such terms as "prediction" and "about" have to do with our interpretation, from the outside, of part of the wiring diagram of the system. Such metalanguage is not required by the computers themselves. I think Steiner has it right. My Math Department colleague Stanley Ocken agrees, although phrasing it differently. He suggests that the problem is not well-posed in that m y ideal-gas particles fog over the issue of setting the whole system up. He challenges me to state the paradox in terms of finite-state machines. I do not see how to do so but that may not be significant. My imaginative facility with computers collapses long before it is hamstrung by logic. Ocken also suggests an alternative route of escape from my Skinnerian nightmare. Complexity theory implies that for many problems, like iteration of a function, methods which exploit size superiority, like parallel processing, cannot be used to compress the number of steps which must be performed in sequence to obtain a solution. (So to build my infinite computer above, we require a sequence of machines of increasing speed rather than size. No problem. Since we are using an infinite number of components anyway, there is no reason to feel bound by the speed of light, either. But back to our universe.) "That means," I told him, "that not only can Skinner not predict me but angels can't either. Of course, God still can because he is exempt from all these rules." "That's right," was his reply. When I looked startled at his certainty, Stanley, who is Orthodox, smiled and added, "I have other sources of information." Postscript: The article exhibited a sample of unpredictability, human or computer. The cartoon illustrating the relationship between Laplace and Dostoevsky was misattributed. It is the work of Samuel Vaughan of Berkeley, California. Ethan Akin Department of Mathematics The City College 137 Street and Convent Avenue New York, NY 10031 USA THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993 5

Painlev6's Conjecture Florin N. Diacu

A-t-on tout ~ fait le droit d'6tablir une s6paration entre les deux grands aspects de la vie de PainlevG son c6t6 scientifique et son c6t6 humain? Ce n'est point certain et, devant nous, r6cemment, l'homme d'l~tat qui a peut-~tre 6t6 le plus pros de sa pens6e et de son action, faisait ressortir l'unit6 secrete par laquelle toutes les manifestations de cette admirable nature sont solidaires les unes des autres. Jacques Hadamard: L'oeuvre scientifique de Paul Painlev6 Revue de Mdtaphysique XLI (1934), 289-325 This is a story about celestial mechanics and mathematics and about a question older t h a n Bieberbach's conjecture; a question that died close to its 100th birthd a y but which--like a n y good question--left behind it m a n y other u n a n s w e r e d questions as well as a universe of intellectual achievements.

that of an initial-value problem for a system of 6n differential equations: Solve

(t = M-lp, # = VU(q)

subject to the initial conditions (q, p)(0) E ( R 3 ~ ) x R 3n, where p = M~1 denotes the momentum of the system. For n = 2, the problem is not difficult, and its solution can be found in any celestial mechanics or astrono m y textbook u n d e r the n a m e of the two-body problem or the Kepler problem (in h o n o u r of the famous German astronomer Johannes Kepler w h o actually provided N e w t o n the inspiration for the inverse-square attraction law). Depending on the initial conditions, the motion of one particle with respect to the other can be an ellipse (including possibly a circle), a parabola, a

The n-Body Problem The roots of the n-body problem get lost s o m e w h e r e in the early history of h u m a n k i n d , but we can easily recognize its m o d e r n birth certificate signed by Isaac N e w t o n in his f u n d a m e n t a l Philosophiae Naturalis Principia Mathematica, published for the first time in 1687. The clear formulation of the problem in terms of differential equations is based on the inverse-square law of m u t u a l attraction b e t w e e n particles a n d can be stated in the following way: Consider n particles in the ambient space w h o s e positions are given by the vectors qi, i = 1, 999, n (with respect to a fixed frame), a n d let q = (ql . . . . . qn) be the configuration of the system. Determine the motion of the n particles by finding the general solution (q, it) of the second-order system /t = M-lVU(q), where U: R3n'~--'> R+, U(q) ~,mimjlqi - qj1-1 is called the potential function (or force function) of the system of particles, A = U{qlq i = qj} is the collision set, a n d M = diag(m 1,m 1,m 1. . . . . m n , m n,mn) i s a 3 n dimensional diagonal matrix, m 1, m2. . . . . mn being the masses of the n particles. The usual formulation is =

6

(1)

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2 9 1993 Springer-Verlag New York

branch of a hyperbola, or a line. This last case, of rectilinear motion, is the only one when collisions between the two particles can take place. It is interesting that the complete solution as described above was not given by Newton as one would expect, but by Johann Bernoulli, and only in 1710 (see [24]). For n I-- 3, the problem is still open even after three centuries of intense efforts to find its solution. Almost all important mathematicians up to the first quarter of this century attacked some aspect of the n-body problem, bringing important contributions to the understanding of the subject. In spite of this, the global image we have today is still far from complete. There are several ways to approach the problem. A modern method for tackling systems of differential equations in 19th-century mathematics was to find first integrals and, consequently, to reduce the dimension of the system. More precisely, a function F: (R3n'~) • R3n ---~ R

is said to be a first integral for Equations (1) if F(q, p) = c (constant), along a solution (q, p) of it. A relation like this between the components of a solution reduces the dimension of the system by 1. It is known that systems Of k equations have (locally) k linearly independent first integrals, and it was an important goal to find as many integrals as possible. For Equations (1), 10 of them were easy to obtain: three integrals of the momentum, three integrals of the center of mass, three of the angular momentum, and one energy integral, namely, Y~Pi = a, Xmiq i - at = b, ?qi x Pi = c, T(p) - U(q) = h, where a, b, c are constant vectors and h is a real constant with T denoting the kinetic energy. Any further attempt to find new ones was unsuccessful, and people started to look for other methods. The decisive result which stopped completely the search for first integrals was published in 1887 by Bruns. In a long paper [2] he proved the following negative statement: THEOREM 1. The only linearly independent integrals of Equations (1), algebraic with respect to q, p, and t, are the 10 described above. This was an important moment in the history of mathematics, which changed the way of thinking prevalent since Galilei. After a long period of quantitative methods, mathematicians understood that the class of problems solvable in this way is very small, and a large w i n d o w on qualitative m e t h o d s was opened. The new era was signaled by Liapunov stability criteria, obtained approximately at the same time, and also motivated by celestial mechanics. Approximately one hundred years ago, interest in the problem reached a high level. Advised by Gustav

Mittag-Leffler (at that time Editor-in-Chief of Acta Mathematica), King Oscar II of Sweden and Norway, a protector and supporter of science and especially of mathematics, established in 1887 an important prize for solving the 3-body problem. The formulation was very precise: one must obtain, for any choice of the initial data, a solution expressing the coordinates as a power series, convergent for all real values of the time variable. The idea of attacking the problem in this way is attributed to Dirichlet (see [19]). Bruns's result was at that time still too fresh to change the belief in quantitative methods. Unexpectedly, nobody could provide the desired solution. In spite of this, the prize was awarded to Henri Poincar~ in 1889 for his memoir Le probl~me des trois corps et les ~quations de la dynamique, published in Acta Mathematica one year later [13]. This was in recognition of this paper's stimulating value for further research in mathematics and mechanics, and indeed this choice was a good one. Poincar~'s interest was aroused by this success and he continued investigation into the mysterious n-body problem for many years. He also wrote the famous Les nouvelles m~thodes de la m~canique c~leste, in three volumes [14], where the idea of chaos appears for the first time. Not only were many mathematical theories born from the study of the n-body problem but also the strength of new theories is checked today by trying to find applications of them to this old problem. It has been studied by classical analysis, differential equations, and sometimes function theory, but nowadays also by new fields like dynamical systems, differential topology, differential geometry, Morse theory, algebraic geometry, algebraic topology, symplectic manifolds, Lie groups and algebras, ergodic theory, numerical analysis and computers, operator theory, and C*algebras.

The Conjecture of Painlev~

In 1895, at 32 years of age, Paul Painlev~ was already one of the most famous mathematicians of his time, and King Oscar II invited him to give a series of lectures at the University of Stockholm in SeptemberNovember of that year. The event was considered of paramount importance and even the King attended the introductory lecture. The notes were published in 1897 in handwritten form under the title Lemons sur la th~orie analytique des ~quations diff~rentielles [11] and can be found today also in Painlev~'s Complete Works [12]. The last pages contain an application of the results to the 3-body problem and an opinion of the author concerning the n-body case, formulated as a statement which was known afterwards as the Conjecture of Painlev~. First, let us try to understand its natural birth. Standard results of differential equations ensure, for any (q, p)(0) E (R3n'~) x R 3n, the existence and THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993

7

uniqueness of an analytic solution of Equations (1) defined locally on (let's say) (t-,t+), with 0 contained in this interval. Due to the symmetry of mechanical laws with respect to the past and future, one can study the problem on (t-,0] or on [0,t+), without loss of generality. Because many scientists have a natural desire to predict future phenomena, let us choose the second interval. We can extend the solution analytically to a maximum interval [0,t*), with 0 < t + ~< t* ~< oo. In case t* = % the motion is called regular, whereas if t* is finite, we say that the solution experiences a singularity. What is the physical meaning of such a singularity and is it important? One obvious possible way for a solution to encounter a singularity is for a collision to occur. Indeed, the configuration vector q will then so tend to the set ,~ that at least two position vectors have the same value, consequently VU tends to infinity and the equations of motion (1) become meaningless. The creation of the prize made the importance of such a study very clear. Because a series expansion of the coordinates convergent for every real value of t was asked, solutions leading to singularities were expected to be extended somehow beyond the singularity. Although very young in 1887, Painlev6 was working on his doctoral thesis and knew about the famous problem. He tried, therefore, to understand whether in the 3-body problem the only possible singularities are collisions. His worry about the occurrence of other singularities was motivated by the possible appearance of large oscillations (suspected already by Poincar6). For example, one particle could oscillate between the other two without colliding but coming closer and closer to a collision at each close encounter. Under such circumstances, one can find a subsequence tn of times converging to a finite t* such that VU(q(tn)) ~ o0. This again makes Equations (1) meaningless, and such t* is also a singularity. In modern terminology, DEFINITION. Let (q, p) be a solution of Equations (1) defined on [0,t*) with t* a singularity. Then t* will be called a collision singularity if q(t) tends to a definite limit when t ~ t*, t < t*. If the limit does not exist, then the singularity will be called a pseudocollision or noncollision singularity. It is clear that these singularities (especially the noncollision ones) are an important obstacle to accomplishing King Oscar's goal. Indeed, one might try to extend a collision as an elastic bounce and possibly obtain a globally convergent power series, but how to do that with pseudocollisions? Painlev6 doubted that pseudocollisions can actually appear and he proved for the 3-body case THEOREM 3. For n = 3, any solution of the Equations (1) defined on [0,t*) with t* a finite singularity, experiences a collision when t ~ t*. 8

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Attempts to extend this result to the n-body problem (n > 3) failed, and the intuition of Painlev6 was that pseudocollisions may, indeed, arise for more than 4 bodies. Thus, his Stockholm lectures end with the following: CONJECTURE. For n ~ 4, Equations (1) admit solutions with noncollision singularities. Painlev6 understood that this is a very hard problem; his subsequent mathematical work contains some papers dealing with singularities, none, however, attempting to prove the conjecture. After 1905, Painlev6's scientific activity becomes less intense because of his deep involvement in politics. Paul Painlev6 was elected several times as deputy, holding the War, then Finance, and finally Air portfolios, and serving as President of the Chamber of Deputies of France. In 1918, he became Pr6sident de l'Acad6mie des Sciences, and in 1927 the University of Cambridge offered him the title of Doctor Honoris Causa. Indeed a remarkable and successful life! His famous conjecture remained open, however, for more than half a century after his death. It is interesting to note that collision orbits are very improbable. Donald Saari proved that in the n-body problem they are of Lebesgue measure zero and of the first Baire category. Moreover, this is true for all singularities in the 4-body problem (see [15,17]). Some of these results were generalized and are expressed in terms of lower-dimensional manifolds [18]. It is also expected that, for any n, singularities are improbable. However, these results did not diminish the interest in the study of singularities.

Singularity Criteria Many of Painlev4's contemporaries tried to find examples of solutions with pseudocollisions but no one succeeded. Their attention was, therefore, directed towards understanding theoretical aspects and especially t o w a r d s criteria for obtaining noncollision singularities. A w a y of finding singularities had already been found, but it is quite hard to discern when and by whom: THEOREM 4. Consider a solution (q, p) of Equations (1). Then t* is a singularity of this solution iff lim inf min qij(t) = 0, t--*t*

(2)

i<j

where qij = Iqi - qjl. Painlev6 himself improved this result [11] in proving Theorem 3. He showed that condition (2) can actually be replaced by lim rain qij(t) = O. t-*t* i<j

,,4

~o",

f~

The first important condition for the occurrence of /2 / noncollision singularities was found and published $ ,' only in 1908 by a Swedish mathematician of German origin, Hugo yon Zeipel [26]. His result has not only a ~_.~;;~, ; ,__ nice formulation but also an unusual history and , . . . . . . ',played a fundamental role in the story of Painlev6's ~.~ L ~ , " ' ",, conjecture, rT . . . . "",i,.~ ~ .-.~. THEOREM 5. If t* is a collision singularity for a solution (q, p) of Equations (1), then J(q(t)) tends to a definite limit when t --~ t*, where J(q) = Emi[qi[2 is the moment of inertia. This implies, of course, that a necessary condition for having a noncollision singularity is that the motion become unbounded in finite time, because the moment of inertia is a measure of the distribution of particles in space. What is obvious is that at a singularity the whole [(q, p)[ has to become unbounded. This always happens at a collision instant because the velocities are infinite. It is not clear what would happen in the configuration space (i.e., for the vector q), and here lies von Zeipel's contribution. His paper appeared in a less famous journal (see [26]) and was, therefore, not well known. Personally I have tried to find it in several good university libraries in Eastern and Western Europe as well as in North America, but without success. An article of Dick McGehee [10], who spent a period in Stockholm and was interested in this subject, makes it less necessary to read the original. The French astronomer Jean Chazy had announced Theorem 5 without making any reference to von Zeipel's paper [3]. Aurel Wintner wrote in 1941 that the proof of the Swedish mathematician has some gaps and there is no complete argument for the theorem [24]. Thirty years later, Hans Sperling gave a detailed proof [20], apparently ending the dispute. However, McGehee's paper cited above provides a translation in modern mathematical language of von Zeipel's proof, showing that it was actually correct from the beginning. Today we know a beautiful generalization of this result which is due to Donald Saari from Northwestern University [16]. He proves that if J ~ q is a slowly varying function as t --~ t* for a solution (q, p) of Equations (1), then the singularity t* is necessarily a collision. Theorem 5 is a fundamental contribution to the subject of singularities in the n-body problem, and the elucidation of Painlev~'s conjecture would have been hard to imagine without it.

The Computer and the Idea As has happened m a n y times, the idea that was to solve Painlev6's conjecture came by looking for something else, and depended on electronic computers.

,/ 05

/ '#o

,/

s#4"

6 /

,,," ~8.a' 60 ' 60

,,, $or .' ,," .~$6

H

Figure 1: Numerical results in the Pythagorean problem. In 1893, Meissel proposed the investigation of a socalled Pythagorean problem, in which three gravitationally attracting particles of masses 3, 4, and 5 are initially located at the vertices of a triangle with sides 3, 4, and 5 such that the corresponding point masses and sides are opposite. Releasing the particles with zero initial velocities from their positions, how will they move in the future? Burrau investigated the problem numerically in 1913 but without reaching important conclusions. Several computer investigations in 1966 and 1967 [21] helped to go much further by showing a surprising qualitative behavior: After passing close to a triple collision, two particles will form a binary while the other one is expelled with high velocity in the opposite direction, as in Figure 1 (see also [1]). The formation of the binary was an interesting point for astronomers, whereas the high-speed escape of the third particle attracted the attention of mathematicians. It provided the idea that it might be possible to construct an example of a noncollision singularity solution. The main reason for this qualitative feature is the triple approach of the particles, as was recognized in [8], [9], [22], [23]. We sketch crucial ideas from Dick McGehee's 1974 paper. He considered the case of the rectilinear 3-body problem, i.e., w h e n the masses ml, m2, m3 move all the time on a fixed line. He was interested in understanding the flow in a neighborhood of a triple collision solution. This was, indeed, a hard problem because previous numerical investigations suggested chaotic behavior near a total collapse (i.e., a simultaneous collision of all bodies). Only qualitatively speaking, the THE MATHEMATICAL INTELLIGENCER VOL. 15, N O . 2, 1993

9

Figure 2: The flow on the collision manifold. r

1

~

2

rrl

3

rfl

4

Figure 3: The example of Mather and McGehee.

particles behave by forming a binary and an escape, numerical investigations showing a highly sensitive dependence with respect to initial data. For example, for some initial conditions the particles m 1 and m2 form a binary and m 3 escapes, whereas, perturbing the data a little bit, it may be that m 1 and m3 form a binary and m2 escapes. Two such solutions look very different in a phase-space picture, in spite of being close to one another at some initial moment of time. McGehee's idea was to restrict the equations of motion to an arbitrary energy level, then to blow up the singularity (by using certain transformations which today bear his name), and finally to introduce a so-called collision manifold which is proved to be independent of the chosen energy level. In the rectilinear 3-body problem, the collision manifold happens to be a sphere missing four points, as in Figure 2. Roughly speaking, the McGehee coordinates are polar coordinates for the configuration vector and a decomposition of the velocity into a radial and a tangential component, rescaled by a suitable transformation of time, which makes the collision manifold be approached asymptotically by the real flow when the new (fictitious) time variable goes to infinity. The equations of motion restricted to the collision manifold do not describe a real physical situation. However, due to the continuity property of the solutions with respect to initial data, a study of the flow on this manifold provides valuable information on the behavior of solutions passing close to a triple collision. Many interesting theoretical results were proved in McGehee's paper using these powerful techniques, including a theorem on the occurrence of solutions with high-velocity escapes. Studies on collision singularities are hard to imagine today without McGehee's transformations. The Example of Mather and McGehee

One of the results McGehee a n n o u n c e d (without proof) in his 1974 paper is the construction of a solution with noncollision singularities in the rectilinear 5-body problem, using the idea of a high-speed escape. The trouble is not that collisions always appear in a ~.0

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993

rectilinear problem but that they always arise before an impending pseudocollision, as was shown by Saari [16]. I may now have confused the reader, for I said before that the solution was defined on a maximal interval [0,t*), t* (finite) representing either a collision or a noncollision singularity. There is no inconsistency. Binary collision solutions can be analytically extended by a mathematical procedure called regularization. There is a vast literature on this subject (see, e.g., [4]). Physically, this means that an elastic bounce, without loss or gain of energy, takes place. I hope the sense of Saari's result is now clear. Mather and McGehee [7] were later able to prove completely that a noncollision singularity can occur in the rectilinear 4-body problem, but only after an infinity of (regularized) binary collisions. Here is their scenario. Four bodies of suitably chosen masses ml, m2, m3, m4 lie on a straight line at some initial moment (see Fig. 3). The initial data (positions and velocities) are such that the particles ml and m 2 stay close together, so we say that they form a binary system. The particle m 3 oscillates between the binary system and the particle m4. The motion is regularized beyond the binary collisions which take place at the instants t 1, t2. . . . . t k. . . . . This sequence converges as k goes to infinity. Meanwhile, the binary m~, m2 goes to - ~ , m4 goes to + ~, and m3 bounces back and forth, with increased velocity after every close passage to a triple collision. This is possible because the distance between ml and m2 tends to zero, the loss of potential energy of the binary being transferred into kinetic energy for the particle m3. The proof of Mather and McGehee is not at all easy. Whatever its mathematical beauty and interest for dynamical systems theory, the above example is not accepted as a proof of Painlev6's conjecture because the pseudocollision a p p e a r s only after (infinitely many) collisions. Gerver's First E x a m p l e

In 1984, Joe Gerver from Rutgers University proposed a solution of a planar 5-body problem in which the particles escape to infinity in finite time [5]. Although he does not give a complete proof, he provides a lot of support for the existence of such a solution. We reproduce his scenario. Consider the planar motion of five particles m 1. . . . . ms, with m 3 = m4, m2 somewhat greater but of the same order of magnitude as m 3, m I much smaller than m2, and m5 much smaller than ml (see Fig. 4). Initially, m 1 is in a roughly circular orbit around m2, whereas m3 and m4 are much further away. The bodies m2, m3, m4 are approximately at the vertices of an obtuse triangle. Initially the triangle is slowly expanding while maintaining its shape. Meanwhile, ms m o v e s rapidly around the triangle, coming close to each of the other

:"

\ m

'% m

2

ITI 5

a /~

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t ~e

/

ss ssSe

~l ~

9 m

/ r

Y

",

4

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Figure 4: Gerver's heuristic example.

Figure 5: The example of Xia.

four bodies, the velocity of m5 being much greater than that of m 1. Each time m s passes close to m 1, it picks up a small amount of kinetic energy. This causes m I to fall into a lower orbit around m2 such that the mean kinetic energy of m I in its orbit actually increases by about the same factor as for m 5. A small fraction of the kinetic energy of m5 is transferred to m2, m3, and m4, causing faster expansion of the triangle. The time required for one trip of ms around the triangle decreases each time (in spite of the expansion) by a factor slightly less than 1. After a finite time, the geometric progression of the time instants t 1, t 2. . . . . tk. . . . measuring a round trip will converge, and m 5 will have travelled an infinite number of times around the triangle. In the meantime, the triangle has become infinitely large.

start to separate. This separation reduces the retaining force on the small particle which consequently moves very fast towards the other binary system. The actionreaction effect forces the binary m3, m4 to move further away from the plane (x,y). The same situation described above is now repeated (in mirror image) for the binary ml, m2. Iterating this procedure with higher and higher accelerations for m5, the two binaries will be forced to tend to infinity in finite time. Simple though this scenario sounds, it is very hard to prove it is possible. For example, because the motion becomes unbounded in finite time, the acceleration effects on the small particle have to become infinitely large. The point masses in each binary must come closer and closer together, making it hard to guarantee nonoccurrence of collisions. There were mistakes in the first attempt of Xia but he was able to correct them. The paper appeared in Annals of Mathematics. His example can be extended to similar symmetric problems for any N > 5. In spite of his youth (not even 30 years old in 1992), today associate professor at Georgia Tech, Xia has already brought a tremendous contribution to the field. He recently proved a new magnificent result, namely, that the very rare (and hard to detect) phenomenon called Arnold diffusion (a kind of chaos) takes place in a very natural system, the elliptical restricted 3-body problem. Arnold himself constructed in the 1960s a very sophisticated and artificial system to show for the first time that such a phenomenon exists. It is expected that Xia will make many other important contributions in years to come.

Xia's Example

In his doctoral thesis written under the supervision of Donald Saari at Northwestern University, Jeff Xia proved in 1988 that a certain type of solution in the spatial 5-body problem leads to a noncollision singularity without involving an infinite number of binary collisions, as was the case in the example of Mather and McGehee. Painlev6's conjecture w a s finally proved. The author considers two pairs of bodies, the particles in the same pair having equal masses, plus a fifth particle of small mass. The bodies in a pair move in highly eccentric orbits parallel with the (x,y)-plane (see Fig. 5). The binaries are on opposite sides with respect to the (x,y)-plane and have an opposite rotation. The motion of the small particle is restricted to the z-axis, so that the total angular momentum is zero. The small particle will oscillate between the two binaries, deterGerver's Second Example mining an u n b o u n d e d motion in finite time. More precisely, suppose the particle m 5 intersects the line con- The idea of using radial symmetry, combined with the necting m 3 with m4 from above, at a moment when experience obtained by trying to prove his previous these particles come near to their closest approach, the heuristic example, led Joe Gerver to the following somotion of m3, m4, and m s thus being close to a triple lution for the planar case. Consider 3n bodies (n sufficollision. The body m 5 goes a little under the line m3m4, ciently large) in a plane as in Figure 5.2n of the partiwhereas the particles m 3 and m4 are at their closest cles are arranged in n nearly circular orbiting pairs and approach. Thus, m 5 is strongly attracted backwards. It all have the same mass. The center of mass of each intersects the line m3m 4 again when these point masses binary lies at one of the vertices of a regular polygon. THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993

11

References

Figure 6: Gerver's planar example. The other n bodies have small equal masses a n d move rapidly from one pair to the other as in Figure 6. W h e n a small particle comes close to the binary it takes some kinetic energy from the pair and transfers some mom e n t u m to it, forcing the binary to move into a tighter orbit and concomitantly to increase its distance from the center of the polygon. Iterating this process for a suitably chosen n, suitable values of the masses, and of the initial velocities, the size of the configuration will increase by each close encounter of a small particle with a binary. The sequence of times from one encounter to the next will converge to a finite value, whereas the system becomes u n b o u n d e d in finite time. The complete proof contains very m a n y computations and is, therefore, quite hard to follow (see [6]). Gerver f o u n d out about Painlev6's conjecture 19 years before he gave the solution. Xia succeeded in proving his example about six m o n t h s before Gerver. However, Gerver's is the first confirmation of the conjecture for the case of planar solutions a n d is also very elementary, using mainly 19th-century mathematics. Seeing the proof, one sees that the conjecture would have been possible for Painlev6's contemporaries to prove, but n o b o d y did it. It was not the first time Gerver attacked a famous problem. As a g r a d u a t e s t u d e n t at Columbia University in 1969, he proved a conjecture of Riemann on t h e n o w h e r e d i f f e r e n t i a b i l i t y of t h e f u n c t i o n ~ = lsin(n2x)/n 2. But this was long before his work on Painlev6's conjecture started. A comparison b e t w e e n the two solutions is hard to make. Each is interesting and valuable in its o w n way. Xia o p e n e d a n e w direction of work bringing fresh air into the field, whereas Gerver used the old m e t h o d s showing that t h e y can be successful too. Surely both achieved a most remarkable feat in an old a n d hard field where good n e w results are not at all easy to obtain.

1. V. I. Arnold, Dynamical Systems III, New York: SpringerVerlag, 1988. 2. H. Bruns, Ober die Integrale des Vielk6rper-Problems, Acta Math. 11 (1887), 25--96. 3. J. Chazy, Sur les singularit6s impossibles du probl~me des n corps, C. R. Hebdomadaires S~ances Acad. Sci. Paris 170 (1920), 575-577. 4. F. N. Diacu, Regularization of partial collisions in the N-body problem, Diff. Integral Eq. 5 (1992), 103-136. 5. J. L. Gerver, A possible model for a singularity without collisions in the five-body problem, ]. Diff. Eq. 52 (1984), 76-90. 6. J. L. Gerver, The existence of pseudocollisions in the plane, J. Diff. Eq. 89 (1991), 1-68. 7. J. Mather and R. McGehee, Solutions of the collinear four-body problem which become unbounded in finite time, Dynamical Systems Theory and Applications (J. Moser, ed.), Berlin: Springer-Verlag, 1975, 573-589. 8. R. McGehee, Triple collision in the collinear three-body problem, Invent. Math. 27 (1974), 191-227. 9. R. McGehee, Triple collision in Newtonian gravitational systems, Dynamical Systems Theory and Applications (J. Moser, ed.), Berlin: Springer-Verlag, 1975, 550-572. 10. R. McGehee, Von Zeipel's theorem on singularities in celestial mechanics, Expo. Math. 4 (1986), 335-345. 11. P. Painlev6, Lemons sur la thdorie analytique des dquations diffdrentielles, Paris: Hermann, 1897. 12. Oeuvres de Paul Painlev~, Tome I, Paris Ed. Centr. Nat. Rech. Sci., 1972. 13. H. Poincar6, Sur le probl6me des trois corps et les 6quations de la dynamique, Acta Math. 13 (1890), 1-271. 14. H. PoincarG Les nouvelles mdthodes de la mdcanique c~leste, Paris: Gauthier-Villar et Fils, vol. I (1892), vol. II (1893), vol. III (1899). 15. D. G. Saari, Improbability of collisions in Newtonian gravitational systems, Trans. Amer. Math. Soc. 162 (1971), 267-271; 168 (1972), 521; 181 (1973), 351-368. 16. D. G. Saari, Singularities and collisions in Newtonian gravitational systems, Arch. Rational Mech. Anal. 49 (1973), 311-320. 17. D. G. Saari, Collisions are of first category, Proc. Amer. Math. Soc. 47 (1975), 442--445. 18. D. G. Saari, The manifold structure for collisions and for hyperbolic parabolic orbits in the n-body problem, J. Diff. Eq. 41 (1984), 27--43. 19. C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, Berlin: Springer-Verlag, 1971. 20. H. J. Sperling, On the real singularities of the N-body problem, ]. Reine Angew. Math. 245 (1970), 15--40. 21. V. Szebehely, Burrau's problem of the three bodies, Proc. Nat. Acad. Sci. USA 58 (1967), 60-65. 22. J. Waldvogel, The close triple approach, Celestial Mech. 11 (1975), 429-432. 23. J. Waldvogel, The three-body problem near triple collision, Celestial Mech. 14 (1976), 287-300. 24. A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton, NJ: Princeton University Press, 1941. 25. Z. Xia, The existence of noncollision singularities in the N-body problem. Ann. Math. (in press). 26. H. von Zeipel, Sur les singularit6s du probl~me des corps, Arkiv f6r Mat. Astron. Fys. 4, (1908), 1--4.

Acknowledgments The author is indebted to Chris Bose a n d Ian P u t n a m for reading the manuscript in detail, for finding some errors, and for proposing several improvements. 12

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993

Department of Mathematics and Statistics University of Victoria Victoria, British Columbia V8W 3P4 Canada

A Visit to Hungarian Mathematics Reuben Hersh and Vera John-Steiner

In July 1988, we visited Budapest to participate in the Sixth International Congress on Mathematical Education. We decided to use this opportunity to try to shed some light on the legendary reputation of Hungarian mathematics. One of us (V.J.-S.) is a native of Budapest and is familiar with the city and its language. Our investigation focused on historical, pedagogical, and social-political aspects of Hungarian mathematical life. We did not attempt to survey Hungarian mathematical research of the present. Even so, our time proved too short for our ambitions. The important Hungarian mathematicians whom we missed are certainly more numerous than those we interviewed. We spoke in depth to a dozen people, and card e d out formal interviews with eight: in Hungary, B61a Sz6kefalvi-Nagy, Pal Erd6s, Tibor Gallai (recently deceased), Istv~n Vincze, and Lajos P6sa; in the United States, Agnes Berger, John Horv~th, and Peter Lax. (While we were in Budapest, two of the leading newspapers carried major articles honoring Sz6kefalviNagy's 75th birthday.) We asked all our interviewees the question, what is so special about Hungarian mathematics? What made possible the production of so many famous mathematicians in such a small, poor country, in the period between the two Wars? In our interviews, and also in our reading, we got two quite distinct kinds of answer. Type 1 was internal. It related to institutions and practices within the world of mathematics. The other kind, type 2, was external. It related to trends and conditions in Hungarian history and social life at large. Perhaps one contribution of this article is to point out the importance of both types of answer. One could conjecture that favorable conditions of both types---within mathematical life and within socio-politico-economic life at large-are necessary to produce a brilliant result such as Hungarian mathematics of the 1920s and 1930s. In the terminology used by Mih~ly Csikszentmih~lyi and Rick Robinson [5] in their study of creativity, perhaps conditions have to be right both in the " d o m a i n " - - t h e area of creative w o r k - - a n d in the "field"--the ambient culture.

Bolyais, Father and Son Hungarian mathematics began, in a sense, with J~inos Bolyai (1802-1860), one of the creators of nonEuclidean geometry, and his father Farkas (1775--1856), also a creative mathematician of importance. In their lifetimes, they were totally ignored, both at home and abroad. "It is a widely accepted opinion that Farkas Bolyai was the first mathematician in Hungary to have

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2 9 1993 Springer-Verlag N e w York

13

original results" [4], page 222. He studied at G6ttingen from 1796 to 1799 and established a lasting friendship with fellow student Carl Friedrich Gauss [4]. He and Gauss were both interested in the "problem of parallels" (independence of Euclid's fifth postulate). Farkas returned to Hungary and, in 1804, became mathematics professor at the Reformed College of Marosv~s~irhely in Transylvania. In 1832-1833, he published a two-volume textbook in Latin entitled Tentamen juventutem studiosam in elementa matheseos introducendi. It was reprinted in 1896 and 1904. J~inos (1802-1860) inherited his father's interest in the problem of parallels. In fact, with one single exception, Farkas was the only human being w h o understood and appreciated J~nos's discovery of nonEuclidean "hyperbolic" geometry. When Farkas sent his son's discoveries to Gauss, Gauss replied, "I cannot praise this work too highly, for to do so would be to praise myself.'" Gauss had anticipated J~nos's discoveries by decades. His decision to withhold his own work from publication made it impossible for J~nos to attain the recognition he knew he deserved. A few years after J~nos Bolyai died in 1860, foreign mathematicians began to get interested in him. In 1868, Eugenio Beltrami in Italy published his discoveries on the pseudosphere. He found that this surface is a model for the Bolyai-Lobatchevsky hyperbolic geometry, and so provides a relative consistency proof for it. In 1871, Felix Klein and, in 1882, Henri Poincar6 published their models of the hyperbolic plane. In 1891, C. B. Halsted of the University of Texas published an English translation of J~nos Bolyai's work on hyperbolic geometry, called the Appendix. He visited J~nos's grave and made strenuous efforts to gain recognition for him. By this time, Hungary began to realize that one of its most illustrious sons was a mathematician. The Hungarian Academy of Sciences established the Bolyai Prize: 10,000 gold crowns, to be awarded every five years, to the mathematician whose work in the previous 25 years had given most to the progress of mathematics. The first prize committee was made up of Gyula K6nig (1849-1913), Guszt~v Rados (1862-1942), Gaston Darboux, and Felix Klein. The first Bolyai Prize went to Henri Poincar6 in 1905; the second, to David Hilbert in 1910. Unfortunately, one consequence of the First World War was the devaluation of the fund from which the prize was to be given. It was never awarded again.

Ausgleich and Emancipation After losing her independence to the Turks in 1526, Hungary was for centuries occupied, first by the Ottoman and later the Habsburg Empires. In 1848, there 14

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993

was a revolution and feudalism was abolished. In 1848-1849, an unsuccessful war for independence was waged against the Austrian Empire. This was followed by years of passive resistance. Then, in 1866, the Austrian Emperor Franz Joseph suffered a humiliating military defeat by Prussia. Faced also with rising nationalism among Czechs, Ruthenians, Romanians, Serbs, and Croatians, the Emperor granted the Hungarians a large measure of economic and cultural independence. In return, the Magyars renewed their allegiance to him. This pact became known as the Ausgleich, "the compromise." A year later, non-Hungarian minorities were granted civil rights. In particular, the Hungarian Jews, 5% of Hungary's population, were emancipated. For the first time, they were permitted to work for the state, including teaching in its schools. Laura Fermi writes [7], "From peasants and peddlers they turned into merchants, bankers, and financiers; they moved into i n d e p e n d e n t businesses and the professions. Soon they entered all cultural fields, giving themselves at last to the intellectual pursuits that are the highest aim of the Jewish people." The Ausgleich was followed by 40 boom years. Along with the commercial and industrial development of Budapest came the creation of an educational system, including universities, college-preparatory schools (gymnasiums), and a technical college. Many of the gymnasiums were denominational--Catholic, Protestant, or Jewish. Most were for boys, but there were some for girls. All this led to the appearance of mathematics teachers and professors. And some of them were brilliant, creative people. Laura Fermi's informants give a vivid picture of intellectual life in Budapest [7]. (See also the recent book [69] of John Luk~cs.) Budapest intellectuals, most of them individualists with no desire to conform, threw ideas at each other in cafes, expounded progressive or eccentric theories in the newspapers, turned their thumbs down in theaters at artists acclaimed in other countries, or made stars of unknown artists... Many students belonged to the Galilei Club of progressive undergraduates founded in 1908 by the philosopher Gyula Pikler and the future sociologist, K~roly Pol~nyi (George P61ya was a member.) . . . Most future emigr6s lived in Budapest or went there for their educat i o n . . . In Budapest, they had to keep mentally alert, to emulate and compete, and in order not to be submerged, they had to develop their capabilities to the full. She goes on: The flowering of Hungarian talent in the generation of the cultural wave was due to the special social and cultural circumstances obtaining in Hungary at the turn of the century. By then a strong middle class had emerged and asserted itself. Having risen in response to needs that the nobility did not feel inclined to fill and the peasants could not fill, it was largely Jewish and was animated by the intellectual ambitions of the Jews. The intellectual portion of this middle class converged upon the capital where it

created a peculiarly sophisticated atmosphere and kept its members under continuous stimulation. The political antiSemitism of the early twenties hit this segment of the population with great vehemence and gave the intellectuals a further reason for striving to excel and stay afloat. Under these circumstances, talent could not remain latent. It flourished. This must definitely be classified as a type 2 (field) explanation. By the time of the First World War, economic strains were affecting Budapest life. Then defeat in the war destroyed the Austro-Hungarian Empire. In Hungary, it was succeeded by a Soviet Republic that survived for only 4 months. The Bolsheviks were overthrown by an invading Romanian army. They were succeeded by Admiral Horthy's clerical authoritarian regime, which in time became one of Hitler's allies. The Allies treated Hungary not as a captive country like Slovakia and Croatia, but as a defeated power like Austria and Germany. The Treaty of Trianon gave twothirds of Hungary to Romania, Czechoslovakia, Austria, and Yugoslavia. Hungary had been primarily agricultural; now it had to live by exporting manufactured goods. But the world market had shrunk, new competitors were busy. Hungary never regained the comfortable prosperity of Franz Joseph's time. Yet, in mathematics its standing after the war would become even more impressive than before. John Horv~th offers a somewhat similar type 2 explanation. You can name the day in 1900 when Fej6r sat down and proved his theorem on Cesaro sums of Fourier series. [This work is described later. R.H.] That was when Hungarian mathematics started with a bang. Until then, there were just a few people who did mathematics. But from then on, every year somebody appeared who became a major mathematician on the international scene. A similar emancipation of the Jews happened in Prussia in 1812. And there you immediately had people like Jacobi, who became a professor in K6nigsberg. In Klein's History of Mathematics in the 19th Century, he has a little remark, that with the emancipation a new source of energy was released. There is one other thing which I sometimes mention. It's quite surprising how many of the mathematicians who came into the profession in Hungary after World War II are sons of Protestant ministers: Szele, Kert6sz, Papp, there's quite a number. And I guess the reason is much the same. Those kids would have become Protestant ministers just as the old ones would have become rabbis. [Note: In Horvfith's analogy between potential ministers and potential rabbis, there is, of course, no suggestion that the social-legal positions of Protestants and of Jews were equivalent or even similar. Peter Lax points out that Gy6rgy Haj6s (see below) started out by studying for the priesthood.] Another type 2 explanation, from John yon Neumann: "It was a coincidence of cultural factors: an external pressure on the whole society of this part of

Central Europe, a feeling of extreme insecurity in the individuals, and the necessity to produce the unusual or else face extinction" [59].

Contest and Newspaper When George P61ya (1887-1985) was asked [1] to explain the appearance of so many outstanding mathematicians in Hungary in the early twentieth century, he gave two sorts of explanations. First, the general one: "Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper. (Hungary never enjoyed the status of a wealthy country.)" Then three specific type 1 explanations:

1. The Mathematics Journal for Secondary Schools (K6z~piskolai Matematikai Lapok, f o u n d e d in 1894 by Dfiniel Arany). "The journal stimulated interest in mathematics and prepared students for the E6tv6s Competition." 2. The E6tv6s Competition. "The competition created interest and attracted young people to the study of mathematics." (This comment is more remarkable because P61ya himself, when a student, refrained from handing in his paper in the Competition!) 3. Professor Fej6r. "He himself was responsible for attracting many young people to mathematics, not only through formal lectures but also through informal discussions with students." We say more about Professor Fej6r later. As to K6zdpiskolai Matematikai Lapok and the E6tv6s Competition, it is virtually impossible to talk to or read about any Hungarian mathematician without hearing tribute to the stimulation and inspiration of these two institutions. In [1], Pill Erd6s was asked: "The great flowering of Hungarian mathematics--to what do you attribute this?" "There must be many factors. There was a mathematical journal for high schools, and the contests, which started already before Fej6r. And once they started they were self-perpetuating to some extent. [Domain, type 1.] Hungary was a poor country--the natural sciences were harder to pursue because of cost, so the clever people went into mathematics. [Field, type 2.] But probably such things have more than one reason. It would be very hard to pin it d o w n . " In our own interview with Erd6s, we pursued this remark. RH: Do you feel that your mathematical development was affected by the high school mathematics newspaper (K6-

zdpiskolai Matematikai Lapok)? Erd6s: Yes, of course. You actually learn to solve problems there. And many of the good mathematicians realize very early that they have ability. THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993

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Our interviewee Agnes Berger, a retired statistics professor at Columbia University, has vivid memories of Krzdpiskolai Matematikai Lapok. "The paper came once a month. It had problems grouped according to difficulty. The solutions were published in the following way: everybody who sent in a correct solution was listed by name, and the best solution or solutions were printed. So here you were taught right away to value, not only the solution, but the best solution, the most beautiful solution. It was called the model solution (minta vdlasz). It was a tremendous entertainment. Also, those people who did well, submitting many solutions, the frequent solvers, had their pictures published at the end of the year!" We asked Tibor Gallai about Krz~piskolai Matematikai

Lapok. Gallai: Nowhere else in the world is there this kind of high school paper, and this more than anything else is responsible for the excellence of Hungarian mathematics. RH: Do you have any idea why this took place in Hungary? What was it in this country that made this possible? Gallai: For part of 1894 and 1895 the Minister of Education was Lor~ind EOtvOs (1848-1919), after whom the University is named. He was deeply committed to the development of Hungarian culture and science. While he was in office there was founded the E6tvOs Collegium, with the purpose of improving the training of high school teachers. So he is part of what stimulated our development. RH: How do you feel about present-day competitions and students compared to years ago? Gallai: The quality is much higher now. When I first participated 60 years ago, the names of the students who solved the problems could easily be published, because there were only 30 or 40 of them. Now there are 600. It's impossible to publish all the names. Vera S6s: Now the problems are more difficult and demanding. There is a whole range of mathematicallyoriented young people who have a more effective foundation. While mathematics education in Hungary for the gifted and talented looks enviable from the perspective of the United States, not all Hungarian mathematics educators are satisfied with their situation. Laios P6sa, who once was one of Erd6s's most promising discoveries, has devoted himself in recent years to mathematics education for the normal or everyday student, not just the brilliant. He feels that the system does not do justice to these students, that the teachers, although supposed to teach by the problem-solving method, often do not feel sure or comfortable about problem-solving, and that many students fail to master mathematics as they could and should. The E6tv6s competition was established in 1894, the same year as Krz@iskolai Matematikai Lapok. The competition was established by the Mathematical and Physical Society of Hungary, at the motion of Gyula Krnig, under the name of "Pupils' Mathematical Competition." This was done in honor of the Society's founder and president, the famous physicist Baron Lor~ind E6tv6s (mentioned earlier by Tibor GaUai), who 16

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became Minister of Education that year. KOnig was a p o w e r f u l personality w h o d o m i n a t e d H u n g a r i a n mathematical life for several decades. His most famous deed in research seems to have been an incorrect proof of Cantor's continuum hypothesis. (He used a false lemma of Felix Bernstein. Except for Bernstein's lemma, K6nig's argument was correct. K6nig's own contribution to the proof survives as an important theorem in set theory.) K6nig wrote an early book on set theory, but its impact was diminished because Hausdorff's famous book on that subject appeared at about the same time. KOnig's son, D~nes (d. 1944), is remembered as the father of graph theory (more details later). Between the two wars, the competition continued under the name, "E6tv6s Lor~ind Pupil's Mathematical Competition." At present, it carries the name of J6zsef K~irsch~ik (1864-1933), who is remembered in particular for his extension of the notion of absolute value to a general field. He was professor at the Polytechnic University in Budapest and a member of the Hungarian Academy. In 1929, he compiled the original Hungarian edition and wrote the preface to Problems of the Mathematics Contests. In 1961, it was published in English as the Hungarian Problem Book [38]. The publication of the original Problem Book honored the tenth anniversary of E6tv6s's death. Winners before 1929 who later became famous include Lip6t Fejrr (1880-1959), Drnes K6nig, Theodore von K~rm~in (1881-1963), AIfrrd Haar (1885-1933), Ede Teller (later known in the U.S. as Edward), Marcel Riesz (1886-1969), G~ibor Szeg6 (1895-1985), L~iszl6 Rrdei (1900-1980), and L~iszl6 Kalm~ir (1900-1976). The English edition [38] contains a preface by G~ibor Szeg6. He wrote: [For a successful mathematics competition] some sort of preparation is essential to arouse public interest. In Hungary, this was achieved by a [high-school mathematics] Journal . I remember vividly the time when I participated in this phase of the Journal (in the years between 1908 and 1912). I would wait eagerly for the arrival of the monthly issue and my first concern was to look at the problem section, almost breathlessly, and to start grappling with the problems without delay. The names of the others who were in the same business were quickly known to me, and frequently I read with considerable envy how they had succeeded with some problems which I could not handle with complete success, or how they had found a better solution (that is, simpler, more elegant or wittier) than the one I had sent in. We get an impressive picture of Hungarian secondary mathematics education early in the twentieth century, including the E6tv6s Competition, from Theodore von K~irm~in, one of the preeminent founders of m o d e m aeronautics. In his autobiography [65], he tells about his high school, the Minta, or Model Gymnasium, which .

.

became the model for all Hungarian high schools. Mathematics was taught in terms of everyday statistics. We looked up the production of wheat in Hungary, set up

tables, drew graphs, learned about the "rate of change" which brought us to the edge of calculus. At no time did we memorize rules from a book. Instead, we sought to develop them ourselves... The Minta was the first school in Hungary to put an end to the stiff relationship between the teacher and the pupil which existed at that time. Students could talk to the teachers outside of class and could discuss matters not strictly concerning school. For the first time in Hungary a teacher might go so far as to shake hands with a pupil in the event of their meeting outside of class. Each year the high schools awarded a national prize for excellence in mathematics. It was known as the E6tv6s Prize. Selected students were kept in a closed room and given difficult mathematics problems, which demanded creative and even daring thinking. The teacher of the pupil who won the prize would gain great distinction, so the competition was keen and teachers worked hard to prepare their best students. I tried out for this prize against students of great attainments, and to my delight I managed to win. Now, I note that more than half of all the famous expatriate Hungarian scientists, and almost all the well-known ones in the United States have won this prize. I think that this kind of contest is vital to our educational system, and I would like to see more such contests encouraged here in the United States and in other countries. After the liberation of Hungary from the Nazis in 1945, the system of contests was greatly enlarged. The Kfirsch~k competition attracts around 500 contestants every autumn. The top 10 contestants are admitted to the university without an admission exam. For seventh- and eighth-graders there is a special 3-session competition. (If they want to, they may also enter the competition for older students.) For first- and second-year high-school students, there is the "D~niel Arany" competition. There are special competitions at teacher-training institutes. Apart from all these prize competitions, the Bolyai Society is aware that some mathematically talented youngsters do not do well under test conditions. Publication in K6zdpiskolaiMatematikai Lapokis another path to recognition. In addition to the problem section, it contains papers by students and young researchers. Erd6s told us, "I did not do terribly well at these competitions," yet a few years later his discoveries in number theory were internationally recognized. At lower age levels, a rich variety of extracurricular activities are offered. For elementary pupils, there is the "Young Mathematicians Friendship Circle," part of the Society for the Popularization of Science. For highschool students, the Mathematical Society organizes monthly "High School Mathematical Afternoons," and for the best (around 60 of them), the "Youth Mathematical Circle." The "Circle" holds a national meeting at Christmas and at Easter. The highest level in the contest hierarchy is the "Mikl6s Schweitzer Memorial Mathematical Competition." This is open to both university and high-school students. It consists of 10 or 12 "very hard" problems, which may be worked at home.

"The Schweitzer competition is an important event in our mathematical life. The problems are discussed for days. It is accepted that those w h o win a prize, or whose results in the competition are published, have proved their wide knowledge of mathematics and their ability to do research. The award ceremony is not just a handing out of prizes. It is a regular scientific session of the Bolyai Society. All the problems are solved at this session" [33]. But who was Schweitzer? Here are some sentences from Commemoration [72], a lecture P~il Tur~in gave in March 1949 to the Bolyai Mathematical Society, in memory of Hungarian mathematicians lost in the war and in the Holocaust: "Mikl6s Schweitzer g r a d u a t e d from s e c o n d a r y school in 1941, and in the same year w o n second prize in the Lor~nd E6tv6s mathematics competition. In 1945, on January 28, near the Cog Railway, he received a German bullet in his body, just a few days before the liberation he so longed for. At that moment he knew that his greatest desire, to be a full-time university student, would never come true. He was granted only a short time to live--a stormy, uncertain time--but he availed of it well." Then Turin goes on for three pages, presenting Schweitzer's discoveries in classical analysis. The Cog Railway is in Budapest. It carries people up and d o w n Freedom Hill.

Hungarian Specialties Hungarian mathematics included many of the major trends and specialties of the twentieth century. But three fields have been characteristically Hungarian: classical analysis in the style of Lip6t Fej6r; linear functional analysis in the style of Frigyes Riesz (1880-1956); and discrete mathematics in the style of P~il Erd6s and P~il Tur~in. Fej6r and Riesz were born in 1880. Each was famous for many important discoveries, and even more for an elegant style, a knack for using simple, familiar tools to obtain far-reaching, unexpected results. Fej6r was b o m in the provincial town of P6cs. His father, Samuel Weisz, was a shopkeeper. (In Hungarian, "white" is "feh6r." "Fej6r" is an archaic spelling.) The family had deep roots in P6cs; Fej6r's maternal great-grandfather, Dr. Samuel Nachod, received his medical degree in 1809. In high school, Lip6t Fej~r became a faithful worker of the problems in KOz~iskolai Matematikai Lapok. It is reported that L~tszl6 R~icz, a secondary school teacher w h o led a problem study group in Budapest, often opened his session by saying, "Lip6t Weisz has again sent in a beautiful solution." [This same R~icz later identified J~inos N e u m a n n (1903-1957) as an outstanding mathematical talent!] In 1897, Fej6r w o n second prize in the EOtvOs competiTHE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993

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tion. Then he studied at the Polytechnic University in Budapest. K6nig, Kfirsch~ik, and E6tv6s were among his teachers. In December 1900, while a fourth-year student, he published his most famous work. This was the use of Cesaro sums (averages of partial sums) to sum the Fourier series of functions which are continuous but not smooth. This method permits one to solve Dirichlet's problem in a disc for arbitrary continuous boundary data. (The use of ordinary partial sums can fail if the boundary data are not piecewise smooth.) This result of Fej6r's is still important wherever Fourier analysis is practiced. It was the core of his Ph.D. thesis. Fourier analysis and summation of series continued as his lifelong interests. For the next 5 years, Fej6r did not find a permanent, full-time job. Among the odd jobs he picked up was one in an observatory, watching for meteors. In 1905, Poincar6 came to Budapest to accept the first Bolyai prize. W h e n he got off the train he was greeted by high-ranking ministers and secretaries (possibly because he was a cousin of Raymond Poincar6, the politician who later became President and four times Premier of the Third Republic). According to the stillcurrent story, he looked around and asked, "Where is Fej6r?" The ministers and secretaries looked at each other and said, "Who is Fej6r?" Said PoincarG "Fej6r is the greatest Hungarian mathematician, one of the world's greatest mathematicians." Within a year, Fej6r was a professor in Kolozsv~ir, in the region of Transylvania. Five years later, mainly by Lor~nd E6tv6s's intervention, he was offered a chair at the University of Budapest. Our interviewee Agnes Berger was one of Fej6r's students. RH: Can you describe Fej6r's teaching? Berger: Fej4r gave very short, very beautiful lectures. They lasted less than an hour. You sat there for a long time before he came. When he came in, he would be in a sort of frenzy. He was very ugly-looking when you first examined him, but he had a very lively face with a lot of expression and grimaces. The lecture was thought out in very great detail, with a dramatic denouement. It was a show. RH: What did you work on? Berger: Interpolation. Tur~in was in fact my real advisor. The way a professor was expected to behave there was very different from the way it is here. I was greatly amazed when I saw that in America a professor would sit down with a graduate student. Nothing like that ever happened in Budapest. You would say to the professor, "I'm interested in this or that." And then eventually you would come back and show him what you did. There was none of the hand-holding that goes on here. I know people here who see their students every week! Have you ever heard of such a thing? Well, I did have Tur~in, who acted for me like an advisor. I don't think of Fej6r as a college teacher. There was only one Fej6r in all of Hungary. And in Szeged there was Riesz. Only two in the whole country. That is a very exalted position. 18

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P~il Tur~in wrote: "A coherent mathematical school in Hungary was created first by Fej6r" [55]. George P61ya said, "Almost everybody of my age group was attracted to mathematics by Fej6r." Besides P61ya, Fej6r's students included Marcel Riesz, Ott6 Sz~isz, Jen6 Egerv~ry, Mih~ily Fekete (1886-1957), Ferenc Luk~cs, G~ibor Szeg6, Simon Sidon, later P~il Csillag (18961944), and still later P~I Erd6s and P~il Tur~in. "Fej6r would sit in a Budapest cafe with his students and solve interesting problems in mathematics and tell them stories about mathematicians he had known. A whole culture developed around this man. His lectures were considered the experience of a lifetime, but his influence outside the classroom was even more significant" [2]. Of course, this brilliant career was not without its shadows. "Naturally, World War I had an impact on him, to which a serious illness added in 1916. The effect of counterrevolutionary times was shown by a three year gap in the list of his papers. He never did overcome the effect of those times, as could be perceived again and again from his hints" [55]. Tur~in's reference to "those times" is clear to Hungarians who lived through them. He means, the "white terror," the early years under Horthy, following the suppression of the Hungarian Soviets. At some time between the two wars, Fej6r was visited in his office at the University of Budapest by a professor seeking Fej6r's assistance in some academic matter. After polite conversation, to be sure Fej6r remembered to do whatever service he wanted, the visitor pressed into Fej6r's hand his "professional card," and left. Presumably, he had forgotten that on the reverse side of the card he had written a reminder to himself: "Go see the Jew." Fej6r kept the card, and showed it to John Horv~th, our informant. It is reported that for some reason Fej6r was not on the best of terms with B61a Ker6kj~irt6 (1898-1946), the topologist who, with Frigyes Riesz and Alfr6d Haar, dominated the mathematical scene at Szeged until he moved to Budapest in the late 1930s. Presumably, it was after some unsatisfactory encounter with Ker6kj~irt6 that Fej6r produced his still remembered cutting remark, "What Ker6kj~irt6 says is only topologically equivalent to the truth." In 1927, due to the political climate of the time, Fej6r did not get enough votes to enter the Hungarian Academy of Science. In 1930, after being elected to societies in G6ttingen and Calcutta, he was finally admitted to the Hungarian Academy. The politics of this period are difficult to grasp today. Horthy accepted the role of Jewish capital in Hungary. He was even on social terms with some upper-class Jews. Nevertheless, he instituted a quota system against Jews seeking to enter a university. No more than 5% of the students could be Jews. As for faculty positions, they became virtually out of the question,

even for someone like Erd6s. The twenties were a time when talented, ambitious Jewish young people in Budapest knew that if they were to achieve what they were capable of, they must leave. Von N e u m a n n went to Berlin, and then to Princeton; P61ya to Zfirich and then to Stanford; Szeg6 to Berlin, K6nigsberg, and then Stanford; von K~irm~n to GOttingen to Aachen and then to Cal Tech; Marcel Riesz to Lund; Mihaly Fekete to Jerusalem; and so on, through Teller, Eugene Wigner, Leo Szil~rd, Arthur Erd61yi, Cornelius L~nczos, and Ott6 Sz~sz (18841952). Fej6r and Riesz, older men with tenured positions, remained in Hungary. Most of these emigr6s left in the 1920s, before the Nazi onslaught. They had time to move in an orderly way, without disrupting their careers or their creativity. In 1944, Fej6r was pensioned off as an alien element to the nation. Late one December night, the residents in his house on T~itra Street were lined up by Arrow Cross " l a d s , " to be marched to the bank of the Danube. They were saved by the phone call of a brave officer. Other Budapest Jews did meet death from a gunshot there by the river bank. After the liberation, Fej6r was found in an emergency hospital on T~tra Street "under hardly describable circumstances." But with the end of the war he again received honors, both from Hungary and abroad. Erd6s reports that in his later years Fej6r was no longer the bubbling, convivial wit of his youth. "He once told Turin, 'I feel I was burned out by thirty.' He still did very good things, but he felt that he didn't have any significant new ideas. When he was 60 he had a prostate operation, and after that he didn't do very much. He kept on an even keel for 15 or 16 years more, and then he became senile. It was very sad. He knew he was senile, and he would say things like, 'Since I became a complete idiot . . . . ' He was happy when he didn't think about it. He continued to recognize my mother and me. In the hospital he was well cared for, till he died of a stroke in 1959."

Frigyes Riesz The other major figure in Hungarian mathematics between the two wars was Frigyes Riesz. His younger brother Marcel was also a famous mathematician, but he lived most of his life away from Hungary. The Riesz brothers were born in the town of Gy6r, where their father, Ign~cz, was a physician. In 1911, Marcel received an invitation from Gosta Mittag-Leffier to give three lectures in Stockholm. He stayed on and became one of Sweden's most influential mathematicians, holding a chair at Lund from 1926 until 1952 and again from 1962 to 1969. Two of his most famous pupils were Lars G~rding and Lars H6rmander. For most of his life, Frigyes was professor at Szeged,

Frigyes Riesz in 1925 (from Riesz's Collected Works, Ak~idemiai Kiad6, Budapest, 1960). a city about 100 miles from Budapest, near the southern border with Yugoslavia. Mainly because of his presence, the University of Szeged became a recognized center of mathematical research. He was known to post-war students of my generation for his great book, Functional Analysis [44], co-authored with his famous student and colleague, B~la Sz6kefalvi-Nagy. The first part of their book is modern real analysis, and the second part is linear operators. Both parts are written with a truly intoxicating elegance. The basic principle is, "Much with little." Results both general and precise, using elementary, concrete tools-trigonometry, plane geometry, first-semester calculus--the true Hungarian style. Ray (Edgar R.) Lorch spent the year 1934 in Szeged working with Riesz. We are indebted to him for an account [26] of how this book came to be. Riesz was a dangerous man with whom to collaborate in writing a paper or a book. He was constantly having new ideas on how to proceed, and the latest brain-child was the favorite. This would lead to disconcerting results for the collaborator, who was perpetually out of step. An example was told me by Tibor Rad6, his ex-assistant. During the academic year, Riesz would lecture on measure theory and functional analysis. Rad6 would take copious notes. When summer arrived, Riesz would depart for a cooler spot (Gy6r). Rad6 would sweat it out for three months, writing THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993

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up at Riesz's request all the material, to be in publishable form in the fall. At the end of September Riesz would put in his first day at the Institute, and Rad6 would come to the library to greet his superior, proudly carrying a stack of eight hundred pages, which he placed in Riesz' lap with great satisfaction. Riesz glanced at the bundle, recognized what it was, and raised his eyes with a mixture of kindness and thankfulness, and at the same time with a spark of merriment, as if he had pulled off a fast one. "Oh, very good, very good. Yes, this is very nice, really nice. But let me tell you. During the summer I had an idea. We will do it all another way. You will see as I give the course. You will like it." This took place many years in a row. The book was not written until Riesz, probably under the pressure of advancing age, wrote the book in collaboration with B61a Sz6kefalvi-Nagy some 18 years later. As we all know, the book, Lemons d'Analyse Fonctionnelle, was an international best-seller for decades. Frigyes did his university studies at the Polytechnic in Zfirich and at the University of G6ttingen, and t h e n e a r n e d his Ph.D. at Budapest. At GOttingen, he was influenced b y Hilbert and H e r m a n n Minkowski, and at B u d a p e s t b y K 6 n i g a n d Kfirsch~k. H e d i d postdoctoral s t u d y in Paris and G6ttingen a n d taught high school in L 6 c s e ( n o w Levite, in Slovakia) a n d in Budapest. In 1911, h e w a s a p p o i n t e d to the U n i v e r s i t y of Kolozsv~r, w h i c h was f o u n d e d in 1872. It was an important center of scholarship, in some w a y s more progressive t h a n the university at Budapest. In 1920, in accord with the Treaty of Trianon, Transylvania was c e d e d to Romania. The t o w n of Kolozsv~r was ren a m e d Cluj. A n e w university was established in H u n gary, at Szeged. T h e H u n g a r i a n - s p e a k i n g s t u d e n t s a n d faculty of Kolozsv~ir were invited to Szeged. Riesz first w e n t to B u d a p e s t in 1918, and t h e n in 1920 to Szeged, along with Alfr6d Haar, w h o h a d also been a professor at Kolozsv~ir. Lip6t Fej6r h a d g o n e from Kolozsv~ir to B u d a p e s t in 1911. In Szeged, Riesz a n d Haar created the Bolyai Institute, and in 1922 the journal, Acta Scientiarum Mathematicarum, w h i c h quickly attained international standing. His greatest research achievement was the t h e o r y of compact linear operators. O n e m u s t also m e n t i o n the Riesz r e p r e s e n t a t i o n theorem, the re-creation of the Lebesgue integral w i t h o u t use of m e a s u r e theory, a n d the introduction of subharmonic functions as a basic tool in potential theory. H e i n t r o d u c e d the function spaces Lp, H p, a n d C and did the basic work on their linear functionals. H e p r o v e d the ergodic theorem. H e p r o v e d that m o n o t o n e functions are differentiable almost e v e r y w h e r e . The Riesz-Fischer t h e o r e m is a central result a b o u t abstract Hilbert space. It is also an essential tool in p r o v i n g the equivalence b e t w e e n Schr6dinger's w a v e mechanics and H e i s e n b e r g ' s matrix mechanics. We quote Istv~in Vincze [63]. As a lecturer Riesz was somewhat unpredictable. He was not always perfectly prepared for the lecture. When that 20

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happened he would ask his assistant, L~szl6 Kalm~r, for help. But Kalm~r wasn't always available. [L~szl6 Kalm~r (1900-1976), like Riesz, was of Jewish ancestry and Calvinist persuasion. A universal mathematician, he was remembered by many as also a superb teacher. R.H.] Nevertheless, we found Riesz a first-class interpreter of science. In his lectures everything appeared naturally in historical perspective. That was highly instructive. When he was not well prepared, he often spent time on very interesting digressions. Once he gave a brilliant explanation of why scientific work is easy. "Everyone has ideas, both right ideas and wrong ideas," he said. "Scientific work consists merely of separating them." Lip6t Fej6r was born only three weeks after Frigyes Riesz (on February 9, 1880; Riesz was born on January 22). There was constant teasing between them. For instance, Fej~r would claim that he actually was older than Riesz, because Riesz was born a month prematurely. Riesz loved a quiet, balanced life. He liked order. He was jovial, even a bit aristocratic. Much of his social life took place in a few fashionable rowing and fencing clubs, where empty-headed "notables" from the city and the military could also be found. He belonged to the most exclusive rowing club in Szeged, and would go there from early spring to late autumn. In the evening he would go to the fencing club and play bridge. He backed L~szl6 Kalm~r very strongly, and hoped Kalm~r would become an outstanding mathematician (which he did). But he expected Kalm~r to remain a bachelor and devote all his life to science. (As Riesz did himself, and as also did Marcel Riesz, Alfr6d Haar, Lip6t Fej6r, D6nes K0nig, and P~l Erd6s.) However, Kalm~r did get married. This made Riesz lose his temper to some extent. For a while he was nervous and impatient to Kalm6r. Then he calmed down. Kalm~r's wife was also an able mathematician, and Riesz liked her, as all of us did. Riesz could see that Kalm~r's scientific goals had not been hurt by marriage. When reading a mathematics journal, he sometimes would heave a sigh: "At last he also understands it." (Meaning, the author at last understands what Riesz and others discovered earlier.) Once Riesz said that a good mathematics book while of course proving all the theorems--should be more than just a sequence of theorems and proofs. It should discuss the significance of the theorems, clarify them from different viewpoints, explain their connections to other parts of mathematics. Fortunately, Riesz did not suffer any injury or imprisonment during the war. Some of his fellow faculty members petitioned to the government that he be exempted from the deportation of the Jews which took place starting in 1943. On advice of friends, he went to Budapest early in 1944. While deportation of the Jews was being enforced in the provinces, he was in Budapest. He returned to Szeged the following summer, and on October 11 Szeged was lucky enough to fall, almost without combat, into the hands of the Soviet Army. (Budapest was not to be so fortunate.) Soviet troops had crossed the Tisza River above and below Szeged and encircled it. So the Germans abandoned Szeged and blew up its bridges. Their Hungarian allies were stranded on the east side of the river. A few years later, a decade-long desire of Riesz was fulfilled: to hold a chair at the University of Budapest. In Budapest Riesz lived a quiet, contented life. He was not completely satisfied with his new social standing, which was much different from what he had enjoyed between the two World Wars. But the changes did not disturb him too much. His new sport became swimming in Gell4rt

Bath or in Palatinus Bath on Marguerite Island. He liked to read crime stories, and smoke cigars occasionally. He did not have many personal students. Edgar R. Lorch, B61a Sz6kefalvi-Nagy, Tibor Rad6, and Alfr6d R6nyi (1921-1970) all became well known. He never refused anyone who came to him for help, but such a thing rarely happened. Nevertheless, he taught every mathematician in the world. Even today, all mathematicians learn from his elegant demonstrations and penetrating ideas. In addition to Riesz, Haar, Sz6kefalvi-Nagy, a n d Kalm~r, two o t h e r mathematicians w h o m w e have alr e a d y m e n t i o n e d p l a y e d important parts at Szeged: Ker6kj~rt6 a n d Rad6. Ker6kj~rt6 was a topologist. Rad6 was an analyst, best k n o w n for his research on surface area. H e was an early mathematical emigrant to the United States. H e became a professor at Ohio State in 1931. In 1932, h e published an article in the American Mathematical Monthly [37] on the E6tv6s competition in H u n g a r y . An anecdote a b o u t the Riesz brothers is told b y both Sz6kefalvi-Nagy a n d John Horv~th. (Horv~th was a long-time friend a n d colleague of Marcel Riesz.) It s e e m s that Marcel o n c e s u b m i t t e d a p a p e r to the Szeged Acta, w h e r e Frigyes was f o u n d e r a n d editor. It Was certainly a g o o d paper, but Frigyes w r o t e to his brother, "Marcel, y o u have written also better things." To be fair, Marcel did publish in the Szeged Acta. In Volumes I and II, 1921-1923, he had four papers. As a n e w journal, Acta m a y have b e e n actively seeking papers in those years. Since these papers of Marcel Riesz are o n Fourier series, h e probably h a d written t h e m years before, while still in H u n g a r y and p e r h a p s u n d e r Fej6r's influence. H e r e is a n o t h e r story Horv~th h e a r d from Marcel Riesz. W h e n Hilbert w r o t e his p a p e r o n the integralequation solution to Dirichlet's problem, he v e r y m u c h w a n t e d F r e d h o l m to read it. But F r e d h o l m n e v e r read it. Then, w h e n Frigyes Riesz wrote his papers, he v e r y m u c h w a n t e d Hilbert to read them. But Hilbert n e v e r read them. A n d finally, w h e n Marcel w r o t e his big p a p e r on the hyperbolic C a u c h y problem, all the time h e was working o n it he tried to write it so that his brother w o u l d u n d e r s t a n d it. But Frigyes n e v e r read it. (Unfortunately, this story is all too typical in mathematics.) I h a d always w o n d e r e d w h y the Riesz-Sz6kefalviN a g y Functional Analysis was first published in French. To this question Professor Sz6kefalvi-Nagy was able to give a simple answer.

Sz3kefalvi-Nagy: We published in French because we had written it in French. First of all, both of us knew French. At least, for writing mathematics. Riesz wrote French very well. Both of us did know German too. But it was just after the war, and Germany was very much compromised by fascism. RH: Sure.

Sz6kefalvi-Nagy: Of course we had nothing against the great mathematicians in Germany. RH: I understand. Sz6kefalvi-Nagy:English? Well, the Cold War already began to ....

RH: I see.

Sz6kefalvi-Nagy: Russian? Neither of us knew Russian. RH: So it had to be French. Anyhow, it was translated very quickly into English. Sz6kefalvi-Nagy: It was translated into German, English, Russian, Japanese, even into Chinese. RH: How did Riesz survive the war? How did he get through those years, '44, '45? Sz6kefalvi-Nagy: It wasn't easy. He was very tolerant. He was greatly esteemed and respected by all kinds of people. During the last year of the war, Hungary was occupied by Hitler. On March 19, '44, from one day to the next, German troops were here in Szeged. After this came bombing by the Allies. Szeged was bombed by British bombers from the north and the south. And then the Jewish people lost a whole population. Although Riesz was of Jewish origin, he was not arrested. But it was not safe for him to leave his apartment until October, when the Red Army surrounded Szeged. Of course, Riesz had a number of very good friends who were not Jewish. I visited him every second or third day. He kept himself ready for a journey, he had his rucksack packed. RH: How did he get food? Sz6kefalvi-Nagy: I told you, he had friends. One was a young lady, the daughter of a medical school professor. The janitor at the Institute came every other day to fix his bath. RH: Was there any risk in bringing him food? Sz6kefalvi-Nagy: That problem existed. Not physically, but mentally. It was very bad to know that your existence depended on some crazy people. RH- Was he able to do mathematical work at home? Sz6kefalvi-Nagy: Yes, but lower in intensity. He listened as much as possible to radio broadcasts, and he received plenty of books and periodicals. He could survive, but under pressure of uncertainty. The period from the beginning of April, '44, till the following October was difficult. Then when the Red Army came in, the professors elected him rector of the university. I was in Budapest during the siege. There it was much worse. My wife's mother and father lived in Budapest, and she was afraid of losing contact with them. Fortunately, we didn't lose anyone. But for several months we had to hide in a cellar with many other people, under conditions far from pleasant. RH: How long did the siege go on? Sz~kefalvi-Nagy: From the middle of December, '44, until February 12th. Some fighting continued even after that. RH: How did people keep from starving? Sz6kefalvi-Nagy:That was a problem which everybody had to solve for himself. I thought ahead of time of storing some potatoes and lard. Even during the siege, if you got up just before midnight and went to a certain place early in the morning, before sunrise, and stood and waited till they opened, then perhaps you had some chance to get a kilogram or two of bread. That was possible almost until the last day. But then there was nothing. The shops were neither open nor shut: their entrances had been bombed out. Many people were starving. It was a war! But in a war there are fallen horses. No doctor had inspected them, but nevertheless, in the morning many people tried to take away a kilogram or so of horse meat. It was very difficult. THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993

21

In the middle of March I came back to Szeged by myself. Partly by train, partly by carriage, partly by horse car, partly just walking. I found Szeged taken over by Soviet troops. Peace banners were on the street and the market was open. And in Szeged I found Riesz. He didn't hate people. He had some sharp, critical words, but he never was too hard. RH: Do you think that was partly why he later decided to go to Budapest, because he had bad feelings about some people in Szeged? Sz6kefalvi-Nagy: No. I think it was because he had never married, and he was getting older. There was a third Riesz brother in Budapest, a lawyer, married. Frigyes lived with him. And he had students in Budapest. Horv~th was one. So was J~nos Acz~l, do you know him? He's in Canada, at Waterloo University. And Akos Cs~sz~r, who is now the president of the J~nos Bolyai Mathematical Society, and was president of the ICME Congress in Budapest. Riesz died in a hospital early in 1956, possibly of blood-vessel problems which had troubled him for some time. It is strange that Hungary's greatest mathematician waited for years for an invitation from his country's leading university. Under Horthy, and much more under Hitler, it was not acceptable to have more than one Jew in an academic department at the P~ter P~zm~ny University (as the Lor~nd E~tv6s University of Budapest was called before 1952). Fej~r had been there since 1911. After the war, such rules no longer applied.

Erd6s and T u r i n The two major streams of Hungarian mathematical research which Fej~r and Riesz inspired were joined in the 1930s by a third--"discrete" mathematics, including combinatorics, graph theory, combinatorial set theory, number theory, and universal algebra. This development began with D~nes K6nig, son of Gyula K6nig. Erd6s and Turin attended his seminar. K6nig wrote the first book about graph theory, Theory of Finite and Infinite Graphs, published in 1936, and until 1958 the only text on the subject. It has recently been reprinted in German and translated into English. According to Mathematical Reviews, "It can truly be called a classic of graph t h e o r y . . , a sound introduction to many branches of the subject, and a valuable source book." In the late twenties and early thirties, a small group of friends met to do mathematics, informally and privately, even after they had left the university. They were interested in combinatorics, graph theory, and other kinds of discrete mathematics. Often they met in Budapest's Liget Park, near a certain statue depicting "King B~la's Anonymous Historian." So they called themselves "the A n o n y m o u s Group." None of the group had jobs; there were no jobs in the early 1930s. Like other unemployed Budapest mathematicians, they put some bread on the table by tutoring gymnasium students. (To mention three 22

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993

others, not part of the A n o n y m o u s Group--R6zsa P~ter tutored Peter Lax, and Mih~ly Fekete and G~ibor Szeg6 tutored J~nos N e u m a n n - - k n o w n later in the United States as John yon Neumann.) The leader of the Anonymous Group, by virtue of his originality, productivity, and total devotion to mathematics, was P~l Erd6s. Erd6s w o n his first fame by an elegant new proof of Chebychev's theorem: "Between any number and its double lies at least one prime." He shared with Atle Selberg the glory of finding the first elementary proof of the prime number theorem. He has led in creating the field of mathematics known as "extremal combinatorics" or "extremal graph theory": "Given some function of a finite set system on n elements, what is the largest value the function can take?" Usually one finds the answer, if at all, only asymptotically for large n. Erd6s left Hungary for England in 1934. He says that by that year it was obvious that Hungary was unsafe. Other members of the Group were M~rta Wachsberger, G~za Grunwald (1910-1943), Anna Grfinwald, AndrOs V~zsonyi, Annie Beke, D~nes L~z~r, Esther (Eppie) Klein, Tibor Gallai, Gy6rgy Szekeres, L~iszl6 Alp~r, and P~l Turin. Esther Klein is credited [10] with first bringing to the group (and solving) a problem on finite sets, of the type considered earlier (as they later learned) by Frank Ramsey in England. "Ramsey theory" became one of the recurrent themes in the work of Erd6s, Turin, Szekeres, and others. Szekeres and Klein married and escaped by w a y of Shanghai to Australia. There they have helped inspire Hungarian-type problem competitions. Gallai became famous both as a researcher and as a teacher. Like Erd6s, he was one of our interviewees. Alp~r became a communist, and was imprisoned in France until the end of World War II. Then he returned to Hungary, to be imprisoned again by the Stalinist Hungarian regime. When released from jail for the second time, he for the first time took up mathematics full time. Turin served in a Fascist labor camp during World War II. Before and after that, he had a brilliant research career. At the time of his death in 1976 he had become a major figure in international mathematics. By the inspiration of leaders such as Erd6s, and by its mutually stimulating relationship with computer science, discrete mathematics has become a recognized part of contemporary mathematics. Discrete mathematics is now the largest mathematics research specialty in Hungary. Hungary is preeminent in this field; it exports combinatorialists to leading mathematics departments in the United States.

Finale In this sample of Hungarian mathematics we have had to neglect some important figures. Jen6 H u n y a d i (1838-1889) and Man6 Beke (1862-1946) were pioneers

who should be remembered. Gy6rgy Haj6s (19121970) won fame by proving Minkowski's conjecture on the lattice-packing of unit cubes. Lajos Schlesinger (1864-1933) became a professor at Leipzig, the first Hungarian mathematician to hold a chair at a German university. He wrote two important books on ordinary differential equations [70, 71]. Mathematicians working today on isomonodromy deformations use "Schlesinger transformations." Peter Lax writes, "Some of Schlesinger's results have become of interest recently because of renewed interest in Painlev6 equations in connection with complete integrabillty. His books are in the spirit of Lazarus Fuchs, whose student Schlesinger must have been and whose son-in-law he was." [For a detailed history of pre-twentieth-century mathematics in Hungary see [74].] We cannot attempt a survey of Hungarian mathematicians since World War II, but there are some we must mention. L~iszl6 Fejes-T6th (b. 1915) is famous for studying packings, coverings, and tessellations in two and three dimensions. He has created a mini-school on these topics. R6zsa P6ter (1905-1977), mentioned earlier as Peter Lax's tutor, was a very special figure. Morris and Harkleroad [32] call her "Recursive Function Theory's founding mother." She was the first to propose (at the International Congress in Ziirich in 1932) that recursive functions warrant study for their own sake. She published important papers about them, and the first book on the subject [35]. Her little book Playing with Infinity [36] is a beautiful presentation of modern mathematics for the general reader. She was a poet, and a close friend of L~iszl6 Kalm~ir, w h o m we mentioned above as Frigyes Riesz's lecture assistant. A brief biography of her is in [32]. L~iszl6 R6dei (1900-1980) was an influential algebraist who worked on algebraic number theory and on PelFs equation. One of his favorite types of problem was to find the algebraic structures (groups, semigroups, rings) all of whose proper substructures possess some particular interesting p r o p e r t y . R6dei earned his Ph.D. at Budapest in 1922, and taught high school in Miskolc, Mez6tur, and Budapest until 1940. While still a gymnasium teacher, he was recognized as part of Hungary's mathematics research community. In 1940, he became department head at Szeged, first in geometry, later in algebra and number theory. From 1967 to 1971 he headed the Department of Algebra at the Mathematical Institute of the Hungarian Academy of Sciences. He published nearly 150 research papers and 5 books, including Lacunary Polynomials over Finite Fields and The Theory of Finitely Generated Commutative

Semigroups. "The main feature of the whole career of L~iszl6 R6dei is hard, stout work; in this he can give an example to every mathematician. Maybe this explains w h y he

was able to go on working even beyond 75. Several times he attacked seemingly hopeless problems, running the risk of complete failure. His efforts were often crowned with success only years later. He had several problems on which he worked continuously for about ten years. He often considered problems in a highly original way, contrary to the expectations of all the other mathematicians . . . He always felt his pupils were his collaborators, and he never refused to learn from them" [68]. Finally, it will be our pleasure to describe a memorable giant whose name is not well enough known among American mathematicians---Alfr6d R6nyi.

Alfr4d R4nyi R6nyi was born in Budapest, the son of an engineer "of wide learning," and the grandson, on his mother's side, of Bern~t Alexander, a "most influential" professor of philosophy and aesthetics at Budapest. His uncle was Franz Alexander, the famous psychoanalyst. He attended a humanistic (rather than scientific) gymnasium and maintained a lifelong interest in classical Greece. In 1944, he was brutally dragged to a Fascist labor camp, but he managed to escape w h e n his company was transported to the West. For half a year he hid with false papers [39]. At that time R6nyi's parents were captives in the Budapest ghetto. R6nyi "got hold of a soldier's uniform, walked into the ghetto, and marched his parents o u t . . . It requires familiarity with the circumstances to appreciate the skill and courage needed to perform these feats" [60]. After the Liberation, he received his Ph.D. at Szeged with Frigyes Riesz. He did postgraduate work in Moscow and Leningrad, where he worked with Yu. V. Linnik on the Goldbach conjecture. There he discovered a m e t h o d which, according to Tur~in, is "at present one of the strongest methods of analytical number theory."

If I feel unhappy, I do mathematics to become happy. If I am happy, I do mathematics to keep happy. From 1950 on, he was director of the Mathematical Institute of the Hungarian Academy of Science. In 1952, he founded the chair of probability theory at Lor~ind E6tv6s University in Budapest. Under his leadership, the Mathematics Institute became an international center of research and the heart of Hungarian mathematical life. He had the rare ability to be equally at home in pure and applied mathematics. He was a leading researcher in probability theory. He was also one of the important number theorists of our time, and he contributed to combinatorial analysis, graph theory, integral geometry, and Fourier analysis. He produced THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993 2 3

more than 350 publications, including several books. "Once when a gifted young mathematician told him that his working ability strongly depended on external circumstances, R6nyi answered: 'If I feel unhappy, I do math to become happy. If I am happy, I do math to keep happy' " [57]. Three of his books are accessible to everybody, including, of course, all mathematicians, regardless of their field or their level. The Dialogues on Mathematics [39] is a remarkable work of philosophy and literature. It contains three dialogues---with Socrates, Archimedes, and Galileo. They deal in profound and original ways with fundamental issues in the philosophy of mathematics, yet their light touch and dramatic flair make them readable by anyone. "For Zeus's sake," asks R6nyi's Socrates, "is it not mysterious that one can know more about things which do not exist than about things which do exist?" Socrates not only asks this penetrating question, he answers it. The Letters on Probability [40] contain four warm personal letters from Blaise Pascal to Pierre Fermat, communicating Pascal's enthusiastic opinions and ideas about the origins and foundations of probability theory. The letters are composed in complex sentences, in the literary style of Pascal and Fermat's day, and display easy familiarity with their lives and work. Nevertheless, as R6nyi makes clear in a "Letter to the Reader," the actual author is R6nyi, not Pascal. This jeu d'esprit must be unique in the writings of modern mathematicians. The fourth letter especially will repay any reader interested in the foundations of probability. Here Pascal, who (like R4nyi) holds the frequentist interpretation of probability, reports in novelistic detail a dispute in the salon of Madame d'Aiguillon with his foppish friend "Damien Miton," an upholder of the subjectivist view. The Diary on Information Theory [41], like the two earlier books, is also written "behind a mask." The diary is kept by one "Bonifac Donat," and contains Bonifac's "lecture notes" on five of "Professor R6nyi's" lectures, plus Bonifac's preparation for a talk of his own. The last diary entry says, "The professor doesn't look too well. I hope it's nothing serious." In fact, the professor was not well enough to finish that last chapter. It had to be completed by one of R6nyi's old pupils, Gyula Katona. R6nyi died on 1 February 1970, at the age of only 49. In view of their hardships, it is amazing how Hungarian mathematicians have been able to persist and create, in poverty and unemployment, in labor camps or under siege. We close with an unforgettable quote from P~il Tur~in: It sounds incredible, but it is true. The story goes back to 1940, when I received a letter from my friend George Szekeres in Shanghai. He described an unsuccessful attempt to prove a famous Burnside conjecture (which was disproved later). The failure of his attempt could have been obtained 24

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993

from a special case of Ramsey's theorem, but Ramsey's paper, beyond its mere existence, was then unknown in Hungary. At that time, most of my income came from private tutoting, and I had to teach my pupils at their homes. While traveling between two pupils, I pondered the contents of the letter. My train of thought soon led me to finite forms, and then to the following extremal problem: What is the maximum number of edges in a graph with n vertices, not containing a complete subgraph with k vertices? Though I found the problem definitely interesting, I postponed it, being then mainly interested in problems in analytical number theory. In September 19401 was called for the first time to serve in a labor camp. We were taken to Transylvania to work on building railways. Our main work was carrying railroad ties. It was not very difficult work, but any spectator would have recognized that most of us did it rather awkwardly. I was no exception. Once one of my more expert comrades said so explicitly, even mentioning my name. An officer was standing nearby, watching us work. When he heard my name, he asked the comrade whether I was a mathematician. It turned out that the officer, Joseph Winkler, was an engineer. In his youth he had placed in a mathematical competition; in civilian life he was a proofreader at the print shop where the periodical of the Third Class of the Academy (Mathematical and Natural Sciences) was printed. There he had seen some of my manuscripts. All he could do for me was to assign me to a wood-yard where big logs for railroad building were stored and sorted by thickness. My task was to show incoming groups where to find logs of a desired size. This was not so bad. I was walking outside all day long, in the nice scenery and the unpolluted air. The problems I had worked on in August came back to my mind, but I could not use paper to check my ideas. Then the formal extremal problem occurred to me, and I immediately felt that this was the problem appropriate to my circumstances. I cannot properly describe my feelings during the next few days. The pleasure of dealing with a quite unusual type of problem, the beauty of it, the gradual approach of the solution, and finally the complete solution made these days really ecstatic. The feeling of some intellectual freedom and of being, to a certain extent, spiritually free of oppression only added to this ecstasy. This beautiful memory appeared in Tur~in's "Note of Welcome" in the first issue of the Journal of Graph Theory [58]. When writing it, he was already battling his last illness. He died on 26 September 1976. The Journal's first issue appeared in 1977.

Acknowledgments: Essential financial support was given by the Soros Foundation. John Horv~ith granted an interview, and painstakingly corrected errors in earlier drafts. Peter Ungar shared his reminiscences of Hungarian mathematics. Istwin Vincze spent hours on being interviewed, and let us use his memoirs. B~la Sz6kefalvi-Nagy, Peter Lax, Agnes Berger, Lajos P6sa, Tibor Gallai, and P~il Erd6s all kindly consented to be interviewed. L~iszl6 Sz6kely gave invaluable help as a translator and advisor. L~iszl6 Fuchs gave important i n f o r m a t i o n about L~iszl6 R4dei. G y 6 r g y Csepeli checked for historical errors. Gy6rgy Sz~pe corrected

20. M. Kac, Enigmas of Chance. Harper and Row, New York 1985. 21. J. P. Kahane, Fej~r ~letmi~v~nek jelent6s~ge. Matematikai Lapok 29 (1981), 21-31. In French: Cahiers du S~minaire d'Histoire des Math. 2 (1981), 67-84. 22. J. P. Kahane, La Grande Figure de Georges Pdlya. Proceedings of the Sixth International Congress on Mathematical Education. J~inos Bolyai Mathematical Society, Budapest (1986). 23. L. Kalm~r, Mathematics teaching experiments in Hungary. Problems in the Philosophy of Mathematics, ed. by I. Lakatos, North-Holland Publishing Company, Amsterdam (1967), 233-237. 24. S. Klein, The Effects of Modern Mathematics. Akad~miai Kiad6, Budapest (1987). 25. K6z~piskolai Matematikai Lapok (1984), Nos. 8, 9, 10. 26. E. R. Lorch, Szeged in 1934, Amer. Math. Monthly (to appear). 27. L. M~rton, Biography of L. E6tv6s, Dictionary of Scientific References Biography, Charles Scribner's Sons, New York (1975), 377-380. 1. D. J. Albers and G. L. Alexanderson, Mathematical People, Birkhauser, Boston (1985). Interview with P. Erd6s, 28. S. M~rton, Matematika-t6rt~neti ABC. Tank6nyvkiad6, Budapest (1987). 81-91; interview with P. Halmos, 120-132; interview 29. W. O. McCagg, Jewish Nobles and Geniuses in Modern Hunwith G. P61ya, 246-253. gary, East European Quarterly, Boulder (1972). 2. G. L. Alexanderson, et al., Obituary of George P61ya. 30. M. Mikol~s, Biography of L. Fej~r, Dictionary of Scientific Bull. London Math, Soc. 19 (1987), 559-608. Biography, Charles Scribner's Sons, New York (1975), 3. L. Alp~fr, Egy ember, aki a szdmok vildgdban dl. Besz~Iget~s 561-562. Erd6s Pdl akaddmikussal, Magyar Tudom~ny 3 (1988), 21331. M. Mikol~s, Some historical aspects of the development of 221. mathematical analysis in Hungary, Historia Math. 2 (1975), 4. J. Bolyai, Appendix: The theory of space, Introduction by F. 304-308. IG'trteszi; supplement by B. Sz6n~ssy, Akad6miai Kiad6, 32. E. Morris and L. Harkleroad, R6zsa P~ter: Recursive FuncBudapest (1987). tion Theory's Founding Mother, The Mathematical Intelli5. M. Csikszentmih~ilyi and R. E. Robinson, Culture, time, gencer 12, 1 (1990), 59-61. and the development of talent in Conceptions of Giftedness (R. J. Sternberg and J. E. Davidson, eds., Cambridge 33. I. Pal~isti, A fiatal kutatdk helyzete a Matematikai Kutat6 Int~zetben, Magyar Tudom~iny 5 (1973), 299-312. University Press, Cambridge (1986). 6. P. Erd6s, The Art of Counting. Selected writings. MIT Press, 34. E. Paml~nyi, A History of Hungary. Corvina Press, Budapest (1973). Cambridge (1973). 7. L. Fermi, Illustrious Immigrants. The University of Chi- 35. R. P~ter, Rekursive Funktionen. Akad~miai Kiad6, Budapest (1951). cago Press, Chicago (1968). 8. L. GSrding, Marcel Riesz in Memoriam, Acta Mathematica, 36. R. P~ter, Playing With Infinity. Dover, New York (1976). 124 (1970). See also: M. Riesz, Collected Papers, Springer- 37. T. Rad6, On mathematical life in Hungary, Amer. Math. Monthly 37 (1932), 85-90. Verlag, New York (1988), 1-9. 9. M. Gluck, George Lukdcs and his generation 1900-1918. Har- 38. E. Rapaport, Hungarian Problem Book I and II. Random House, New York (1963). vard University Press, Cambridge (1985). 10. R. L. Graham and J. H. Spencer, Ramsey theory, Scien- 39. A. R~nyi, Dialogues on Mathematics. Holden Day, San Francisco (1967). tific American (July 1990), 112-117. 11. I. Grattan-Guinness, Biography of F. Riesz, Dictionary of 40. A. R~nyi, Letters on Probability. Wayne State University Press, Detroit (1972). Scientific Biography, Charles Scribner's Sons, New York 41. A. R4nyi, A Diary on Information Theory. Akad~miai (1975), 458-460. Kiad6, Budapest (1984). 12. G. Hahisz, Letter to Professor Paul Turin (1978). 13. P. Halmos, Riesz Frigyes munkdssdga, Matematikai Lapok, 42. C. Reid, Hilbert. Springer-Verlag, New York (1970). 43. C. Reid, Courant in G6ttingen and New York. Springer29 (1981), 13-20. Verlag, New York (1976). 14. A. Handler, The Holocaust in Hungary. University of Ala44. F. Riesz and B. Sz6kefalvi-Nagy, Functional Analysis. Unbama Press, University, Ala. (1982). gar, New York (1955). 15. S. J. Heims, John von Neumann and Norbert Wiener. MIT 45. F. Riesz, Oeuvres completes. Acad~mie des Sciences de Press, Cambridge (1980). Hongrie (1960). 16. P. Hoffman, The Man Who Loves Only Numbers. Atlantic 46. F. Riesz, Obituary, Acta Scientiarum Mathematicarum Monthly, November (1987). Szeged 7 (1956). 17. J. Horv~th, Riesz Marcel matematikai munkdssdga. Matematikai Lapok, 26 (1975), 11-37; 28 (1980), 65-100. French 47. P. C. Rosenbloom, Studying under P61ya and Szeg6 at Stanford, The Mathematical InteUigencer 5, 3 (1983). translation: Cahiers du S6minaire d'Histoire des Math. 3 48. G. Szeg6, Collected Papers. Birkh/iuser, Boston (1981). (1982), 83-121; 4 (1988), 1-59. 18. A. E. Ingham, Review of P. Erd6s, On a new method in 49. B. Sz~n~ssy, A magyarorszdgi matematika t6rt~nete. Akad~miai Kiad6, Budapest (1970). elementary number theory which leads to an elementary proof of the prime number theorem. Mathematical Re- 50. B. Sz6kefalvi-Nagy, Riesz Frigyes tudomdnyos mun~ss4fgdnak ismertet~se, Matematikai Lapok 5 (1953), 170-182. views, 595-596. 19. A. C. J~nos, The Politics of Backwardness in Hungary. 51. B. Sz6kefalvi-Nagy, Riesz Frigyes ~lete ~s szem~lyis~ge, Princeton University Press, Princeton (1982). Matematikai Lapok 29 (1981), 1-5. errors of spelling a n d accents. Barna Sz6n~issy of Debrecen sent helpful information and advice about the history of H u n g a r i a n mathematics. C h a n d l e r Davis helped arrange our interview with B61a Sz6kefalviNagy. Erzs~bet Be6thy very kindly helped us with the H u n g a r i a n u m l a u t (short and long). We heartily thank t h e m all. We especially thank Vera S6s, lifelong friend, a n d m e m b e r of the Mathematics Institute of the Hungarian A c a d e m y of Science. W i t h o u t her help this s t u d y would have been impossible. She arranged most of our interviews in H u n g a r y , and let us use historical and biographical articles by P~/1 Turin.

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993 2 5

52. L. Takfics, Chance or Determinism? The Craft of Probabilistic Modelling. Springer-Verlag, New York, 137-149. 53. K. Tandori, Fejdr Lip6t dlete ds munkdssdga, Matematikai Lapok 29 (1981), 7-11. 54. E. Tettamanti, The Teaching of Mathematics in Hungary. National Institute of Education, Budapest (1988). 55. P. Turfin, "Leopold Fej6r's Mathematical Work," lecture to the Hungarian Academy of Sciences, 27 February 1950. 56. P. Tur~in, The Fiftieth Anniversary of Pdl Erd&, Matematikai Lapok 14 (1963), 1-28 (Hungarian). English trans, pp. 1493-1516 of [73]. 57. P. Turfin, The Work ofAlfrdd Rdnyi, Matematikai Lapok 21 (1970), 199-210 (Hungarian). English trans, pp. 21152127 of [73]. 58. P. Tur~in, A note of welcome, J. Graph Theory 1 (1977), 7-9. 59. S. M. Ulam, Adventures of a Mathematician. Charles Scribher's Sons, New York (1983). 60. F. Ulam, Non-mathematical personal reminiscences about Johnny, Proc. Symp. Pure Math. 50 (1990), 9-13. 61. P. Ungar, Personal communication. 10 October 1989. 62. I. Vincze, Az MTA Matematikai Kutat6 Int~zet~nek husz6not eve, Magyar Tudomfiny 2 (1976). 63. I. Vincze, Vallomfisok Szegedr61, Somogyi K6nyvtari miih~ly, 2-3 (1983). 64. I. Vincze, Eml&ezes Riesz Frigyes Professzor l~Irra. Unpublished manuscript.

26

THE MATHEMATICAL 1NTELL1GENCERVOL. 15, NO. 2, 1993

65. T. von Kfirmfin and L. Edson, The Wind and Beyond. Little, Brown, Boston (1967). 66. N. A. Vonneumann, John von Neumann as seen by his brother. Meadowbrook, Pa. (1987). 67. A. A. Wieschenberg, The Birth of the E6tv6s Competition, The College Mathematics Journal 21, 4 (1990), 286-293. 68. L. Mfirki, O. Steinfeld, and J. Sz6p, Short review of the work of Lfiszl6 R6dei, Studia Sci. Math. Hungar. 16 (1981), 3-14. 69. J. Lukfics, Budapest 1900. Weindenfeld and Nicolson, New York (1988). 70. L. Schlesinger, Handbuch der Theorie der linearen Differentialgleichungen, Vols. 1, 2:1, 2:2. Teubner, Leipzig (1895, 1897, 1898). 71. L. Schlesinger, Einfiihrung in die Theorie der Differentialgleichungen mit einer unabh~ingigen Variabeln, 2nd ed. (Sammlung Schubert, Vol. 13). G6schen, Leipzig (1904). 72. P. Turfin, Megeml6kez6s, Matematikai Lapok 1 (1949), 3-15. 73. P. Tur~in, Collected Papers. Akad6miai Kiad6, Budapest (1990). 74. B. Sz~nfissy, History of Mathematics in Hungary until the 20th Century (English trans, by J. Pokoly). SpringerVerlag, New York (1992).

University of New Mexico Albuquerque, NM 87131 USA

Illl

Projective Ranks of Hermitian Symmetric Spaces Amassa Fauntleroy

For many mathematicians, the lure of mathematics began with a gradual recognition of the beauty of Euclidean geometry. It attracts us with its simplicity and clarity. Later, with the discovery of Cartesian coordinates, we see the power and precision of algebra in describfng geometric ideas. Such seeds having been sown, I have found myself deeply enthralled for many years now with algebraic geometry. Recently, however, a yearning for a return to a more concrete geometry overcame me. It was a desire to understand the "geometry" in algebraic geometry which has led to a study of those classical examples which seem to be well understood by the old masters. With a background in algebraic groups, I found the symmetric spaces of Cartan to be a good starting point. Currently, the "geometry" in algebraic geometry is represented most effectively by the theory of complex manifolds, many of the most popular of which are projective algebraic varieties. (For one who thought theorems should be proved in a characteristic-free environment, this represents a significant shift in perspective.) The compact Hermitian symmetric spaces can be represented as homogeneous spaces for reductive algebraic groups (so are objects of study in algebraic geometry) and also as homogeneous spaces for compact Lie groups (so are objects of study in Riemannian geometry). Among these spaces are the complex projective spaces and the Grassmann manifolds which play a fundamental role in the study of algebraic manifolds and the geometry on them. This article is concerned with recent characterizations of the Hermitian symmetric spaces using the curvature of the natural K~ihler metrics on each. In particular, the only ones with everywhere positive biholmor-

phic sectional curvature are the complex projective spaces. A motivating problem for what follows is: Find a similar characterization of Grassmann manifolds in terms of their curvature.

Notions and Notations Recall that a complex manifold, say M, of dimension n is a topological space together with an open covering M = U~V~ and homeomorphisms % from V~ onto an open set in C n. It is required that in the diagram pre-

This p a p e r is a written version of the NAM William Claytor Lecture delivered at the 1991 A n n u a l Meeting of the AMS-MAA-NAM, Janu a r y 1991 in San Francisco, CA. THE MATHEMATICALINTELLIGENCERVOL. 15, NO. 2 9 1993 Springer-VerlagNew York

27

A c o n n e c t i o n is called the metric connection if d(Sl,S2) = ( D s l , s 2 )

(Sl,DS 2)

+

for Sl,S 2 in F(M,T(M)). T h e m e t r i c c o n n e c t i o n is unique.

%

Example 1. Let M = C". T h e n with the usual metric (x,y) = t-~y, the c o n n e c t i o n d e f i n e d by D(O/az,,) = D(0/O~) = 0 is the metric connection. Let 3

o q 21 J

J

and Figure 1.

a

sented in Figure 1 the composition % o ~0~-1 is to be analytic. If TM is the t a n g e n t b u n d l e of M, t h e n each tangent space TmM, m E M, has a natural structure of c o m p l e x v e c t o r space. O f course, M is a real 2ndimensional manifold. A n Hermitian metric ( , ) on M is a s m o o t h l y assigned Hermitian metric (,)m o n TmM for all m in M. The pair (M, (,)) has the structure of a Riemannian manifold. Let Zl . . . . . z , be complex coordinates in a neighb o r h o o d of s o m e p o i n t m E M. T h e n z I . . . . . z,, zl . . . . . ~, can stand in for real coordinates in this n e i g h b o r h o o d in the sense that xi = (zi + zi)/2 and Yi = (zi - ~i)/2i are the usual cartesian coordinates on the u n d e r l y i n g ~2,. The complexified t a n g e n t space at m has a basis over C,

J

Then

D.Sl =

3

3Zl " -

3 9

9

9

3 9

" 3zn 3Zl

Ds,(s2) = (

J

9 daj | ~jzj (s2) +

(i) D(s 1 + s2) = Ds 1 + Ds2,

(ii) D(f . s) = d / | s + fDs for all s, s 1, s 2 in F(M,T(M)) and all C~-functions f on M. 28 THE MATHEMATICALINTELLIGENCERVOL. 15, NO. 2, 1993

0

+ E bj(s2) 0F J

0

E s2(aj) + E s2(bj)0F' J

satisfying

dbj | ~z/ (s2)

3

defined as the basis dual to

D : r(M,T(M)) ~ r(M,T(M)* | T(M))

0

E aj(s2) J

3Zn

Let ~ , ~ g~,~ dz~ | d-~ r e p r e s e n t the H e r m i t i a n metric in this n e i g h b o r h o o d . The form 1~ = (V'-L-1/2) E~,~ g~,~ dz~,/~ d ~ is called the f u n d a m e n t a l 2-form. The complex manifold (M, (,)) is said to be K~hler if the exterior derivative vanishes, that is, if df~ = 0. In this case, 1~ is called the Kdhler form. For a real or complex vector b u n d l e E o n M, d e n o t e b y F(M,E) the space of C~176 of M, a n d let E* d e n o t e the dual of E (with respect to C if E is complex). Consider T(M) a n d E = T(M)* | T(M) as real bundles o n M. T h e n a connection o n M is a m a p

D

so that

O

cl~..

+ !

1 . . . 1 - - 1

dz,, d-~1. . . . .

aj

J

--

dz 1. . . . .

D !

--

-

J

J

where 3 J

J

Thus, the connection D enables us to differentiate the vector field s 1 in the direction of s 2.

Example 2. Let G be a c o n n e c t e d Lie group. D e n o t e b y B(X,Y) the Killing form of the Lie algebra g = TeG. If G is compact a n d semisimple, t h e n Q = - B is a positive definite symmetric form o n g a n d can, thus, be u s e d to define a metric on G. There is a u n i q u e connection D o n G w h i c h is invariant u n d e r left a n d right translations a n d u n d e r the m a p G ~ G-1. For left-invariant vector fields X,Y, the connection is d e t e r m i n e d b y the formula

DY(X) = V2[X,Y], w h e r e [X,Y] is the bracket in g.

Next we define the curvature operator. The connection D can be extended to a map P

P+I

D:F(M,(AT(M)* | T(M))) ~ F(M,( A T(M)*) | T(M)) by the formula D(o~ |

V t of ~(t) and a vector field X on V t such that XIv,n~ = X on ~. This holds in particular if X = T, the unit tangent vector to % For a C~-function f in the neighborhood Vt,

d

S) = d00 | S + (-1)Po~ 9 DS.

T(t) = dt f(~l(t) . . . . .

~n(t)) = ~

The curvature operator is the map O = D2. For X , Y , Z in F(M,T(M)), define R(X,Y)(Z) = O(Z)(X,Y), where if O(Z) = ~ o~ | Z~, ~% a 2-form, then O(Z)(X,Y) = ~ ~%(X,Y)Z~. This is the usual curvature operator. Thus, if D corresponds to differentiating sections, then the curvature operator corresponds to a second derivative. If X,Y in TmM are orthonormal relative to the metric (,), the sectional curvature at the plane spanned by X and Y is

=

Of

~/j(t) "~jXj(t) ~j(t)

.

The parametrized curve ~/is called a geodesic when VTT = 0. [Here VxY = DX(Y).] In case ~7 is the fiat connection, that is, D(O/Oxj) = O, we see that

0

(R(X,Y)X,Y). We say R is positive (respectively, non-negative) if the sectional curvature is positive (respectively, nonnegative) for all choices of X and Y. The sectional curvature determines O. Returning to the case of complex manifolds M, when the plane P = span (X,Y) is invariant under the complex structure J in TraM, then

R(P) = (R(X,Y)X,Y) is called the holomorphic sectional curvature at the plane P. In this case, X,JX is an orthonormal basis for P, so R(P) = (R(X,JX)X,JX). If P and P' are two J-invariant planes (representing therefore a 2-dimensional complex subspace), then

R(P,P') = (R(X,JX)Y,JY) (for X E P, Y E P' unit vectors) is called the holomorphic bisectional curvature. The last notion I want to describe is that of a geodesic segment. Let (M,~) be any Riemannian manifold with metric connection D. If ~/: (a,b) --) M is a parametrized curve in M and X is a vector field along M (see Fig. 2), then for each t E (a,b) there is a neighborhood

Xt

Figure 2.

So in the flat case, VTT = 0 says @j = 0 for each j and ~l/(t) = a/t + bj , aj , bi constants. Hence, a geodesic is a straight line segment. Let N C M be a submanifold. We say N is totally geodesic in M if every geodesic in N is also a geodesic in M. For example, in Rn with the Euclidean metric and fiat connection, the r-fiats are the totally geodesic submanifolds.

Hermitian Symmetric Spaces Let us fix a K/ihler manifold M. The manifold M is Hermitian symmetric if for each point m E M, there is an involutive holomorphic isometry Sm of M having the point m as an isolated fixed point. The universal covering space of a compact Hermitian symmetric space is again Hermitian symmetric and may be decomposed into a product C k x N , where Ck and N are Hermitian symmetric and N is compact. The manifold N may be written as a product N 1 x 9 9 9 x N r with each Nj, 1 j ~ r, irreducible. An irreducible compact Hermitian symmetric space N has the form N = G/K, where G is a compact simple Lie group with trivial center and K is a maximal compact proper subgroup of G having nondiscrete center. These may be classified into four infinite series and two exceptional types as follows (here we use the universal covering group G for G): AIII(n): G = SU(n,C), K = S [U(p,C) • U(q,C)], p + q = n. This is the set of p-dimensional subspaces of C n or the Grassmann manifold G(p,n). BDI(m): G = SO(m + 2), K = SO(m) • SO(2). This is the complex quadric in P~ + 1 given by Zo +

+

..-

THE MATHEMATICAL

2

+ Zm§

= O.

INTELLIGENCER VOL. 15, NO. 2, 1993

29

CI(n): G = Sp(n,C), K = U(n). H e r e G/K is the subspace of G(n,2n) consisting of n-dimensional subspaces W of C 2" which are annihilated by the skew-symmetric

Let

forrn J, = (0_i. ~)' thatis, v T j , w = O f o r a l l v , w i n W.

k =

DIII(n): G = SO(2n),/< = U(n), G/K is the set of n-dimensional subspaces W of C 2" which are annihilated

m =

b y the symmetric form In 0 " EIIh G is a c o m p a c t form of the exceptional Lie g r o u p of type E6, K is a maximal compact s u b g r o u p corres p o n d i n g to a h o m o m o r p h i s m of Spin(lO) x SO(2) into E6. EVIl: G is a compact form of the exceptional Lie g r o u p of type E7, K is a maximal compact s u b g r o u p corres p o n d i n g to a h o m o m o r p h i s m of E6 x SO(2) into E 7. The series AIII(n) contains the projective spaces P~ = G(r,r + 1). In a sense, G r a s s m a n n i a n s are the next " m o s t linear m a n i f o l d s " to projective spaces. In the case of h o m o g e n e o u s spaces M = G/K, the g e o m e t r y can be described effectively using the Lie algebra ~ of G. W h e n G is a compact semisimple Lie group, and K is a maximal compact s u b g r o u p , then the Lie algebra ~ of G can be d e c o m p o s e d :

Tr A + Tr B =

{(0 :)z _tz

=

1/411 [X, YImll2

g = so(p + 2),

m =

w h e r e IlZll2 = - B(X,X) and Z = Zm + Zk is the d e c o m p o s i t i o n of Z E g into c o m p o n e n t s in m a n d k. . W h e n M is s y m m e t r i c a n d 0 is the involution of i n d u c e d b y the s y m m e t r y at 0, t h e n k is the + 1eigenspace o f 0 a n d m is t h e - 1 - e i g e n space. In the H e r m i t i a n symmetric case, the complex structure is d e t e r m i n e d b y an e n d o m o r p h i s m of w h o s e square is O. Let us look at the above data in the case of the four classical series of Hermitian symmetric spaces. 1. AIII(n): Let p + q = n and g = su(p + q)be the space of n • n skew-Hermitian matrices. The involution

O(X) = Ip,q XIp,q, w h e r e Ip,q = 30

THE MATHEMATICAL INTELLIGENCERVOL. 15, NO. 2, 1993

o)

Iq "

.

0(X) = Ip,qXIp,q,

_tX

a real p x q matrix .

If

I Xl,yl X

+ 3/411[X,Y]kll 2,

}

2. BDI(n)

=

I Xn, y , t h e n the complex structure is given by

x1 + iyl ]

(i) The geodesics in G/K t h r o u g h 0 = {K} are ~/x(t) = exp(tX) 9 0, w h e r e X E m . (ii) If X,Y E rn are orthonormal, the c o r r e s p o n d i n g sectional c u r v a t u r e of G/K is given b y

R({X,Y})

a p x q complex matrix

The complex structure o n M comes via its identi-

= mGk, w h e r e k is the Lie algebra of K. If B(X,Y) d e n o t e s the Killing form o n ~, that is, B(X,Y) = Trace(adX o adY), t h e n B is negative definite, so g = - B induces a metric on G a n d in a natural w a y on m, which is taken to be the orthogonal c o m p l e m e n t of k in ~. T h e n

'

X~-~

I

Xn + iyn

3. CI(n)

g = sp(n),

O(X) = JnXJn 1,

w h e r e ], =

-I,

"

k = sp(n) n so(2n) w h i c h is isomorphic to u(n), m =

Z2 - Z l ] tZ2 symmetric, pure imaginary j "

Let S = ]~/2. T h e n the complex structure is given b y X ~ SXS -1. 4. DIII(n)

g = so(2n),

O(X) = JnXJ21,

k = so(2n) n sp(n) ~- u(n), X2

x,) X, in

As in series 3, the complex structure is given by X--->

SXS-1, S = jdl2. In the case of series 3 and 4, one may verify that M is stable under the involution of series 1. This implies [A] that CI(n) and DIII(n) are totally geodesic subspaces of G(n,2n).

S o m e Motivations The classical uniformization theorem from the theory of Riemann surfaces has served as a model to many g e o m e t e r s for a desired classification of higherdimensional complex manifolds. In brief, it asserts that the simply connected covering spaces of compact Riemann surfaces are three in number and are classified according to whether the curvature is positive ($2), negative (unit disk), or zero (R2). Of course, one must choose a suitable metric, but there is a standard way of doing so. In the early 1970s, Goldberg and Kobayashi [4], Bishop and Goldberg [2], and Ochiai [8], as well as several other mathematicians, began to make some progress on the problem of classifying positively curved manifolds. Around the same time, Hartshorne was led to make a related conjecture concerning "positive" vector bundles on algebraic varieties. This activi t y led to two outstanding conjectures.

Conjecture 1 (Frankel's Conjecture). If M is a compact K/ihler manifold with everywhere positive holomorphic bisectional curvature, then M is biholomorphic to a complex projective space.

Conjecture 2 (Hartshorne's Conjecture). If M is a smooth projective algebraic variety w h o s e tangent bundle is ample, then M is biregular to a complex projective space. It was known that when M is a projective manifold in P~, the assumption that the curvature is positive implies the ampleness of the tangent bundle. Thus, Hartshorne's conjecture implies the Frankel conjecture. In 1980, both conjectures were settled affirmatively. Mori [7] proved Hartshorne's conjecture is true, and, independently, Siu and Yau [9] proved that the Frankel conjecture holds. Mori's proof involved two main ideas: A. The ampleness of T(M) implies the existence of a large family of projective lines in M, parametrized by a projective space of dimension d i m M - 1. B. Using algebraic geometry and deformation theory, the family of lines "fill u p " M in a linear way. Essentially, every point in M may be joined to a fixed point 0 in M by a projective line. Naturally, the question of generalizations arose. What about R (P,P') everywhere non-negative? In the early 1980s, Howard, Smyth, and Wu [5, 10] in their

attempts at generalizations were led to make the following conjecture: (IV,). Let M be an n-dimensional compact K/ihler manifold with non-negative holomorphic bisectional curvature and let M be its universal covering space. Then M is biholomorphic to C k x N, where N is a compact Hermitian symmetric space. In 1986, Mok [6] settled (W,) affirmatively. His proof is, in spirit, close to Mori's proof of the Hartshorne conjecture and uses in a crucial w a y Mori's theorem on the existence of a rational curve in M. Earlier, in 198283, this author and A. Babakhanian began to look for a characterization of the Grassmann manifolds along the lines of the Mori-Hartshorne characterization of P~. Our attempts were not successful, though later Babakhanian and Hironaka [1] gave a characterization in terms of the generators of the intersection theory of Grassmannians. Much later, I began to look at the curvature condition; hoping for a more differential geometric characterization. This leads to the results which I describe next.

H o w Close to P" Is G/K? Let M be a compact irreducible Herrnitian symmetric space and write M = G/K, with G the group of holomorphic isometries of M, and K the stability group of the point 0 E M. Write ~ = rn + k with m identified with the tangent space ToM and k identified with the Lie algebra of K. If S C m is a subspace such that IX, Y] = 0 for all X,Y in S, then there corresponds to S a totally geodesic subspace N(S) C M such that the induced sectional curvature on N(S) is identically zero. The dimension of the largest such subspace S is called the rank of M. The rank of M is always positive, so it follows that there are submanifolds of M containing 0 which are flat in the sense of having curvature tensor identically zero. For P', this rank is 1 and the corresponding submanifolds are circles. We consider now the question: What are the maximal closed complex subrnanifolds of M containing 0 having everywhere positive biholomorphic sectional curvature? By the theorems of Mori and Siu and Yau, these submanifolds will be biholomorphic to complex projective spaces. We assume for technical simplicity that such manifolds N C M are totally geodesic. Then, as in the flat case, the tangent space n = ToN C m = ToM has a special property:

[[n,n],n] C_ n. Such subspaces are called Lie triple systems, and, in such a case, n + In,n] is a Lie subalgebra of ~ and In, n] C k. The corresponding Lie group, up to a possible central factor, will be SU(r,C,) for some r. Thus, the existence of such an N gives rise to a representation

SU(r,C,) --> G. THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993 3 1

For the classical series of compact Hermitian symmetric spaces, the description of these maximal totally geodesic complex projective subspaces can be given in terms of linear algebra. Here are the results. For AIII(p + q = n), we assume 0 < n ~ 2p and take for M the G r a s s m a n n manifold G(p,n). Fix a (p + 1)dimensional subspace W of C". Then the set G(p,W) of all p-dimensional subspaces of W is a totally geodesic subspace of G(p,n). Because it is also isomorphic to G(1,W), it is biholomorphic to a complex projective space. For n > 4, these are the maximal ones. In the case BDI, M = Qm with equation

Z~ + Z~ + . . .

2 Zm +l

= 0,

the linear subspace of pm+ 1 defined by

,,:

1

is a maximal totally geodesic subspace. For the spaces of type CIII(n), consider the totally isotropic subspaces W1 9Z 1 . . . . . W2"

Zn =

Z 1 q- Z n + 1 . . . . .

0, Z n q- Z 2 n = O.

Then Wl N W 2 = (0). If L is any (n - 1)-dimensional subspace of W 1, t h e n L • f3 W2 is 1-dimensional and L + (L" A W2) is totally isotropic in C". Thus, the assignment: L ~ L + (L • N W2) exhibits an injective holomorphic m a p p i n g ~ : G(n - 1,W1) = p , - 1 CIII(n), and the image is totally geodesic in CIII(n). This gives a m a x i m a l complex projective space in CIII(n). Finally consider DIII(n). Here let the totally isotropic subspaces be W1 : Z 1 . . . . . Z n = 0, W 2 : Z ~ - iZ,~+, = 0 , l~ot~n. For a n y (n - 1)-dimensional subspace of W1, L • N W2 is 1-dimensional a n d L + L ~- N W2 is totally isotropic. For example, let L C W 1 be defined by Z , + 1 = 0. Then v E L • if a n d only if

s v, :v iO, for all ~ ~ L. So if vi denotes the l*h c o m p o n e n t of v, t h e n v2 . . . . . v, = 0. If v E W 2, t h e n vi = 0 for 2 i~nandn + 2 ~ < i ~ < 2 n a n d v l = V ' - l v , + l . Thus, L l A W2 is 1-dimensional. Also, for a n y f , f ' E L and v,v' ~ L • N W2, S(v + e, v' + e ' ) = S ( v , v ' ) + S(e,v') + ,/S(v,e') + s(e,e') = o. The assignment L --* L @ L" N W2 defines a totally 32 THE MATHEMATICAL INTELLIGENCERVOL. 15, NO. 2, 1993

geodesic e m b e d d i n g of p , - 1 _-__ G(n - 1, W1) --* DIII(n), a n d this space is maximal. The above results for AIII(n) and BDI(n) have been well k n o w n for some time. The fact that CI(n) a n d DIII(n) are totally geodesic subspaces of G(n,2n) a n d the fact that the projective rank of G(n,2n) is n gives an u p p e r b o u n d on the projective ranks of CI(n) a n d DIII(n). Investigation of this, that is, trial a n d error, led me to discover the results for CI(n) and DIII(n). The actual proofs use a computation of the degrees of the irreducible representations of SL(r,C) of small degree. The projective ranks of the exceptional spaces are computable using more special techniques from Lie theory and root systems (see [3]). An early conjecture of mine was that the equality dime M = rank(M) 9 projective rank(M) characterized G r a s s m a n n manifolds. H o w e v e r , this equality also holds for Qm w h e n m is even. The equality falls to hold for each of the remaining classes of irreducible Hermitian symmetric spaces and also fails in the exceptional cases. Thus, a simple characterization of Grassmannians analogous to the Mori-Siu-Yau characterization of complex projective space remains to be found.

General References on Symmetric Spaces A. S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, New York: Academic Press (1978). B. J. Wolf, Fine structure of Hermitian symmetric spaces, Symmetric Spaces (Boothby and Weiss, eds.), New York: Marcel Dekker, (1972).

References 1. A. Babakhanian and H. Hironaka, On the complex Grassmann manifold, Illinois J. Math 33 (1989), 170-179. 2. R. L. Bishop and S. I. Goldberg, On the second cohomology group of a K/ihler manifold of positive curvature, Proc. AMS 16 (1965), 119-122. 3. A. Fauntleroy, On the projective rank of compact Hermitian Symmetric spaces, preprint. 4. S. I. Goldberg and S. Kobayashi, On holomorphic bisectional curvature, J. Diff. Geom. 1 (1967), 225-233. 5. A. Howard, B. Smyth, and H. Wu, On compact IG~hler manifolds of nonnegative bisectional curvature, I, Acta Math. 147 (1981), 51-56. 6. N. Mok, The uniformisation theorem for compact K/ihler manifolds of nonnegative holomorphic bisectional curvature, J. Diff. Geom. 27 (1988), 179-214. 7. S. Mori, Projective manifolds with ample tangent bundle, Ann. Math. (2) 110 (1979), 593-606. 8. T. Ochiai, On compact IG~hler manifolds with positive holomorphic bisectional curvature, Proceedings of Symposia in Pure Mathematics, Providence, RI: American Mathematical Society (1975), Vol. 27, Part 2. 9. Y. T. Siu and S. T. Yau, Compact K/ihler manifolds of positive bisectional curvature, Invent. Math. 59 (1980), 189-204. 10. H. Wu, On compact K/ihler manifolds of nonnegative bisectional curvature, II, Acta Math. 147 (1981), 57-70. Department of Mathematics North Carolina State University Raleigh, NC 27695 USA

Ramanujan and Harish-Chandra V. Kumar Murty

The purpose of this article is to discuss a central theme of mathematics to which both Ramanujan and HarishChandra have made significant contributions. In 1916, Ramanujan published a paper which was innocuously tiffed " O n certain arithmetical functions." In it, he studied [13, p. 151] the discriminant function oc A(Z) =

e 2~iz

H

(1 --

Ramanujan begins his study by multiplying out the product formula for A(z): oc

A(Z) = E "r(n)ea~inz" n=l

Here ~(n) E Z, and the first few values are

e2~rinz)24.

n=l

If z ~ C and Im(z) > 0, then this product converges absolutely and, thus, defines ~l(z) as an analytic function in this domain. This function had been studied before Ramanujan. In particular, Dedekind had considered

9(1) = 1, ~(2) = - 2 4 , T(4) = - 1472 = - 2623, $(5) = 4830 = 2 - 3 . 5 . 7 - 2 3 .

-r(3) = 252 = 22327,

oo "q(Z) = r

H

(1 -- r

n=l

which is a 24th root of a(z). This function transforms in a particular way under the action of the group

That, in turn, it implies the following transformation formula for the discriminant function:

then

AI'az| ~+ )b\ \

= (CZ + d ) 1 2 A ( z ) .

! THE MATHEMATICALINTELLIGENCERVOL 15, NO. 2 9 1993 Springer-Verlag New York 33

Though the function A(z) had been studied earlier, Ramanujan was the first to realize that the Fourier coeffidents held interesting and deep arithmetic information. His study focused on three aspects which we shall discuss in turn. First, there are the multiplicative properties. Ramanujan conjectured that if ra and n are relatively prime, then

'r(mn)

"r(m)'r(n).

=

Thus, for example, "r(6) = (-24)(252) = -6048. He suggested, moreover, that if p is a prime, then, for any natural number a, ,r(pa + 1) = ,r(pa),r(p) _ pll,r(p~- 1). Thus, for example, r(9) = 1"(3) 2 - 311 = ( 2 5 2 ) 2 - 311 = 34 9 23 9 61. Both of these conjectures were proved by Mordell in 1917. His proof was later generalized by Hecke and led to the definition and theory of what are now called Hecke operators. We may formally consider the series 0r

L(s) = 2

r(n)

It follows easily from this that the Dirichlet series defining L(s) converges absolutely for Re(s) > 13/2. As a consequence, L(s) is analytic in this half-plane. Consider the polynomial T2 - "r(p)T + p11. Its roots are given by "r(p) +- 3v/'r(p)2 - 4p 11

Thus, the roots are complex conjugate if and only if 4p 11. One supposes that some numerical calculations led Ramanujan to conjecture that this was so for all primes p:

"r(p) 2 <

(*)

I~(p)l

This estimate was proved by Deligne as the culmination of the efforts and ideas of many mathematicians. He showed [3] in 1971 that the Weil conjectures in algebraic geometry imply the Ramanujan conjecture, and, in 1974, he succeeded [4] in proving the Weil conjectures. Following Hecke, let us consider the integral

I(s)

ns

~< 2 P 11/2-

dy

=

f f A(iy)y~ dO Y

n=l

for Re(s) sufficiently large. Note that for s E C. By the multiplicative property and the unique factorization of integers in prime powers, we have the formal identity

)

p

Furthermore, by the recursion formula for the x(pa), one can show that

A(iy) = ~ "r(n)e-2r n=l

As "fin) has at most polynomial growth in n, it is easy to deduce that

A(iy) = O(e-2"rrY). Hence, the integral I(s) actually converges absolutely for Re(s) > 0. If we substitute the series expansion of dl(iy) into the integral and interchange the summation and integration (as we may), we find that

Second, Ramanujan considered the growth of r(n) as a function of n. He proved that

IT(n)l

~<

C1n7

I(s) = ~ "r(n) f o e-2~nyY~ dy Y n=l oc

for all n. (Here and below, C1, C2 . . . . denote positive absolute constants.) This bound implies that the formal manipulations of L(s) made above are actually valid for Re(s) > 8. Hardy improved this bound t o C2n6. His proof also shows (see [8, pp. 170-172]) that "r(n) 2

<~ C3x12.

n~x

34

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993

= (2~r)-SF(s) ~

"fin)

n s = (2,rt)-SF(s)L(s).

n=l

These manipulations are valid for Re(s) > 13/2. On the other hand, we may write

dy

Note that/2 is entire. Using the modular transformation property of A, we may rewrite 11 as follows. We have A ( - - ! ) = A((01 o 1 ) Z ) =

The function A~(s) =

rr

has an analytic continuation for all s with a simple pole at s = 1 and it satisfies the functional equation

zl2A(z)

A~(s) = hr

and so -12

/

Hence,

dy 7 =

r,(sl = fo

du

Note that the last expression for 11 is absolutely convergent for all values of s. Thus, we see that for Re(s) > 13/2,

- s).

We think of Aa(s ) as a two-dimensional analogue because the Euler factor at p is a quadratic polynomial in p-S. The above similarities with the Riemann zeta function should already arouse hopes that Aa(s ) may contain interesting arithmetical information. This is the third aspect that Ramanujan explores, in the form of congruences satisfied by the -r(n). For example, he shows [13, p. 230] that

"r(n) =- mr(n) (mod 5), where (r(n) = Eel, d, and

(2~r)_Sr(s)L(s) = f ~ A(iu)(u12_s + uS)dU,u and the right-hand side is entire and invariant under the substitution s ~-> 12 - s . This, therefore, gives the analytic continuation of A(s) = Aa(s) =

(2"rr)-SF(s)L(s)

as an entire function of s and shows that it satisfies the functional equation

-fin) --- crn(n ) (mod 691), where (rn(n) = Eel, d11. There were also other congruences modulo 7 and 23. From today's vantage point, we see that these congruences can best be understood through f-adic representations. To explain this, it is necessary to say a few words in general about algebraic number theory. The fundamental problem of algebraic number theory is factorization. For example, consider the inclusion

A(s) = A(12 - s). Z C R = Z[V2] = {a + bV~: Thus, L(s) provides a two-dimensional analogue of the Riemann zeta function. Recall that for Re(s) > 1, the zeta function is defined by ov

1

(;)1

a,b, E 7/}.

For p a prime, we may consider the lift pR of the prime ideal p77. In general, it will no longer be a prime ideal. However, as R is a Dedekind domain, there is unique factorization of ideals. Thus, we have 7R = (1 + 2 V 2 ) R . (1 - 2V~)R.

The equality of the sum and the product, and the divergence of the harmonic series, were used by Euler to give an analytic proof of the infinitude of primes. In some sense, this was the birth of analytic number theory. The product on the right, and more generally a product of the form

H pp(p-s)-i P

with each Pp(T) ~ 77[T], P(0) = 1, is called an Euler product, and Pp(p-s)-1 an Euler factor.

More generally, let F be an algebraic number field (that is, a field of finite degree over Q). Let 0 denote the integral closure of Z in F. By definition, this is the ring of elements of F which satisfy a monic polynomial with coefficients in ?7. A rule that tells us h o w to determine the factorization of p0 is called a reciprocity law. Let us suppose that F is a Galois extension of Q, and set G = GaI(F/Q). For each prime p, we may associate a conjugacy class (rp in G which holds the information about the splitting of p in F. For example, if ~rp = {1}, then p splits completely in F: THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993 3 5

P~

= ~1

9 9 9 ~r,

r = [F : Q ] ,

~i

distinct.

g= There is a result of Deligne, Serre, and SwinnertonDyer (see [16] and the references therein) which implies that for any prime 1~ > 691 and/~ ~ 1 (rnod 11), there is a Galois extension Fe/Q with the following properties: (a) Gal(Fe/Q) ~- GL2(77/~,), (b) if p #/?, then (rp is a conjugacy class which when viewed in GL2(77/e,) has characteristic polynomial

~-->g.i-ci+

d.

The stabilizer of i under this action is the group ~(COS 0 - s i n 0~ } K = SO2(R) = L \ s i n 0 c o s 0 / : 0 ~ < 0 < 2 r r . Thus, we get an identification SL2(R)/SO2(R ) ~ ~.

T2 - (a'(p)(mod f))T + (pll(mod f)). Thus, the "r function shows up in a remarkable way in the reciprocity law for the extension Fe/Q. Deligne obtains such an extension for all primes 2. What he actually constructs is a representation

We may, therefore, consider functions f on f~as functions ~b on SL2(R) which are invafiant under the translation by elements of K: d~(gh) = ~b(g), g ~ SL2(R), h E K. If f satisfies a transformation law

Pe : Gal(Q/Q) ~ GL2(Ze) / a z + b\

for each 17, with the property that if p # f, then pe(frp) has characteristic polynomial T 2 - r(p)T + p~l. (Here Ze is the ring of f-adic integers.) The above field extension is obtained as the fixed field of the kernel of the composition of Pe with the map

for (~ b) E F, some discrete subgroup of SL2(R), then ~b satisfies 4(.~g) = (cz + d)%tg) for ~/ = (* ~), and z = g 9 i. Let us modify 6 as follows. For g = (* ~), let us set

GL2(Ze) ~ GL2(77/e ) j(g,z) = (cz + d) k given by reduction mod f. The family {Pe} of f-adic representations is constructed [3] out of the cohomology of a certain algebraic variety. Using Deligne's extensions, Swinnerton-Dyer has found interpretations for Ramanujan's congruences in terms of the size of the Galois group. Having seen the numerous and deep arithmetic and analytic properties arising from A, it is natural to seek the most general context in which these phenomena persist. This separates itself into two problems: 1. Find the most general functions which are analogues of di. 2. Extract arithmetic information from these functions. These are really different problems requiring different techniques and approaches. In the remainder of this article, we shall confine ourselves to the first question, as it is here that Hafish-Chandra has made profound contributions. Let us begin by reinterpreting functions on the upper half-plane ~ = {z E C:Im(z) > 0}. The group SL2(R) acts transitively on ~ by fractional linear transformations, and so we can define a surjective map SL2(R) t} by 36 THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993

and define r

= f(g" i)j(g,i) - k.

Then 6" is left F-invariant, and under fight translation by K, it transforms by the character cos 0 - s i n 0~ sin0 cos0 ] ~ eik~ More generally, we may consider functions : SL2(R) ~ C which are left F-invariant and are fight K-finite (in the sense that the fight K-translates of 6 lie in a finite number of K-eigenspaces), and in addition satisfy some conditions: (i) qb is C ~ and slowly increasing: There is a constant c > 0 and a positive integer n such that Id~(g)l ~< ciIm g . iI" for all g E SL2(~ ). (ii) (b is an eigenfunction of the Laplacian V. (iii) (b lies in L2(FLqL2(R)).

This defines the notion of an automorphic form on SL2(R). More generally, let G be a semisimple Lie group over R. The concept of an automorphic form for G was defined and studied by Harish-Chandra in a fundamental paper in 1959. Now let F be a discrete subgroup of G. The group G acts on the space L2(F\G) by right translation. This representation R, the right regular representation, is unitary for the usual norm on L2(F\G). Given an automorphic form • E L2(F\G), we may consider the closure of the subspace generated by the set {63} of right G-translates of ~b [thus, ~g(h) = 6(hg)]. In this way, we obtain a subrepresentation of G in L2(F\G). Conversely, if we consider an irreducible representation of G in L2(F\G) and choose from it a K-finite vector (where K is a maximal compact subgroup), we get an autornorphic form. Thus, the study of autornorphic forms is intimately connected with the study of the decomposition of R. This connection between autornorphic forms and (in general, infinite-dimensional) representations was first discovered by Gelfand and Fornin in 1952. (For an excellent exposition of the theory of autornorphic forms and autornorphic representations for the case G = GL2(~ ), the reader may consult .[51.) From the very outset, Harish-Chandra's mathematical work concerned the infinite-dimensional representations of sernisirnple Lie groups. One of his very important contributions was to u n d e r s t a n d the role played by K-finite vectors. Specifically, let ~ be an irreducible unitary representation of G on a Hilbert space ~ . Consider the vector subspace ~ K of K-finite vectors in ~:~. Then ~ K is dense in ~ . Moreover, by differentiation of ~, the Lie algebra g of G acts on ~ . Harish-Chandra showed that it preserves ~K, and moreover, ~/K is an irreducible g-rnodule. This led to the concept of a (g,K)-rnodule V. It is a vector space on which both g and K act in a compatible way such that V = ~r162 where or ranges over the irreducible (finitedimensional) representations of K and V~ is a subspace on which K acts by or. The direct sum is an algebraic direct sum (and not a Hilbert space direct sum). The (g,K)-module V is unitary if there is a positive definite inner product (,) on V so that (Xu,v) = - (u,Xv) for all X E g and u, v E V. Harish-Chandra proved that if G is simply connected, there is a bijection between isomorphism classes of irreducible unitary representations ~ of G and isomorphism classes of irreducible unitary (~,K)-rnodules V, with the map being given by ~ ~K. What is gained by this is that (g,K)-rnodules are essentially algebraic objects. The implications of this result are more pronounced w h e n one considers the representation theory of p-adic groups, for here it leads to the important notion of admissible representations. To understand this in the context of autornorphic forms, it is first necessary to globalize the concept.

The adele ring A o v e r Q consists of vectors (x=,x2,x3,x 5. . . . ) with x= ~ R, Xp E Qp, and Xp E Zp for all sufficiently large primes p. (Here, Qp is the field of p-adic numbers and Zp is the ring of p-adic integers.) It is locally compact and a convenient base on which to build a global theory of automorphic forms. Let G be the algebraic group GL2. Then we may still consider the right regular representation R of G(A) on L2(G(Q)\G(A)). An automorphic representation ~r is an irreducible constituent of R. As shown by Jacquet and Langlands [9], there is a factorization of "rr

into a (restricted) tensor product of representations ~p of the local groups G(Qp) and "rr~of G(0~). As a first step towards understanding the automorphic representations, it is therefore necessary to extend some of the classification theory established for G(0~)to these p-adic groups G(Qp). As shown by Jacquet and Langlands [9], the local components Tip a r e admissible in the sense that the subspace of GL2(Zp)-finite vectors is dense and is a representation space for a certain associative algebra (the so-called Hecke algebra). This algebra plays the same role in the p-adic theory as the Lie algebra (or rather the universal enveloping algebra) plays in the real theory. Any finite-dimensional irreducible admissible representation of G(Qp) must be one-dimensional and factor through the determinant:

c(%)

Q?

c•

A way of constructing some infinite-dimensional admissible representations of G(Qp) is as follows. Let P-1 and P-2 be two continuous characters of Qpx : ~1,.2 : Qp ~ C • We may define a representation of the Borel subgroup

by

where I" I is the p-adic norm. Then w e induce this from Bp to GL2(Qp) to get a representation, denoted P(~x,~2). This representation is admissible, and usually it is irreducible. The only exceptions are if ~1~2 = I " I or I " I-1. In those cases, one can find either an irreducible subspace of codimension I or an irreducible quotient by a one-dirnensional invariant subspace. This irreducible representation is denoted by or(~x,~-2). Thus, this produces two sets of representations, the so-called principal series and special representations. THE M A T H E M A T I C A L INTELLIGENCER VOL. 15, NO. 2, 1993 3 7

In addition to these and the one-dimensional representations, there are the supercuspidal representations. These are representations whose matrix coefficients are compactly supported modulo the center. For general reductive linear algebraic groups G which are defined over Q, an important result of Jacquet and Harish-Chandra (the subquotient theorem) implies that every irreducible admissible representation of G(Qp) is constructed in a similar fashion. Though many people have contributed to laying the foundations, the principal architects of the modern theory of automorphic forms are Langlands and Selberg. In [14], Selberg introduced the fundamental technique of the trace formula. Roughly speaking, this is a formula that relates the spectral theory of a group to its geometry. It was a key tool in the work of Jacquet and Langlands mentioned above, and along with Arthur's generalization, it has maintained a central position in much of the subsequent work on automorphic representations. Conjectured functorial properties of the general trace formula are expected to lie at the heart of several reciprocity laws. For some indication of this relationship in the case of number fields, the reader is referred to the expository paper of Gelbart [6] and to the important work of Arthur and Clozel [1, Chap. 3]. In [11], Langlands considers the notion of an automorphic representation in great generality and prescribes a method of attaching an L-function to such a representation. This method depends on the theory of the constant term of Eisenstein series and HarishChandra's theory of spherical functions. In many cases, Langlands has succeeded in establishing that the L-function has a meromorphic continuation and satisfies a functional equation. In the case of G = GL2, for example, the L-function attached to a (cuspidal) ~r is of the form

L(s,,rr) = L(s,'rr~) H L(s,'rrp), P

where

(~i - Ixl(P)P-S)-I (1 - p,2(p)p-S) -1

L(s,'trp) =

I

if

"/Tp = p(la, l , ~ 2 )

pq(p)p-S)-I if "trp or(iXl,tX2) i f "rrp

is supercuspidal.

Returning to our function A, if we denote by 1ra the associated automorphic representation, then

Shahidi [15] has recently succeeded in determining the complete list of L-functions which can be analyzed by this approach of Langlands. For general automorphic representations -rr, the analogue of the Ramanujan conjecture (*) and the exis38 THE MATHEMATICALINTELLIGENCER VOL. 15, NO. 2, 1993

tence of r representations are open questions and the subject of intensive study. We have remarked that reciprocity laws for some number fields are related to the values of Ramanujan's r function. In Langlands's vision, the general reciprocity law for number fields should arise from automorphic representations. In our brief discussion, we have made no mention of the discovery by Ramanujan of the circle method, or of his collaboration with Hardy in which the method was refined and applied to various problems in additive and analytic number theory. We have also not discussed the construction and classification by HarishChandra of the discrete series representations of a real semisimple Lie group, nor the Plancherel formula. These are among their most important mathematical contributions, and the interested reader is referred to their respective collected works [7,13] and the introductory articles therein (see also [18,19]). In studying Harish-Chandra's work, we are largely aided by the fact that he wrote thoroughly and carefully, and the conceptual moorings of his feats of technical prowess are clearly visible. In Ramanujan's case, however, an enormous obstacle is encountered. The vast majority of his work is unpublished and in the form of conclusions with little or no argument in the sense to which we are accustomed. It is no exaggeration to say that even today we have very little idea of the conceptual meaning of this work. This is notwithstanding the heroic efforts of several mathematicians (see, for example, the volumes of Berndt [2]) to go through his assertions systematically and to offer precise formulations and rigorous arguments. Although Harish-Chandra's work can be seen as a successor to the earlier work of Cartan and Weyl, the work of Ramanujan at once transports us to a terrain unknown and unfamiliar. The delight of the explorer is commingled with the longing to find a way home. Ramanujan and Harish-Chandra had quite different educational backgrounds. While the one failed his college exams, the other received a doctorate from Cambridge University. But in some sense, both were selfmade mathematicians. Contrasting Ramanujan's creativity with his mathematical ignorance, Hardy [13] writes: The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations and theorems of complex multiplication to orders unheard of . . . . who had found for himself the functional equation of the zeta function... ; and he had never heard of a doubly periodic function or of Cauchy's theorem, and had indeed but the vaguest idea of what a function of a complex variable was. Similarly, Langlands [12, p. 206] writes of HarishChandra that he came to mathematics relatively late and, in spite of enthusiastic initial attempts, there were broad domains of mathematics that he never assimilated in any serious way, 9

.

.

although he learned all that he needed . . . it is not too much of an exaggeration to say that he manufactured his own tools as the need arose, and that one of the grand mathematical theories of this century has been constructed with the skills with which one leaves a course in advanced calculus. H a r i s h - C h a n d r a (as q u o t e d b y L a n g l a n d s [12, p. 206]) himself said: I have often pondered over the roles of knowledge or experience, on the one hand, and imagination or intuition, on the other, in the process of discovery. I believe that there is a certain fundamental conflict between the two, and knowledge, by advocating caution, tends to inhibit the flight of imagination. Therefore, a certain naivete, unburdened by conventional wisdom, can sometimes be a positive asset. In their o w n w a y , R a m a n u j a n a n d H a r i s h - C h a n d r a s u c c e e d e d in cultivating a n d e m p l o y i n g t h a t asset. H o w e v e r , it s h o u l d be r e m e m b e r e d that their imagination w a s stirred a n d fired b y a f o u n d a t i o n of serious study. In R a m a n u j a n ' s case it w a s Carr's S y n o p s i s (see [8, pp. 2-3]), a n d in H a r i s h - C h a n d r a ' s case, Chevalley's t h r e e - v o l u m e treatise on Lie groups. O f the two, R a m a n u j a n p r e s e n t s m o r e of an enigma. H e did n o t go t h r o u g h the usual educational process. His m e t h o d s of d i s c o v e r y are not k n o w n . This raises the inevitable question: H o w did he d o it? W a s it his p e n c h a n t for s a m b a r (a delicious South I n d i a n dish) or the i m p o r t a n c e he ascribed to s u p e r s e n s u a l forces? Authors in b o t h the East a n d the West h a v e b e e n speculating o n this for the last century. 1 N o t w i t h s t a n d i n g these u n r e s o l v e d speculations, w e m a y safely assert that w h a t e v e r else w a s involved, R a m a n u j a n w o r k e d v e r y v e r y h a r d to m a k e the discoveries that he did. In the infinite o c e a n of time, e v e n great thinkers m u s t s e e m as floating w i s p s of straw. But t h r o u g h our 1 The latest attempt is a biography [10] by R. Kanigel entitled The Man Who KnewInfinity. Whatever the title means, this work certainly indulges the reader who is interested in such speculations. It contains a fairly detailed account of Ramanujan's life, interspersed with a good deal of interpretation. As it is written for the general public, the author attempts to convey some idea of Ramanujan's mathematics in nontechnical terms. This is complicated by the fact that Kanigel is describing a subject that is not really understood even by mathematicians. At best, we get a view of Ramanujan's mathematics as seen by Hardy, and this is a view which is already half a century old. Moreover, there is an unfortunate tendency in the book to emphasize the mystical. We do not deny the significance and importance of a certain inscrutable element that pervades Ramanujan's life and work. Moreover, it is clear that Hardy is trivializing this when he says [8, p. 4] that Ramanujan was no mystic and . . . religion except in a strictly material sense played no important part in his life. However, we do not seem to gain very much by naive speculations on that which the Upanishad describes as something " . . . from which mind and speech return baffled" [17, II.9.1].

h u m a n perspective, w e see t h e m as great flagships a n d spacious ferries, a n d d r a w inspiration f r o m their life a n d w o r k . Mathematical contributions aside, it is clear that b o t h R a m a n u j a n a n d H a r i s h - C h a n d r a h a v e inspired m a n y m a t h e m a t i c i a n s all a r o u n d the w o r l d a n d will continue to do so in the d e c a d e s to come.

References 1. J. Arthur and L. Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Princeton, NJ: Princeton University Press, 1989. 2. B. Berndt, Ramanujan's Notebooks, New York: SpringerVerlag, I (1985), II (1989), III (1991), IV (to appear), V (in preparation). 3. P. Deligne, Formes modulaires et representations e-adiques, Lecture Notes in Mathematics, No. 179, Berlin: SpringerVerlag, (1971), 139-172. 4. P. Deligne, La conjecture de Weil I, Publ. Math. IHES 43 (1974), 273-307. 5. S. Gelbart, Automorphic Forms on Adele Groups, Princeton, NJ: Princeton University Press, 1975. 6. S. Gelbart, Automorphic Forms and Artin' s Conjecture, Lecture Notes in mathematics, No. 627, Berlin: SpringerVerlag, (1977), 242-276. 7. Harish-Chandra, CollectedPapers, 4 volumes (V. S. Varadarajan, ed.), New York: Springer-Verlag, 1981. 8. G. H. Hardy, Ramanujan--Twelve Lectures on Subjects Suggested by His Life and Work, New York: Chelsea, 1978. 9. H. Jacquet and R. Langlands, Automorphic Forms on GL(2), Lecture Notes in Mathematics, No 114, Berlin: Springer-Verlag, 1970. 10. R. Kanigel, The Man Who Knew Infinity, Toronto: Collier Macmillan, 1991. 11. R. Langlands, Euler Products, New Haven, CT: Yale University Press, 1967. 12. R. Langlands, Harish-Chandra, Biographical Memoirs of Fellows of the Royal Society 31 (1985), 197-225. 13. S. Ramanujan, Collected Papers (G. H. Hardy, P. V. Seshu Aiyar, and B. M. Wilson, eds.), New York: Chelsea, 1927. 14. A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. 20 (1956), 47-87. (See also Collected Papers, Berlin: Springer-Verlag, 1989, Vol. 1, pp. 423-463.) 15. F. Shahidi, On the Ramanujan conjecture and finiteness of poles for certain L-functions, Ann. Math. 127 (1988), 547-584. 16. H. P. F. Swinnerton-Dyer, Congruence properties of T(n), Ramanujan Revisited (G. Andrews et al., eds.), San Diego: Academic Press, 1988, 289-311. 17. Taittiriya Upanishad, Volume 1 of Eight Upanishads (Sw. Gambhirananda, transl.), Mayavati: Advaita Ashrama, 1986. 18. V. S. Varadarajan, Harish-Chandra 1923-1983, J. Indian Math. Soc. 56 (1991), 191-215. 19. V. S. Varadarajan, Harish-Chandra (1923-1983), Mathematical Intelligencer, vol. 6 (1984), no. 3, 9-13.

Department of Mathematics University of Toronto Toronto, Ontario M5S 1A1 Canada

THE MATHEMATICAL 1NTELLIGENCER VOL. 15, NO. 2, 1993

39

Karen V. H. Parshall*

Embedded in the Culture: Mathematics at the World's Columbian Exposition of 1893 Karen V. H. Parshall and David E. R o w e

Chicago, Illinois, Monday, 21 August, 1893. It has been almost four months since the President of the United States, Grover Cleveland, opened the World's Columbian Exposition, and the Fair continues to draw tens of thousands of people a day. On the Midway, a dazzling array of sights and sounds confront the visitor: the Streets of Cairo; the Irish, German, and Javanese villages; the Japanese Bazaar; the Hagenbeck Animal Show with its performing lions, tigers, elephants, and bears; and, of course, the gigantic Ferris Wheel. A stroll down the Midway's central promenade brings with it encounters with Bedouins in flowing robes, German cavalry officers in full regalia, barechested South Sea Islanders, and sword-toting Japanese warriors. Moving from the Midway to the myriad buildings surrounding the Court of Honor, technology, art, and architecture reign. Machinery Hall, 850 feet by 100 feet in size, houses the great boilers which generate much of the power for the Fair along with some 77 engines of all types and uses. This structure is dwarfed by the great Manufactures and Liberal Arts Building. With its Corinthian-style fagade, this building covers some 30 acres and contains thousands of displays of manufactures---jewelry, precious stones, ceramics, textiles, glassware, timepieces, telescopes--and the liberal arts--books, journals, mathematical models, portraits--far too numerous to list. Modes of transportation from locomotives to sailing vessels to bicycles are found under the roof of the Transportation Building; * Column Editor's address: D e p a r t m e n t s of Mathematics and History, University of Virginia, Charlottesville, VA 22903 USA.

40

THE MATHEMATICALINTELLIGENCERVOL. 15, NO. 2 9 1993Springer-VeflagNew York

great trees from every climate grow under the ll4-foothigh glass dome of the Horticulture Building; electric lights in flashing displays of color greet the visitor to the Electricity Building; exquisite works of art from every country in wood, marble, bronze, oil on canvas, and countless other media adorn that paragon of Ionic architecture, the Art Building; and everything from the art of Mary Cassatt to fashions of the French couturiers, from handiwork to state work celebrates the role of w o m e n in society at the Woman's Building. All of this and much, much more is surrounded on the Jackson Park site by acres of man-made waterways and lagoons in which float replicas of Columbus's caravels, the Nifia, the Pinta, and the Santa Maria, Venetian gondolas, and a Viking ship. Industrial, technological, and artistic achievements are not the only cultural aspects in evidence at the Fair, however. M o d e m intellectual life is celebrated north of the Jackson Park grounds on Michigan Avenue at Lake Shore Park. There, the World's Congresses meet in two large auditoriums, the Halls of Columbus and Washington, for the purpose of discussing broader topics of international interest and import. Congresses on Women, Education, and Authors have already met .and the much-anticipated World's Parliament of Religions will take place beginning on 11 September. Today, the week-long Congresses on Science and Philosophy open. In particular, the Congress on Mathematics meets in its room in the Hall of Columbus to set up its schedule of lectures and e v e n t s . . . A reporter sending back copy to the h o m e t o w n newspaper might have noted these among many other sights and events at the World's Columbian Exposition on 21 August 1893.1 For the throngs w h o attended on that day, the amusements on the Midway, the ethnological displays at some of the national pavilions, the technological wizardry everywhere in evidence, and the sheer size of the spectacle, undoubtedly made the most vivid impressions. This was certainly no accident. Governments of various nations vied not only in i There were many contemporary accounts, both written and photographic, of the World's Columbian Exposition. See, for example, J. Seymour Currey, A Century of Marvelous Growth, 5 vols., Chicago: Clarke Publishing Co., 1912, especially volume 3, Chicago and Its History and Builders; and James William Buel, The Magic City: A Massive Portfolio of Original Photographic Views of the Great World's Fair and Its Treasures of Art, Including a Vivid Presentation of the Famous Midway Plaisance, St. Louis: Historical Publishing Co., 1894; reprint ed., New York: A m o Press, 1974. For more modem interpretations of the Fair, consult Stanley Appelbaum, The Chicago World's Fair of 1893. A Photographic Record, New York: Dover Publications, Inc., 1980; Reid Badger, The Great American Fair: The World's Columbian Exposition and American Culture, Chicago: Nelson Hall, 1979; Robert W. Rydell, All the World's a Fair: Visions of Empire at American International Expositions, 1876-1916, Chicago: University of Chicago Press, 1984, pp. 38-71; and Jeanne Madeline Weimann, The Fair Women: The Story of The Woman's Building, World's Columbian Exposition, Chicago 1893, Chicago: Academy, 1981.

their individual pavilions but also in their displays in the thematically organized buildings to stress the cultural attainments of their citizenry. Such nationalistic concerns particularly motivated the government of the United States as the host of the Fair, whereas the city of Chicago strove to establish itself in national as well as international eyes as a center of culture. One contemporary writer expressed the sense of both American national and Chicago civic pride in these words: Many of our elders across the water, while they admit that Americans may be quite strong upon their legs, still think in their souls that they are scarcely out of swaddling clothes--at most, are but green youth, who have yet to bear ripe fruit. Our own city is such a bounding type of life that some good people even in the United States, who are stiff-jointed, tire of the buzz, the bustle and the rush, and call the atmosphere simply wind. Now, what the Government proposes to do, and what Chicago has set out to accomplish, is to warm the world up to the blood-heat of youth--to prove that there is bottom to American speed . . . . The Great Republic and Republican America are on trial. 2 The writer's implication here is clear: The United States and the city of Chicago would come through this trial with flying colors; American culture equaled, and in many ways surpassed, the European culture from which it developed. 3 Furthermore, in Curler's view, that culture extended far beyond business and industry, technology, and invention, to the more sublime, intellectual aspects of the human existence. Here, too, the United States in general and Chicago in particular would shine at the Fair: "World's congresses of art, of science, of politics, of philosophy, of religion, will meet in the throbbing heart of this young nation, a n d - - w e will warm them. America shall stretch forth her hand in such a way that the world must grasp it. ''4 As one of the sciences represented in these congresses, mathematics thus took its place amid the myriad expressions of late nineteenth-century culture featured at the Fair, and the mathematicians on the local organizing committee took advantage of the occasion both to bring the mathematical endeavor into sharper focus within the United States and to highlight mathematics in the Chicago area, especially at the newly formed University of Chicago. s The American mathe2 H. G. Cutler, The World's Fair: Its Meaning and Scope, Chicago: Star Publishing Company, 1891, p. 9. 3 From Cutler's late nineteenth-century perspective, the Eurocentrism of American culture was unquestioned. 4 Cutler, The World's Fair, p. 9. s The University of Chicago had opened in October of 1892, less than a year prior to the convening of the Mathematical Congress. On the formation of this department per se, see Karen V. H. Parshall, The one-hundredth anniversary of mathematics at the University of Chicago, Mathematical Intelligencer vol. 14 (1992), no. 2, 39-44. We discuss the Mathematical Congress of the World's Columbian Exposition in detail, but with a somewhat different emphasis, in our forthcoming book, The Emergence of the American Mathematical Research THEMATHEMATICALINTELLIGENCERVOL.15, NO. 2, 1993 41

maticians were not the only participants in the Mathematical Congress with cultural motives, however. G6ttingen University's Felix Klein came to Chicago as the Congress's invited keynote speaker and as an official representative of the Prussian Ministry of Culture. He regarded the Congress as a venue for underscoring the dominance of German mathematics at the close of the nineteenth century and for cementing his own reputation as his country's leading spokesperson for and standard-bearer of that field. No less than the exhibition areas in the Manufactures or Electricity Building, the Mathematical Congress of the World's Columbian Exposition served as a stage on which to play out cultural agendas and nationalistic competitions. The members of the local organizing committee--Eliakim Hastings Moore, Oskar Bolza, and Heinrich Maschke of the University of Chicago and Henry Seely White of Northwestern University--had, in a sense, the largest vested interest in bringing the Mathematical Congress to a successful realization, and of these men, it was perhaps E. H. Moore who had the most to gain from its success. Untried and unproven as a mathematical researcher, the 31-year-old Moore had nevertheless been given the reins of the Department of Mathematics at the University of Chicago the year before by its president, William Rainey Harper. Harper saw in the World's Columbian Exposition, with its fairgrounds adjacent to the site of his new university, a prime opportunity. In particular, the associated congresses in the various areas of intellectual life provided ideal occasions for his assembled faculty to take center stage in realizing the stated goal of the World's Congress Auxiliary, namely, "to bring the leaders of human progress from the various countries of the world together at Chicago . . . . to review the achievements already made in the various departments of enlightened life. . . . to make a clear statement of the living questions of the day which still demanded attention, and to receive from eminent r e p r e s e n t a t i v e s . . , suggestions of the practical means by which further progress might be made and the prosperity and peace of the world advanced. ''6 As key participants in the congresses, the Chicago faculty members placed themselves, at least symbolically, among those "leaders of h u m a n progress." In the case of the Mathematical Congress, E. H. Moore and his colleagues thus stood shoulder to shoulder with Felix Klein, even if their reputations by no means invited any direct comparison with his. In putting together the program for the Congress, Community (1876-1900): James Joseph Sylvester, Felix Klein, Eliakim Hastings Moore, AMS/LMS Series in the History of Mathematics, Providence: American Mathematical Society, and London: London Mathematical Society, Chap. 7. 6 Currey, A Century of Marvelous Growth, vol. 3, p. 68. 42 THE MATHEMATICALINTELLIGENCERVOL.15, NO. 2, 1993

Moore, Bolza, Maschke, and White solicited papers from their American colleagues as well as from mathematicians abroad. Once Klein's direct participation was assured in the spring of 1893, the job of securing contributions from Germany fell to him. In all, 39 papers were presented (most in absentia) before the 42 men and 3 women actually in attendance at the event. 7 American-based mathematicians provided 13 of these, whereas the Germans supplied 16, and the remaining 10 came from mathematicians in Austria (2), France (3), Italy (3), Russia (1), and Switzerland (1). Although, numerically speaking, the American contribution to the content of the Congress roughly matched that of the Germans, mathematically speaking the Germans clearly outclassed their hosts. David Hilbert summarized his recent, seminal work (including the finite basis theorem) in the theory of algebraic invariantsS; Hermann Minkowski sketched some of his results at the interface between number theory and geometry, in particular, his theorem stating that if V is a closed, convex subset of ~3 which is symmetric about the origin and if the volume of V I> 23, then V contains a point, other than (0,0,0), with integral coordinates9; and Eduard Study surveyed the theory of hypercomplex number systems (that is, finite-dimensional algebras over C or ~), to which he himself had contributed. 1~ Still, the Americans did make a respectable showing. Frank Nelson Cole of the University of Michigan conveyed his discovery of a new finite simple group [PSL2(8) in modern notation] of order 504 within the context of a complete enumeration of all finite simple groups of order 660 or less11; Irving Stringham of 7 The Chicago Mathematical Congress is briefly discussed in Donald J. Albers, G. L. Alexanderson, and Constance Reid, International Mathematical Congresses:An Illustrated History 1893-1986, New York: Springer-Verlag, 1987, pp. 2-3. The proceedings of the Congress, together with the complete list of participants and attendees, were subsequently published. See Eliakim Hastings Moore, Oskar Bolza, Heinrich Maschke, and Henry Seely White, ed., Mathematical Papers

Read at the International Mathematical Congress held in Connection with the World's Columbian Exposition Chicago 1893, New York: Macmillan & Co., 1896 (hereinafter denoted CongressPapers). Only four of those in attendance came from abroad for the event: Klein, Eduard Study of Marburg, Norbert Herz of Vienna, and Bernard Paladini of Pisa. s David Hilbert, Ueber die Theorie der algebraischen Invarianten, Congress Papers, pp. 116--124. On Hilbert's work in this area and its implications for the nineteenth-century approach to the subject, see Karen V. H. Parshall, The one-hundredth anniversary of the death of invariant theory?, Mathematical Intelligencer, vol. 12 (1990), no. 4, 10-16. 9 Hermann Minkowski, Ueber Eigenschaften von ganzen Zahlen, die dutch r/iumliche Anschauung erschlossen sind, Congress Papers, pp. 201-207. x0 Eduard Study, Altere und neuere Untersuchungen fiber Systeme complexer Zahlen, Congress Papers, pp. 367-381; and, for example, Eduard Study, Complexe Zahlen und Transformationsgruppen, Be-

richte fiber die Verhandlungen der k6niglichen sdchsischen Gesellschafl der Wissenschaflen zu Leipzig 41 (1889), 177-228. 11 Frank Nelson Cole, On a certain simple group, Congress Papers, pp. 40--43.

the University of California, Berkeley c o m p a r e d the more traditional a p p r o a c h to elliptic function t h e o r y of Jacobi with the t h e n m o r e fashionable techniques of Weierstrass and d e m o n s t r a t e d the a d v a n t a g e s of the older t h e o r y within the context of the tranformation t h e o r y of elliptic integrals12; and H e n r y T. E d d y of the Rose Polytechnic Institute in Terre Haute, Indiana surv e y e d recent work, including his own, in graphical m e t h o d s in applied mathematics. 13 Moreover, the m e m b e r s of the D e p a r t m e n t of Mathematics at the University of Chicago---Moore, Bolza, and Maschke---each gave substantial p a p e r s a n d so placed the research issuing from their n e w university's p r o g r a m squarely within the Congress's r e v i e w of notable mathematical a c h i e v e m e n t s J 4 Bolza was the first of the trio to speak. In the afternoon session o n Thursday, 24 August, h e gave an exposition of one of the areas in which h e actively worked, the t h e o r y of hy~2Irving Stringham, A formulary for an introduction to elliptic functions, CongressPapers, pp. 350--366. 13Henry T. Eddy, Modern graphical methods, Congress Papers, pp. 58-71. 14Bolza and Maschke were, of course, German and were proteges of Felix Klein. Nevertheless, they were permanent faculty members at the University of Chicago. They thus formed part of the American-and more particularly the Chicago--mathematical scene. 15Oskar Bolza, On Weierstrass' systems of hyperelliptic integrals of the first and second kind, Congress Papers, pp. 1-12.

perelliptic integrals, and featured particular contributions of Riemann, Clebsch, Weierstrass, a n d Klein. 15 Maschke s t e p p e d before the assemblage o n the Congress's closing day, Friday the 25th, to detail some of his invariant-theoretic findings. H e p r e s e n t e d a complete determination of the absolute invariants of a certain s u b g r o u p G (of order 336 = 2 9 168) of GL4 (C), which h a d arisen in the context of some of Klein's earlier w o r k o n m o d u l a r f o r m s . 16 Finally, M o o r e b r o u g h t the Congress to its conclusion w i t h w h a t w e r e u n d o u b t e d l y the most significant n e w mathematical results to be a n n o u n c e d a n d p r e s e n t e d at the Chicago meeting. H e not only e x t e n d e d the result Cole h a d discussed in his Congress p a p e r to obtain a w h o l e n e w class of finite simple g r o u p s (today d e n o t e d PSL2(qn), for q a prime a n d n E 77+), but also p r o v e d that e v e r y finite field is, in fact, a Galois field. 17 At least a m o n g 16Heinrich Maschke, The invariants of a group of 2 9 168 linear quaternary substitutions, Congress Papers, pp. 175-186. See, also, Felix Klein, Ueber die Aufl0sung gewisser Gleichungen vom siebenten und achten Grade, MathematischeAnnalen 15 (1879), 251-282, or Felix Klein, GesammelteMathematische Abhandlungen, 3 vols., Berlin: Springer-Verlag, 1921-1923, 2:390-425 (hereinafter abbreviated Klein: GMA); and Zur Theorie der allgemeinen Gleichungen sechsten und siebenten Grades, Mathematische Annalen 28 (1886--1887),499532, or Klein: GMA, 2:439-472. 17 Eliakim Hastings Moore, A doubly-infinite system of simple groups, CongressPapers, pp. 208-242.

Congress of Mathematicians, World's Columbian Exposition, 1893. Bottom row, left to right, James E. Oliver and William E. Story; second row, William B. Smith, Henry S. White, Felix Klein, Harry W. Tyler, and Thomas F. Holgate; third row, Arthur G. Webster, C. A. Waldo, E. Study, J. M. Van Vleck, H. T. Eddy, J. B. Shaw, James McMahon, and Professor of Mathematics at Hope College (John Kleinheksel); top row, E. M. Blake, H. G. Keppel, Frank Loud, Henry Taber, Oskar Bolza, E. H. Moore, and Heinrich Maschke. Reprinted from "American Mathematics Comes of Age: 1875-1900/" by Karen H. Parshall and David E. Rowe, in A Century of Mathematics in America, Part III, edited by Peter Duren with the assistance of Richard A. Askey, Harold M. Edwards, and Uta C. Merzbach, History of Mathematics, Volume 3, page 4, by permission of the American Mathematical Society.

the American-based participants in the Congress, Harper's mathematicians unquestionably took center stage. They proved that the United States and the city of Chicago nurtured one of the more rarefied expressions of higher culture, research-level mathematics. As the representative of what was clearly the West's strongest country mathematically at the close of the nineteenth century, Felix Klein had very different objectives to fulfill at the World's Columbian Exposition. First, he aimed to show off German mathematics in the strongest possible light at both the Mathematical Congress in particular and the Fair in general. To this end, Klein had solicited papers for the Congress proper from some of the biggest names in late nineteenthcentury German mathematics, so the German dominance of the content of the Congress was essentially assured in advance. The Prussian Ministry of Culture, however, also mounted a German Universities' Exhibit in the mammoth Manufactures and Liberal Arts Building down on the fairgrounds in Jackson Park, and Klein w a n t e d to highlight G e r m a n mathematical achievements there as well. Initially, though, he was frustrated in these efforts by what he viewed as a lack of organization and direction from the exhibit's overall coordinators. In a confidential letter addressed to Bolza in Chicago, Klein worried "that the German University Exhibition will only come together very imperfectly: the whole thing was taken up too late and lacks the right personalities to lead it. ''18 Nevertheless, he continued, "I am sticking together with my friends so that at least the mathematics [exhibit] will succeed. ''19 His friends, principally Walther yon Dyck, had helped him to bring together an impressive array of German mathematical paraphernalia for display: mathematical models in plaster and string2~ mathematical apparatus of all sorts; a gigantic bust of Gauss and portraits of Dirichlet, Jacobi, and Riemann; copies of every mathematical doctoral dissertation or Habilitationsschrift written at a German university since 1750; copies of some 500 textbooks by German authors; and complete sets of seven German journals and the publications of the scientific academies of Berlin, Gbttingen, Leipzig, and Munich. 21 Although the broader cultural message of 18 Draft of Felix Klein to Oskar Bolza, 6 June 1893, Klein Nachlass VIII, Nieders/ichsische Staats- und Universit/itsbibliothek, Gbttingen (hereinafter cited as NSUB, GOttingen). Our translation. 19 Draft of Klein to Bolza, 6 June 1893. 2o Many of these were manufactured by the Darmstadt firm of L. Brill. For illustrations and mathematical explanations of the Brill firm's models, see Gerd Fischer, ed., Mathematische Modelle, 2 vols., Braunschweig/Wiesbaden: Fr. Vieweg & Sohn, 1986. Dyck himself published an account of the models in Walther yon Dyck, Katalog

mathematischer und mathematische-physicalischer Modelle, Apparate und Instrumente, Munich: C. Wolf, 1892. 21 Walther von Dyck, ed., Deutsche Unterrichtsausstellung in Chicago, 1893: Special-Katalog der mathematischen AussteUung, Berlin: n.p., 1893, and Amtlicher Bericht fiber die Weltausstellungen in Chicago 1893, erstattet von Reichskommisar, 2 vols., Berlin: Reichsdruckerei, 1894, p. 988. 44 THEMATHEMATICAL1NTELLIGENCERVOL.15, NO. 2, 1993

this display in its enormous glass cases was most probably lost on the general public which streamed past it, the American mathematical audience from the Congress could not have failed to be impressed by---and perhaps more than a bit envious of---the physical embodiment of a century of German mathematical contributions. Klein's second goal at the Fair was more personal. By virtue of his position as his country's official mathematical emissary to the Exposition, Klein knew that he was held in high regard in the political circles which controlled German higher education. He hoped, in effect, to capitalize on this to secure permanently the "position" of official spokesperson for mathematics in Germany. This would allow him to campaign more effectively for the vision of mathematics he alluded to in his remarks at the opening of the Congress on 21 August. 22 Speaking on "The Present State of Mathematics," Klein argued for the unity--as opposed to the increasing specialization--of mathematics as a discipline. He noted that in recent years the concept of a group, for example, had served to bring together the seemingly disparate mathematical areas of geometry and number theory. As he saw it, "It]his unifying tendency, originally purely theoretical, comes inevitably to extend to the applications of mathematics in other sciences, and on the other hand is sustained and reinforced in the development and extension of these latter. ''23 Klein believed that this sort of mutual interaction and reinforcement between mathematics and the sciences maintained the vitality of all of the areas, and he did not fail to point out that his great German forebear, Gauss, held the same view. "Speaking as I do, under the influence of our Gbttingen traditions, and dominated somewhat, perhaps, by the great name of Gauss," Klein allowed, "I may be pardoned if I characterise the tendency that has been outlined in these remarks as a return to the general Gaussian programme. A distinction between the present and the earlier period lies evidently in this: that what was formerly begun by a single master-mind, we now must seek to accomplish by united efforts and cooperation."24 Clearly, Klein intended to play an active role in directing those efforts and securing that cooperation. His trip to the United States only underscored his intentions. 2s 22 Klein also pushed this vision hard at the two-week-long series of Evanston Colloquium lectures he gave at Northwestern University immediately following the Fair. See Felix Klein, The Evanston Colloquium Lectures on Mathematics, New York: Macmillan & Co., 1893; reprint ed., New York: American Mathematical Society, 1911. 23 Felix Klein, The present state of mathematics, Congress Papers, pp. 133-135, p. 134, or Klein: GMA, 2:613-614. 24 Klein, The present state of mathematics, p. 135. Klein's emphasis. Indeed, only a year later, Klein was already busy laying plans for one such collective undertaking, the massive Encyclopadie der mathematis-

chen Wissenschaften. 25 Of course, Klein had already been actively engaged in training American, and other foreign, students at the doctoral level. To get an

The German University Exhibit at the World's Columbian Exposition, 1893.

Klein also saw in the United States a tremendous opportunity for Germany and German mathematics. Although he recognized certain shortcomings in the American system of education which posed problems for the development of mathematics at the research level there, he knew that the Americans were well aware of these problems, were working to correct them, and had a very good chance of effecting the necessary changes. He outlined his thoughts and observations on these matters in an unpublished report intended for the authorities who had sponsored his trip to the Exposition. 26 There, Klein bluntly suggested a highly nationalistic program for German science. "Without question," he wrote, "'at the present time and for the immediate future, America represents the richest and most promising object [das gr6sstm6glichste und gliicklichste Objekt] for scientific colonization. ,,27 He viewed his presence at the Fair and at the Mathematical Congress as an important early step in the development of what might become Germany's exclusive scientific sphere of influence in the United States. The World's Columbian Exposition of 1893 served as a grand stage upon which all sorts of cultural and intellectual agendas were played out. First, and foremost, however, it was an event animated ostensibly by all of the high ideals associated with any self-conscious display of the world's progress. One contemporaneous idea of his impact on the American mathematical scene, see the list of his American students in Karen Hunger Parshall and David E. Rowe, American mathematics comes of age: 1875-1900, in A Century of Mathematics in America Part III, Peter Duren, et al. eds., Providence: American Mathematical Society, 1989, pp. 12 and 15. 26 Felix Klein, Bericht fiber die Reise nach Chicago zwecks Teiln a h m e am mathematischen Congresse, Klein Nachlass IC, NSUB G6ttingen. As always, we thank the Handschriftenabteilung at the NSUB GOttingen for its hospitality and for permission to quote from its archives. 27 Bericht fiber die Reise nach C h i c a g o . . . Klein's emphasis.

commentator, caught up in the Fair's idealism, described it as "an epitome of all that was extant in the world as the outcome and the evidence of its advancement in every department of human effort; it was a condensed history of the successive epochs through which the human race has pursued its long and toilsome march toward the realization of its nobler destinies. ''28 Among those "nobler destinies" was mathematical research as chronicled at the Mathematical Congress and on the fairgrounds and as embodied in its assembled practitioners. 29 The mathematical displays may not have been as eye-catching as the fullscale model of the 2400-ton steam hammer from the Bethlehem Iron Works or the 46-foot-long, 16V2-inchcaliber gun from the Krupp Armory, but their very presence put the mathematics they represented at least on a cultural par with such technological innovations. Likewise, the Mathematical Congress may not have generated the interest of the Congress of Architects or Authors, but it implicitly conveyed an assumption that mathematics and mathematical research were embedded at least as deeply in culture as architecture or literature or any other h u m a n intellectual endeavor. Through their representation of their discipline, mathematicians--irrespective of the cultural or personal objectives motivating their participation--celebrated their subject's niche in human culture one hundred years ago at Chicago's World's Columbian Exposition.

Departments of Mathematics and History FB 17 Mathematik University of Virginia Universitf~t Mainz Charlottesville, VA 22903-3199 USA 6500Mainz 1, Germany 28 Currey, A Century of Marvelous Growth, 3:96. 29 Another emerged from the lectures that formed the Historical Congress. At that meeting, Frederick Jackson Turner set forth his thesis that the westward expansion of the United States was a crucial cause of its egalitarianism and democratic institutions. THE MATHEMATICALINTELLIGENCERVOL. 15, NO. 2, 1993 45

Ian Stewart* The catapult that Archimedes built, the gambling-houses that Descartes frequented in his dissolute youth, the field where Galois fought his duel, the bridge where Hamilton carved quaternions-not all of these monuments to mathematical history survive today, but the mathematician on vacation can still find many reminders of our subject's glorious and inglorious past: statues, plaques, graves, the cafd where the famous conjecture was made, the desk where the

famous initials are scratched, birthplaces, houses, memorials. Does your hometown have a mathematical tourist attraction? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks. Please send all submissions to the Mathematical Tourist Editor, Ian Stewart.

Pentagonal Decoration in Granada Istv in Hargittai The uniquely rich geometrical decorations of the Alhambra in Granada, Spain are widely known [1]. They have inspired artists [2] and have been thoroughly studied for their symmetries [3]. One of the smaller palaces adjacent to the Alhambra is the Generalife. Visitors to Granada usually start with the Alhambra and may or may not get to the General* Column Editor's address: Mathematics Institute, University of Warwick, Coventry CV4 7AL England.

ife. During a recent (October 1991) one-day visit, I followed the usual routine and completed my tour in the farthest tower of the Generalife, where I noticed a wall decoration reproduced in Figure 1. I am aware of no mention, of this particular decoration in the literature. Its special interest lies in the pentagonal/decagonal character of the pattern. Fivefold symmetry has gained prominence recently (see, e.g., [4,5[) due to discoveries in solid state science (quasicrystals) and in chemistry (fullerenes).

Figure 1. Photograph of wall decoration in the Generalife (Granada, Spain) taken by the author in October 1991.

Figure 2. Line drawing from David Wade's Pattern in Islamic Art [6], p. 88.

A line drawing analogous to the pattern of Figure 1 is reproduced here (Fig. 2) from David Wade's excellent collection Pattern in Islamic Art [6]. Wade gives no reference to the origin or location of this pattern. This is an interlacing version of another noninterlacing pentagon/decagon-based pattern analyzed by Wade [6] and shown here in Figure 3. The Generalife pattern has only local fivefold symmetries of the pentagons and decagons; the entire decoration does not have fivefold symmetry. Due to the interlacing, both the patterns shown in Figures I and 2 are chiral and happen to be heterochiral to each other.

_A

x

G

Acknowledgment: I am grateful to Dr. Emil Makovicky (Copenhagen) for introducing me to David Wade's book.

References

1. M. Vela Torres, ed., La Alhambra, Granada: Grafsur (1987). 2. D. Schattschneider, Visions of Symmetry. Notebooks, Periodic Drawings, and Related Work of M. C. Escher, New York: Freeman (1990). 3. B. Grtinbaum, Z. Griinbaum, and G. C. Shepard, Symmetry in Moorish and other ornaments, Symmetry Unifying Human Understanding, (I. Hargittai, ed.), Oxford: Pergamon Press (1986), 641-645. 4. I. Hargittai, ed. Quasicrystals, Networks, and Molecules of Fivefold Symmetry, New York: VCH (1990). 5. I. Hargittai, ed., Fivefold Symmetry, Singapore: World Scientific (1992). 6. D. Wade, Pattern in Islamic Art, Woodstock, NY: The Overlook Press (1976). Technical University Budapest Szt. Gell&t t~r 4 H-1521 Budapest, XI Hungary l~igure 3. Construction of the noninterlacing version of the pattern of Figure 2 according to David Wade ([6], p. 56). The decagons are surrounded by pentagons. The centers of the

sides of the pentagons provide the intersection points for the lines of the secondary pattern. THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2 9 1993 Springer-Verlag New York

47

The Stadtgottesacker in Halle Manfred Stern

Near downtown Halle is an old graveyard, the socalled Stadtgottesacker (God's acre of the town). Halfway between the main railway station (Hauptbahnhof) and the Marktplatz (the very center of town) is the Leipziger Turm, a tower from the fifteenth century and part of the old fortifications. From there, walk some 100 meters on the Hansering, then turn right. After climbing a few steps you will find the entrance to the graveyard through a gate in the Stadtgottesackerstrasse. The graveyard has an ample layout built in Renaissance style (1557-1594) and comparable to the Italian

Camposanto. 1 It is surrounded by a wall of elaborate masonry, and on the interior side the wall opens into niches under strainer arches (Schwibbogen). These niches were the burial places of prominent people: The theologian and pedagogue August Hermann Francke (1636-1727) and the philosopher Christian Thomasius (1655-1728) are buried here. The Stadtgottesacker is also a memorial place for the history of mathematics: Johann Andreas Segner (1704-1777), Eduard Heine (1821-1881), and the lesser known Friedrich Meyer (1842-1898) found here their everlasting rest. The Hungarian Segner was born in or near PozsonyPressburg (today Bratislava, the capital of Slovakia) on October 9, 1704 and died in Halle on October 5, 1777 (the year Carl Friedrich Gauss was born). From 1735 to 1755 he taught mainly physics, mathematics, and 1For a history of the StadtgottesackerHalle, see [10].

The Stadtgottesacker (above). Tombstone and portrait of Edward Heine (right).Portrait and tombstone of Johann Andreas Segner (below).

FriedrichMeyer and details from his tombstone.

chemistry at the University of G6tfingen, where he also founded the observatory. 2 From 1755 until his death, he was a professor of physics, mathematics, and astronomy at the University of Halle. He invented the "Segner-wheel," a precursor of the turbine. Leonhard Euler, who visited Segner in 1761 in Halle, used Segner's results in his own mechanical investigations. 3 In honor of Segner's astronomical activities a lunar crater was named after him. Segner was buried in a niche under one of the strainer arches (Schwibbogen No. 83). The new headstone of his tomb was sponsored by the Hungarian Segner Society and emplaced in 1977 on the occasion of the 200th anniversary of his death. The text on the headstone is in both Hungarian and German. Eduard Heine (of Heine-Borel fame) was born on March 16, 1821 in Berlin and died in Halle on October 24, 1881. Heine dedicated his dissertation [3] (Berlin, 1842) to Dirichlet; Kronecker was one of the opponents. The dissertation concludes with five "Theses" clearly proving Heine a staunch proponent of the theory of limits. From 1856 until his death, Heine was a full professor (Ordinarius) of the University of Halle. His book [4] on spherical functions has become a classic. Heine also had an influence on Georg Cantor, who came to Halle in 1867. 4 Both Heine and Cantor were actively engaged in the further development of the program initiated by Weierstrass. 5 Friedrich Meyer was born on March 5, 1842 in Mlinsk (Western Prussia) and died in Halle on December 5, 1898. He was a teacher of the Stadtgymnasium Halle and remained on close terms with the university as well as with Eduard Heine and Georg Cantor. He already used their ideas from 1880 on, e.g., in his book [8]. 6 It is interesting to note that in 1894 the Honorary 2 See [6]. 3 More informationon Segner's life and work is given in [7,12]. 4 For memorialplaces of Georg Cantor in Halle, see [11]. 5 This is witnessed, for instance, by Heine's paper [5]. Detailedinformationconcerningthe mathematicalrelationship between Heine and Cantor can be found in [1]. 6 Meyer's merits are also mentioned in Fraenkel's biography of Georg Cantor ([2]). More details can be found in [9].

Doctor's Degree of the University of Halle was conferred upon Beltrami and Meyer.

Acknowledgments: The photograph of F. Meyer is taken from [9]. All other photographs are from the Archives of the Martin-Luther-Universit/it Halle which is gratefully acknowledged. In particular, I am indebted to Mrs. M. Heinrichsdorff for taking the photographs at the Stadtgottesacker. I am also grateful to Dr. G. Betsch and Dr. W. Lagler (both Tfibingen), and last but not least to Professor B. Sz~missy (Debrecen) for several valuable remarks. References

1. J. W. Dauben, Georg Cantor. His Mathematics and Philosophy of the Infinite, Cambridge: Cambridge University Press (1979). 2. A. Fraenkel, Das Leben Georg Cantors, Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Berlin: Springer-Verlag (1990) (reprint of the 1932 edition). 3. E. Heine, De aequationibus nonnuUis differentialibus, Berlin, 1842. 4. E. Heine, Handbuch der Kugelfunktionen, Berlin: G. Reimer (1861, 2nd edition, 1881). 5. E. Heine, Die Elemente der Funktionenlehre, J. Reine Angew. Math. 74 (1872), 172-188. 6. F. Hund, Die Geschichte der G6ttinger Physik, G6ttingen: Vandenhoeck & Ruprecht (1987), 22-23. 7. W. Kaiser, Johann Andreas Segner, Leipzig: Teubner Verlagsgesellschaft (1977). 8. F. Meyer, Elemente der Arithmetik und Algebra, 2nd ed., Halle: Verlag H. W. Schmidt (1885). 9. G. Riehm, Friedrich Meyer (obituary), Jahresber.DMV (1899), 59--61. 10. G. Sch6nermark, Beschreibende Darstellung der dlteren Bauund Kunstdenkmdler der Stadt Halle und des Saalkreises, Halle: Verlag O.Hendel (1886). 11. M. Stern, Memorial places of Georg Cantor in Halle, Mathematical Intelligencer 14, No. 4 (1988), 48-49. 12. B. Sz6missy, History of Mathematics in Hungary until the 20th Century, Berlin: Springer-Verlag (1992).

Kiefernweg 8 D-0-4050 Halle Germany THE MATHEMATICAL INTELLIGENCERVOL. 15, NO. 2, 1993 4 9

Kepler, Einstein, and Ulm Dirk J. F. Nonnenmacher, Theo F. Nonnenmacher, and P. F. Zweifel

The beautiful city of Ulm, Germany lies on the Danube River some 130 km west of MOnchen, just on the border of W(irttemberg and Bavaria. It is worth visiting in its own right, not only for its wonderful 600-year-old

Gothic cathedral (the "MOnster") but also for its beautiful old city next to the river, with timbered medieval houses still in use, a spectacular city wall, and sparkling torrents rushing down to the Danube. The M~inster can be seen for miles around, because its 165meter-high steeple--the tallest church steeple in the world--dominates the countryside. It is possible to climb to the top of the tower, for an unforgettable panorama. Be sure also to visit the inside of the cathedral, and especially to study the wonderful carvings which adorn the choir stalls.

Ulm's theatre is the oldest city theatre in Germany, dating from 1641. One of the most famous musicians of all t i m e . . . Herbert von Karajan, conducted the Ulm orchestra from 1929 to 1934.

Figure 1. The bust of Einstein on the left was a gift from the people of India; the sculpture to the right marks the location of the house in which Einstein was born. Figure 2. Kepler's "'Ulmer Kessel" in the Ulm Museum.

50

It was in this lovely city on March 14, 1879 at 11:30 a.m. that Albert Einstein was born, the first child of Hermann and Pauline (ne~ Koch) Einstein. The family was then living in a house on Bahnhofstrasse, quite near the station, having moved from another house on the M~insterplatz only a few months before. The house on Bahnhofstrasse was destroyed in an air raid in 1944, but today the site of the home is commemorated by a monument; nearby is a sculpture donated by the people of India (Fig. 1). To arrive at the monuments, leave the train station by the underpass. Shortly after arriving at street level you will see a McDonald's restaurant on your left. This restaurant is just out of view to the left of Einstein's bust in Figure 1. Einstein was not the only famous scientist to live in Ulm. Johannes Kepler (1571-1630) stayed in Ulm during 1626/27 to assist in the printing of the tabulae Rudolphinae which he had finished in Linz in 1624. Begun by Tycho Brahe, they replaced the "Alfonsinean" and "Prutenean" tables, and were to become the most important auxiliary material for astronomical calculations in the forthcoming centuries. During his stay in Ulm, Kepler had to make money for living expenses, so he accepted a job offered by the city of Ulm to create the famous "Ulmer Kessel" (Fig. 2). In this kettle he unified all the length, weight, and

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2 9 1993 Springer-Verlag New York

volume measurements which were in use in Ulm at that time in order to protect them from manipulations by unethical merchants. The kettle can be seen in the Ulm Museum. This museum is not too far from the beautiful Rathaus (worth a visit in its o w n right), which in turn is a short walk from the M~inster. On the wall of the Rathaus one can see a memorial plaque dedicated to Kepler (Fig. 3). As is well known, Kepler worked during the last years of his life in the city of Sagan for Albrecht Wallenstein, the Duke of Friendland. The famous General of the Empire had also been in Ulm, and a memorial tablet can be found at the Weinhof near the Rathaus (Fig. 4). It is generally believed that Kepler was working for Wallenstein as an astrologer, but this is fallacious. What Kepler actually did was to make his planetary data available to Wallenstein's astrologers---it was they, and not Kepler, who made astrological predictions of the most propitious startling times for some of the battles of the 30-year War (1618-1648). Finally, we note that Ulm's theater is the oldest city theater in Germany, dating from 1641. One of the most famous musicians of all time (and the most recorded), Herbert von Karajan, conducted the Ulm orchestra from 1929 to 1934, his first professional job. Music was significant both to Einstein, an accomplished violinist, and to Kepler, who while working for Wallenstein developed an astronomical model in which planetary orbits were related to the Pythagorean ratios of musical tones (hence "Music of the Spheres"). The data on the birth of Albert Einstein are found in the book Subtle is the Lord by Abraham Pais (Oxford University Press, 1982). The data on Kepler and Wallenstein are from Johannes Kepler by Walther Gerlach and Martha List (Piper, Mfinchen, 1987).

Figure 3. Memorial tablet on the wall of the Rathaus, commemorating Kepler's stay in Ulm.

Abteilung Mathematik II Helmholtzstr. 18 University of Ulm Ulm, Germany Abteilung Mathematische Physik Albert Einstein-Allee 11 University of Ulm Ulm, Germany Center for Transport Theory and Mathematical Physics Virginia Polytechnic Institute and State University Blacksburg, VA 24061-0435 USA

Figure 4. Memorial tablet dedicated to Wallenstein on the side of the Weinhof.

Roman Dodecahedra Benno Artmann

Facts

Conjecture

Bronze dodecahedra were popular in Roman Imperial times. More than 50 of them, mostly from the third or fourth century A.D., can be seen in varius museums in Western Europe. Most of them come from the northeastern part of France, from Switzerland, or from the Roman parts of Germany along the Rhine river. They are generally of about fist size or a little bigger, and in the vast majority of cases look similar to the one in Figure 1, which is displayed in the Landesmuseum in Mainz (Germany). They are hollow with circular holes of differing sizes in the faces and with knobs at their vertices. Many of them have impressed rings around the holes. One specimen has been excavated from a female burial, which is important for archeologists. There are a few exceptions to the rule. The first one was described by Lindemann (of ~r) in 1896 [1]. He reports on a dodecahedron cut from soapstone with unintelligible markings on the faces, which was found in northern Italy. It is supposedly of Etruscan origin, dating from about 500 B.c. The precise circumstances of its excavation have been lost and we have no way of being certain of Lindemann's claims. There are, however, two Etruscan bronze dodecahedra in the Museum of Antiquities in Perugia (Italy). They are mounted on bronze sticks and are without any further decorations of mathematical nature. Another singular example has been excavated quite recently in Geneva (Switzerland); see Figure 2. Its edge length is about 1.5 cm. Its faces are made of silver and are inscribed with the names of the signs of the zodiac. The core is solid lead.

The meaning of these dodecahedra is an open problem in archeology. Various hypotheses have been proposed, some of them quite fanciful, like a "surveying instrument," but nothing is k n o w n for certain. I think two conjectures look promising.

Figure 1. Roman dodecahedron from the Museum of Mainz (Germany).

1. The association of the dodecahedron with fire. The mineral pyrite (FeS2) crystallizes in almost regular dodecahedral shapes. Pyrite has been used--as the Greek p y r = fire suggests in the Stone and Bronze Ages as a fire starter. (This is well-known in archeology; see [3].) When hit with a stone, pyrite gives relatively long-lasting sparks which are caught by tinder. Hence, there is "fire inside a dodecahedron." F. H. Thompson [4] and others think of the Roman dodecahedra as a sort of candlestick. 2. The dodecahedron from Geneva strongly suggests some connections to the zodiac. Deonna [5] quotes Plutarch as saying that the dodecahedron is a sort of image of the zodiac or the year because all three have twelve parts.

Figure 2. Dodecahedron from Geneva [21.

Illustrations from Saint-Venant, 1907. Mathematics and Philosophy

Please H e l p

As far as we know, the dodecahedron enters mathematics about 480 B.c. One of Pythagoras's students was Hippasus of Metapontum, w h o belonged to a group of the master's followers who were later called "the mathematicians." Iamblichus, writing about A.D. 300, says: "About Hippasus in particular they report that he belonged to the Pythagoreans, but because he was the first to make public the secret of the 'sphere of the twelve pentagons,' he perished at sea. The fame of the discoverer, however, was his . . . . " Aristotle mentions in his Metaphysics (A 3, 984a7) that Hippasus regarded fire as the first principle. This may be another reference, however vague, to the connection between fire and the dodecahedron. A hundred years after Hippasus, the mathematician Theaetetus in Plato's academy defined the concept of regularity and proved that there are exactly five regular solids. Theaetetus's constructions are preserved in the last book of Euclid's Elements. (For details, see [6--8].) Plato himself speaks about the five solids in his Timaeus (55a-c). The element fire consists of particles in the shape of tetrahedrons; similarly, air consists of octahedra, water of icosahedra, and earth of cubes. The dodecahedron is used by the Creator for "the whole." Plato's words at this particular place (55c 5/6) are somewhat obscure; he says something to the effect that the Creator drew figures on the faces of the dodecahedron. This could very well point in the direction of the zodiac and the one dodecahedron found in Geneva. For the great majority of the "standard" Roman dodecahedra, however, we have so far no plausible explanation.

R. N o u w e n of the museum in Tongeren (Belgium) is preparing a new list of ancient dodecahedra. The ones from France, Germany, and Switzerland are relatively well-documented. There may, however, be hidden treasures in other museums. If you k n o w of one, please write to me giving as specific information as possible. (And a photo?) Especially to our friends of Etruscan descent in Italy, please have a look at your local museums! References 1. F. Lindemann, Zur Geschichte der Polyeder und der Zahlzeichen, Sitzungsber. math.-phys. Klasse Kgl. Bayerischen Akad. Wiss. XXVI (1896), 625-783. 2. I. Cervi-Brunier, Le dod6ca~dre en argent trouv6 ~ Saint Pierre de Gen~ve, Z. Schweizerische Arch~ologie Kunstgeschichte 42 (1985), 153-156. 3. M. Ebert, Reallexikon der Vorgeschichte. 3 Bd. 4. F. H. Thompson, Dodecahedrons again, The Antiquaries Journal 2 (1970), 93-96. 5. W. Deonna, Les dod~ca~dres gallo-romains en bronze, ajour6s et boulet6s, Bull. Assoc. Pro Aventica 16 (1954), 19-89. 6. E. Sachs, Die fiinf platonischen KOrper, Berlin: (1917). 7. W. C. Waterhouse, The discovery of the regular solids, Arch. Hist. Exact Sci. 9 (1972-1973), 212-221. 8. B. Artmann, Hippasos und das Dodekaeder, Mitt. Math. Sem. GieJ3en 163 (1984), 103-121. Arbeitsgruppe Fachdidaktik Fachbereich Mathematik Technische Hochschule Darmstadt SchloJ3gartenstr. 7 D-6100 Darmstadt Germany

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2 9 1993 Springer-Verlag New York

53

David

Gale*

For the general philosophy of this section see Vol. 13, no. 1 (1991). Contributors to this column who wish an acknowledgment of their contributions should enclose a self-addressed postcard.

The ant lives in the plane which is divided up into cells by a square grid. There are two kinds of cells, white and black (later we will also introduce gray cells). Initially the ant is sitting on a cell, call it the origin, heading in one of the four compass directions. It proceeds to travel from cell to cell according to the following rule: It moves one cell in the direction it is heading. When it lands on a white (black) cell it rotates

The Industrious Ant Most readers of this column are no doubt familiar with John Conway's famous Game of Life. Life is an example of what have come to be called cellular automata. I will discuss here another such automaton, called the ant, and though it is very easy to describe, its behavior is interesting and somewhat mysterious. 2o-

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Figure 9. THEOREM. An ant's trajectory is always unbounded.

Figure 10. its heading 90~ to the right (left) and the cell then reverses its color. This is all there is to it. The game is then to start out with some given distribution of black and white cells and see h o w the ant behaves. The special case where all cells are initially white and the ant is heading, say, east, is typical of what happens generally. In the early stages of its travels, say the first 500 steps, the ant, at intervals, returns to the origin, leaving behind centrally symmetric patterns of black and white cells as shown in Figures 1-4. As far as I know no one has come up with an explanation of w h y these patterns occur. After a while, however, things become rather chaotic for about 10,000 or so steps, but then the ant suddenly seems to make up its mind where it wants to go and heads off resolutely due southwest, leaving behind the periodic pattern shown in Figure 5, which Jim Propp, who first called attention to the phenomenon, calls a highway. On a highway, the ant takes 104 steps, ending up two units southwest of its starting point, and then repeats the process ad

infinitum. If the initial position includes some black cells, the ant will, of course, pursue a different course, but in hundreds of experiments it has always ended up building a diagonal highway in one of the four possible directions. Must this always happen? No one knows, but there is one quite charming result due to X.P. Kong and E.G.D. Cohen.

Proof: First note that the ant rule is reversible. The pattern of black and white cells and the ant's current position and heading determines where it came from. Thus, if a trajectory was bounded, then eventually a black-white pattern would be repeated and, hence, by the preceding observation, the path would have to be periodic, so every cell which was visited would have to be visited infinitely often. N o w the key observation is that the ant's moves are alternately horizontal and vertical. This means that the cells of the plane are partitioned, checkerboard fashion, into h-cells which are always visited horizontally, from the right or left, and v-cells which are always visited vertically, from above or below. N o w consider a "maximal" cell M that was visited by the ant, meaning a cell such that no cell above or to the right of it has been visited. Suppose M is an h-cell. By maximality, it must have been entered from the left and exited downward, so the cell must have been white, but then it turns black, so on the next visit from the left, the ant must go up, contradicting maximality. If M is a v-cell the argument is similar. Neat, don't you think? A variation on the game is to introduce a third type of cell, a gray cell, with the property that when the ant lands on such a cell it simply continues on in its current direction. Gray cells do not change their type but remain gray forever. Note that in this model the unboundedness theorem fails to hold because the partition into h- and v-cells is no longer valid, and, indeed, Cohen has found a fairly simple example of an initial configuration, shown in Figure 6, which yields a trajectory which repeats every 52 steps. One gets some rather pretty patterns by starting with an initial line of gray cells, leaving all remaining cells white. Figures 7-10 show how, initially, the ant seems to be behaving like a spider spinning a web, but gradually asymmetries appear, and highway construction begins at about step 9000. The ant was invented by Chris Langton who studied their behavior in some detail, including cases where several ants are moving simultaneously. THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993 5 5

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Straightedge Constructions The best w a y to learn a n e w subject, as everyone knows, is to teach it. Sometimes, if things go well, one even ends up making an original contribution. I was surprised, however, when this happened to me recently in a course I was designing for fourth- and fifthgrade schoolchildren. It was to be about geometric construction, not with the traditional ruler and compass but with a "markable" straightedge. That is, the only equipment supplied was to be a pencil, an eraser, and a rectangular strip of white cardboard on which one could make (and erase) marks.

Figure 16. ments with care. This last exercise leads to the first example of a construction problem. Given a line segment, h o w can we use the straightedge to find its true midpoint? My proposed solution, which will probably have to be shown to the students, is to construct perpendiculars of equal length at the two end points of the segment and connect them as shown in Figure 15. (I am assuming at this point that the straightedge is a true rectangle so that constructing perpendiculars is immediate. I will return to this later.) This can lead to some significant discussion. One asks the students to explain h o w they know that this construction really does find the midpoint, which leads into the important subject of symmetry. (As an interesting digression, one may change the rules and allow the students to use an ordinary sheet of paper along with their pencils to find the midpoint and see how many of them come up with the idea of marking and folding the sheet appropriately. In my limited experience a child is just as likely as an adult to figure this out.) The next step is to divide a segment in 3 equal parts using the construction in Figure 16. Again there is the opportunity for some pertinent discussion as to w h y one believes the length of the right-hand segment is one-third of the total length. After this, the students could be asked to divide a segment into 5 equal parts on their o w n and so on.

liiiiiiiiiiiil i !i i !i i i i i i i i i i i i i i i i i i i i i i i i i i i i i iJ By w a y of a warm-up, the students were to be asked to use this equipment to determine which of the pair of segments in the Figures 11-13 was longer. As a variation, they could then be asked to make a dot at the point which they consider to be the midpoint of the arrow shaft in Figure 14, and then use the straightedge to find out whether their guess was to the left or right of the true midpoint. The idea here is to see if the students are able to use the straightedge suitably marked to compare the distance of their guessed point from the two ends of the arrow. The well-known optical illusions used here are intended to convince them that there is some point in making these measure56

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993

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One can now do a variety of things. A nice exercise, for example, is to give the students a horizontal line segment on a sheet of paper and ask them to construct the equilateral triangle having this segment as a base. The idea is to construct the perpendicular bisector of the segment and then mark the end points of the segment on the straightedge (Fig. 17). It is n o w necessary to be a bit more explicit about which operations are permitted with the marked straightedge. As illustrated by this last example, one is able to find the intersection of a given line with a circle of given center and given radius, even though one is not able to draw the circle. Otherwise stated, given a line L and a point P not on L and given two marks, A and B, on the straightedge, one may obtain a new point Q on L by placing the straightedge so that A falls on P and B lies on L, assuming, of course, that this is possible. Indeed, all the classical ruler-and-compass constructions are possible because it is not hard to see h o w one can construct "square roots of segments." The next exercise was to show how to bisect an angle (by making marks on each ray at equal distances from the vertex and constructing the perpendiculars at these points and connecting their intersection to the vertex). I was about to write some remarks to the effect that although, as was shown earlier, we can "trisect" segments with our straightedge, the analogous construction for angles has been proved to be impossible, and then I realized that this assertion was false! Herewith a trisection, attributed to Pappus, which uses only a marked straightedge (Figure 18). The angle BOQ is one-third of POQ. Proof: As indicated, the length of segment AB is twice that of OP. N o w draw the line from P to the midpoint M of AB. Then the length of PM is d because--and you can take it from there. So the question is, w h a t is it that the marked straightedge can do that a ruler and compass cannot? The construction above proceeds by choosing the point P arbitrarily and then dropping the perpendicular from P onto OQ. N o w one makes marks, X and Y, 2d units apart on the straightedge, and one must place the straightedge so that (1) it passes through the point O, (2) the mark X lies on PR, and (3) the mark Y lies on PS.

1/2~ u r e 20. 0

This is the crucial maneuver. More generally, one is given a line L and a point P not on L and two marks X and Y on the straightedge. N o w consider the locus traced out by Y when the straightedge is placed so that it passes through P and the point X lies on L. This locus, known as the conchoid of Nichomedes, is given by a 4th-degree equation. Typical graphs are shown in Figure 19. Of course, one is not allowed to draw the conchoid, but one can find its intersection with a line, as in the Pappus construction. Manually, this amounts to placing the straightedge so that two given marks, X and Y, lie on a pair of given lines, and then "sliding" the straightedge, keeping the marks on the lines, until it passes through a given point, a fairly easy operation to perform. N o w the interesting fact is that this conchoid maneuver allows one not only to trisect angles but, in fact, to find the roots, real or complex, of any polynomial of degree at most 4. The trick is to show that it is possible to construct cube roots of positive numbers. Figure 20 shows h o w to do it. The claim is that x is the cube root of a (boldfaced letters are used here for lengths). This can, of course, be proved analytically, although the calculations can get rather tangled if one does not set things up in a convenient manner. There is a slick proof making use of the theorem of Menelaus. In making the construction, one chooses the unit of length, in this case OP. N o w lay off the segment AB of length a, construct the isosceles triangle APB, the point O, and the prolongations of the segments AB and OB. With marks at distance !/2 apart on the straightedge, one then uses these two lines together with the point P to locate the point Q by means of the "conchoid maneuver." Finally, using trisections and cube roots, one is able THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993 ~ 7

to take cube roots of complex numbers (de Moivre's theorem), and these together with square roots are sufficient to find the roots of any polynomial of degree at most 4. The point of all this is the observation mentioned earlier that using this seemingly primitive tool one can make constructions which are impossible with the classical ruler and compass. Thus, for example, from Galois theory one sees that it is possible to construct a regular heptagon as well as a 13-gon and a 19-gon, but not an 11-gon. In general one can construct a p-gon for those primes p such that p - 1 has only 2 and 3 as prime factors. Returning n o w to the subject of constructing perpendiculars, one may ask whether the straightedge needs to be equipped with built-in right angles. Suppose, for example, it looked like this:

Can one using only such a straightedge erect perpendiculars? It turns out one can. Namely, to erect a perpendicular to a given line at a given point . . . . but on second thought, let me leave this construction as an exercise. My solution involves drawing two auxiliary lines and two auxiliary points, and making two marks on the straightedge. On the other hand, to drop a perpendicular from a given point to a given line costs me four lines and three straightedge marks, but no additional points. As a further diversion, suppose one is to draw a line through two points A and B whose distance apart is greater than the length of the straightedge. I will assume that by taking careful aim one can draw a line from A which misses B by no more than the length of the straightedge. I then have a somewhat cumbersome construction; I would welcome and publish any tidier ones. Needless to say, none of the foregoing results is new, with the possible exception of the very elementary facts of the preceding paragraph. One source for the material is an interesting book by the Danish geometer J. Hjelmslev called Geometriske Eksperimenter which has been translated into German as a Beihefte to the Zeitschrifl fiir Mathematischen und Naturwissenschafllichen Unterricht, 1915. According to Hjelmslev, the cube-root construction given above appears without proof in Newton's Arithmetica Universalis (Cambridge, 1707), but Hjelmslev says Newton's proof is only a slight variation on one given by Nichomedes. I suppose one could carry this further and consider constructions which can be made using both a marked straightedge and a compass. This would produce loci of degree at most 8. Perhaps such studies have been made. Any information on the subject will be gratefully received and subsequently reported. 58

THE MATHEMATICAL INTELUGENCER VOL. 15, NO. 2, 1993

Why the Ant Trails Look So Straight and Nice* Alfred M. Bruckstein

Ira "'pioneer" ant shows the way to the food along a random path it marked, and other ants follow in a row, each ant pursuing the one in front of it, their path becomes a straight line connecting the anthill and the food location.

Introduction in the chapter titled "The Amateur Scientist" of Surely You're Joking, Mr. Feynman!, Feynman [1] describes a series of experiments he has done to study the behavior of ants and, in particular, the way they communicate the information regarding the location of food. After realizing that ants can leave some sort of trail on the ground, he goes on to ask a very interesting question:

Suppose an ant finds some food by walking around at random. It then traces a wiggly path back to the anthill and does what in the entomology literature is called "group recruitment" (see, for example, Sudd and Franks [2], p. 113, or Holldobler and Wilson [3], p. 265): "Follow me!" the pioneer ant tells the others, and they do, one after the other, at some small distance, so that each ant walks straight toward the one in front of it. With this rule of pursuit, the ants' paths converge, for any initial trace, to the straight line connecting the anthill with the food.

One question that I wondered about was why the ant trails look so straight and nice. The ants look as if they know what they are doing, as if they have a good sense of geometry. Yet the experiments that I did to try to demonstrate their sense of geometry didn't work. (p. 79) After some further experimentation, Feynman did find an explanation: The ants try to follow the original randomly found and marked path, but they coast on the wiggly path and leave it here and there, soon to find the trail again. Each ant straightens the trail a bit, and their collective effort makes up for their lack of any sense of geometry. Attempts to formalize this argument raise many questions: What determines the points where coasting occurs? What happens when an ant fails to return to the trail? etc. In this short article a simple model of local interaction is proposed that could lead ants, or other natural or artificial creatures with little or no sense of global geometry, to find the straight path from the anthill to the food.

* This w o r k w a s s u p p o r t e d in part by the F u n d for Promotion of Research at the Technion, Haifa, Israel. THE MATHEMATICALINTELLIGENCERVOL. 15, NO. 2 9 1993 Springer-VerlagN e w York 59

Formalization o f the Pursuit P r o b l e m

Proof. First note that due to the type of pursuit, the

Suppose that the pioneer ant, let us call it A o, follows an arbitrary piecewise regular path, from the ant hill located at (0, 0), to the food, located at (L, 0), with unit speed. The initial path, Po, is described by the parametric curve Po(t) = [xo(t), yo(t)], w h e r e time, or arclength, is the parameter, running from 0 to To = Lo, the length of the initial path. For t > T o, Po(t) = (L, 0). At time "q I> ~ > 0 the next ant, A1, starts walking so that its velocity vector is always pointing toward the first ant. The second ant's path, P1, will be described b y Pl(t) = [xl(t), yl(t)], w h e r e t will run from "rI to T 1 = "rl + L1, L1 being the length of the pursuit path. At time "r2 I> "q + 8, ant A 2 will begin following ant A 1 the same way, describing the next path, via P2(t), and so on (see Fig. 1). If an ant catches up with the one ahead, it joins the pursued ant on its path. The path of ant A n + 1 is the solution of the following differential equation, implied by the rule of pursuit:

distance b e t w e e n chaser and p u r s u e d is a nonincreasing function of time. Indeed, since An + 1 always directs its velocity toward A n, the best A n could do is to evade optimally, i.e., to use all its s p e e d to run directly a w a y from its chaser, and this w o u l d keep the distance between t h e m constant. As s h o w n in Figure 2, denoting by ~,(t) the time-varying angle b e t w e e n the velocity of A n and that of A n + 1, we have at time t > "rn + 1

d

d

Denote the initial distance b e t w e e n A. + 1 and A n by

Ai(n + 1) = IPn+1(%+1) - P.(rn+31

Af(n + 1) = IPn+I(Tn)

(1)

w h e r e tP, +a(t) - Pn(t)l denotes the Euclidean distance b e t w e e n the points Pn+ l(t) and P,(t). Equation (1) is valid for t t> "rn+ 1, for IPn+l(t) -Pn(t)l ~ 0. The initial condition is P,+l('rn+0 = (0, 0), and if the pursuer catches u p with, or "captures," the p u r s u e d , they walk together, from that m o m e n t on following the path pn. Given the initial path P0, and the time sequence %, for n = 1,2,3 . . . . . w e have defined a sequence of pursuit paths Pn, for all n E N, all starting at (0, 0) and ending at (L, 0). We claim that the paths Pn become smoother and straighter with increasing n, converging to the straight line from source to destination.

(.qo>

I

Tn+ 1 --

Tn =

60

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993

~

Jr-

(5)

En,

J

A htl

/

t%.~,J

Figure 1. A.+ 1 starts following A..

(4)

Af(n + 1) ~ IP.+l(t) - P,(t)[ <<-ai(n + 1)

J

Figure 2. Local pursuit geometry.

I

- Pn(Tn)I.

Then from (2) it follows for all t e [% + 1, Tn] that

T H E O R E M . The sequence of pursuit paths, Pn, converges

to the straight segment connecting (0, O) to (L, 0).

(3)

and the distance b e t w e e n the chaser and p u r s u e d w h e n An reaches the destination (at time Tn = "rn + Ln) by

1

d-~ P,+fft) = IPn+l(t) - P.(t)l (Pn(t) - P.+l(t)),

(2)

d--t IP"+l(t) - P"(t)l = cos(~n(t)) - 1 ~< O.

I

since it was assumed that the sequence % obeys % + 1 -- % = 8 + % for some % I> 0. From the problem definition, it is clear that the time it takes ant A , + 1 to complete the journey from the origin to the destination, a time equal to the length of its path, is L,+I = L. + ~'.

-

"r.+ 1 + Af(n + I).

(6)

Note that if A.+I intercepts A., we have ~r + 1) = 0. The infinite sequence of positive numbers L, is therefore nonincreasing, hence convergent to some L~. Because all % + 1 - .r. are not less than 8, the number of times an ant can capture the one it follows must be finite, thus, for all n > N 0, ~/(n + 1) = ('r,+ 1 - %) + (L.+I - L,).

(7)

As (L.+I - L.) --~ 0, the sequence Af(n + 1) will track the sequence 8 + % very precisely, implying by (5) that

d

1 Y n + l ( t ) = ~- [ Y m a x i m u m ( ' )

-

~ln+l(t)],

~n+l(0)

-- 0

(12) evaluated at time L0, longer than L,+ 1, provides an upper bound on the excursions of 9,+ l(t) 9 Hence, Yma•

+ 1 ) ~ Yma•

- e -LdK) (13)

and the maximal projection of P, + 1 is less than that of P, by a constant factor strictly less than one, implying that it decreases exponentially to zero. An identical argument applied to the minimal projection yields the same conclusion for the negative excursions of yn + l(t). This shows that in the limit the projection of P,(t) on the y-axis converges to the point (0, 0). Q.E.D.

Other Interesting Pursuit Problems

The problem we dealt with in the previous section requires one to prove a result concerning the limiting behavior of solutions of a differential equation describAi(n ) - 5f(n) --) 0 as n ~ ~. (8) ing a sequence of pursuit paths. The problem of actuFrom this, it follows that ally solving the differential equation (1) for the pursuit path of the chaser, given the trajectory of the evader, is f r . [1 - cos(~,(~))] at--) 0 (9) a well-known difficult problem in nonlinear differenJTn+I tial equations. The problem has been solved in only and the angle function ~,(t), defined for all n > No two cases: the case of "linear pursuit," i.e., when the where captures do not occur, converges to 0 almost pursued object travels on a line at constant speed; and everywhere. Because one can easily show that the an- when the path of the pursued is a circle, see, e.g., gle functions v~,(t) have uniformly b o u n d e d deriva- Boole [5], p. 251, and Davis [5], p. 113. tives for n > N 0, the angle converges to zero everyThe history of pursuit problems is very interesting; where. Thus, in the limit, A , walks straight away from some even claim that Leonardo da Vinci posed the first A,+ 1, until it reaches the destination (L, 0), and the question concerning pursuit paths. Attempts to solve path of A, + 1 will be a straight line, in pursuit of A,. (Of the differential equations describing pursuit paths date course, in case % + 1 ~> T,, the pursuit path becomes a back to the eighteenth century (for some historical restraight line right away!) marks, see Morley [6]). More recently, in the context of Another way to prove that the pursuit paths ap- differential games, pursuit games have been the subproach the straight line from (0, 0) to (L, 0) is the fol- ject of much research [7]. lowing: Consider the differential equation describing Various "pursuit puzzles" have swept the mathethe evolution of y,+l(t) for t E [~',+1, T,]: matical communities of the world. A very famous one deals with tactics of pursuit in a closed arena, where a d 1 lion wants to catch a man having the same speed, see d--t yn+l(t) = ipn+l(t) _ Pn(t)l [yn(t) - yn+l(t)], Littlewood [8], p. 114, and Croft and Stewart [9]. Another well-known pursuit puzzle is to analyze the Yn+l('rn+1) = O. (10) problem in which four dogs [10], D1, D2, D3, and D4 Denote by Ym~-n~m(n) and by Yminimum(Y/) the maximal or in recent reincarnations, four bugs or "turtles" (unand minimal values of the y-projection of the path P,. der the influence of the LOGO programming language [11])---located at the vertices of a square, start moving Then clearly with the same speed, each one following the dog/bug/ d 1 turtle to its right. In this problem, one easily realizes dt yn+i(t) ~ Af(n + 1) [Ymaximum(n) - yn+i(t)] that, due to symmetry, each evader moves perpendicularly to the direction of its chaser, hence in Equation 1 ~. [Ymaximum(r/) - yn+fft)], (11) (2) one has cos 9 = 0 during the entire pursuit. Thus, the total path of pursuit, as the players spiral toward because for a large enough n, 8 + % > K for some the point of encounter at the center of the square, is strictly positive constant K < 8. By a well-known com- equal to the length of its edge. This "circular pursuit" parison theorem (see, e.g., Arnold [4]), the increasing problem may be generalized to involve different numbers of dogs (or ants) at arbitrary starting positions (see solution of the equation THE MATHEMATICAL INTELLIGENCER VOL. 15, NO, 2, 1993

61

&

Figure 3. Cyclic pursuit.

Fig. 3), with variations in speed and local pursuit laws, and so on. Graphical means of solving pursuit problems have always been popular [5,6] and became even more so after the development of computer graphics and LOGO-type programming languages. In a book entitled Turtle Geometry, popularizing LOGO as a medium for exploring mathematics, the authors encourage readers to play with pursuit projects, and ask, among other questions: what kinds of initial conditions, speeds, and following mechanisms will ensure that all bugs eventually meet at one point? [11], p. 76. For a simple circular pursuit problem, one can see that the following result holds. THEOREM. If K players start a circular pursuit from any initial configuration, they always converge to a point of encounter. Proof Sketch. We assume, as before, that w h e n a chaser catches the player it pursues, they unite and continue chasing the (circularly) next player. Call the players D1, D2 . . . . . DK. Define their positions in time by Pi(t) and the distances between chasers and pursued by Ai(t ) -- IPi(t) - Pi+l(t)l, where PK+I(t) ~-- Pl(t). By the arguments of the previous section, it is clear that the distances Ai(t) are differentiable, nonincreasing functions of time, hence they converge to some nonnegative limit, Ai~. Suppose some of these limits are nonzero. Then the corresponding players tend to move in the same direction. Applying this argument "circularly," the players must approach a "limit" configuration in which they are all moving in the same direction. However, the last player must chase the first one, hence it must be moving in the opposite direction, strictly reducing AK(t ). Thus, all pursuit paths must converge to the same point. Q.E.D. Discussion The ant-pursuit problem analyzed indicates that a simple interaction of players can solve the problem of finding the optimal path between the source and destination. This result generalizes to pursuit in higherdimensional space too. One could also try to find other 62

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993

trail-following models, based on simple local interaction, that provide convergence to a straight path. It is of much interest to have results of this type, showing that globally optimal solutions for navigation problems can be obtained as a result of myopic cooperation between simple agents or processors. We have also seen that a simple law of circular pursuit is enough to ensure that all agents will eventually get together. However, their pursuit paths may be quite complex. The analysis of global behavior that results from simple and local interaction rules is a fascinating subject of investigation and may even lead to a better understanding of natural and artificial animal colony behavior (see, e.g., Braitenberg [12]). Such ideas could also be of use in problems arising in robotics, for example. As a concluding remark, it seems appropriate to quote a saying from long ago [13]: "Go to the ant, thou sluggard; consider her ways, and be wise." Acknowledgments

It is my pleasure to thank Nir Cohen, Nachum Kiryati, Michael Heymann, Ciprian Foia~, and Allen Tannenbaum for interesting discussions on ants and related topics, and Aviezri Fraenkel for calling my attention to the very suitable final quotation. References

1. R. P. Feynman, Surely You're Joking, Mr. Feynman!, Toronto: Bantam Books (1985). 2. J. H. Sudd and N. R. Franks, The Behavioral Ecology of Ants, New York: Chapman and Hall (1987). 3. B. Holldobler and E. O. Wilson, The Ants, New York: Springer-Verlag (1990). 4. V. I. Arnold, Ordinary Differential Equations, Boston: M.I.T. Press (1973). 5. G. Boole, A Treatise on Differential Equations, 5th ed., London (1859), or H. T. Davis, Introduction to Nonlinear Differential and Integral Equations, New York: Dover (1962). 6. F. V. Morley, A curve of pursuit, Amer. Math. Monthly 28 (1921), 54-61. 7. O. Hajek, Pursuit Games: An Introduction to the Theory and Applications of Differential Games of Pursuit and Evasion, New York: Academic Press (1975). 8. J. E. Littlewood, Littlewood's Miscellany, (B. Bollobas, ed.), Cambridge: Cambridge University Press (1986). 9. H. T. Croft, Lion and man: A postscript, J. London Math. Soc. 39 (1964), 385-390; and I. Stewart, All paths lead away from Rome, Scientific American (April 1992). 10. H. Steinhaus, Mathematical Snapshots, Third American Edition, New York: Oxford University Press (1969), 136. 11. H. Abelson and A. A. diSessa, Turtle Geometry, Cambridge, MA: M.I.T. Press (1980). 12. V. Braitenberg, "Vehicles," Experiments in Synthetic Psychology, Cambridge, MA: M.I.T. Press (1984). 13. The Bible, Proverbs, vi, 6. Computer Science Department Technion Haifa Israel

Jet Wimp* A Course in Number Theory and Cryptography by Neal Koblitz N e w York: Springer-Veflag, 1987. ii + 210 pp. US $35.00. ISBN 0-387-96576-9

Reviewed by J. H. Loxton " A n d what do you do for a living?" asked Acacia above the party babel. This question, as it always did, provided Bondi with an anxious moment, but, feeling bold, he replied, "I do mathematics at the University." In fact, Bondi had spent a fretful day arguing about the relative merits of analysts and algebraists for a lectureship, and hoped that the conversation would follow the usual route, reflecting on the difficulties of school mathematics, particularly algebra, and moving on to less emotionally sensitive territory. Bondi was not to escape easily. "I always wanted to find out more about mathematics. Tell me, what is the use of it? Don't we know all we need to know for building bridges and making explosions?" Acacia probed. Bondi bridled, then attacked by quoting from Hardy's Apology:

ematicians of yours already done everything worthwhile? Is there anything n e w in mathematics today?'" In less time than it takes to say tautological, Bondi retorted, "Computational number theory." Elementary 2 number theory, Bondi tried to explain, revolves around computation and delights in massive sums done on the back of an envelope. It is an ancient tradition and its problems have become household words such as "perfect numbers" and "Fermat's Last T h e o r e m . " These are w h o l e s o m e problems about whole numbers, originating in the old days w h e n arithmetic was only done with whole numbers. A less well-known example is the infamous cattle problem attributed to Archimedes: "Compute, O friend, the host of the oxen of the sun . . . . '" The problem he posed comes down to finding the numbers W, X, Y, Z, of white, black, spotted, and yellow bulls and the numbers w, x, y, z, of white, black, spotted and yellow cows, naturally all whole numbers, such that 5 W = 7X + Z, o

9 X = ~-~Y ,'u + Z,

13 7 It is undeniable that a good deal of elementary mathematY = + z, w + x), ics--and I use the word "elementary" in the sense in which professional mathematicians use it, in which it in9 11 cludes, for example, a fair working knowledge of the difX = + y), y = y 6 ( z + z), ferential and integral calculus--has considerable practical utility. These parts of mathematics are, on the whole, 13 rather dull; they are just the parts which have least aesz = ~ ( W + w), W + X = square, thetic value. The "real" mathematics of the "real" mathematicians, the mathematics of Fermat and Euler and Gauss Y + Z = triangular number. and Abel and Riemann, is almost wholly "useless" (and this is as true of "applied" as of "pure" mathematics). It is The solutions can be found by solving a "Pellian" not possible to justify the life of any genuine professional mathematician on the grounds of the "utility" of his work. 1 equation (u 2 - 4729494v 2 = 1). The answers are huge---several hundred thousand digits. 3 Elsewhere in the Apology, Hardy writes, "A matheTraditionally, a course in elementary number theory matician, like a painter, is a maker of patterns"; "rr = begins with the fundamental theorem of arithmetic: 3.14159265 may be "enough for an engineer, and he every number has a unique decomposition as a prodcan be perfectly happy without the rest," but there is uct of prime numbers. Then, the greatest common distill much serious mathematics and unresolved pattern visor (a, b) of two numbers a and b can be obtained by in ,rr that transcends practical utility. Bondi happily inspection, for example, expanded on this theme, mentioning the many digits 12183 = 3 x 31 • 131, 15327 = 32 • 13 • 131, of w now known, and still other numbers such as e and ~/. (12183, 15327) = 3 • 131 = 393. But Acacia attacked again. "Haven't those real mathi G. H. Hardy, A Mathematician's Apology (Cambridge University Press, 1940). The book u n d e r review proves once again that it is impossible to write about applied n u m b e r theory without quoting from the Apology. * Column Editor's address: Department of Mathematics, Drexel University, Philadelphia, PA 19104 USA.

2 This equally traditional use of elementary has almost exactly the opposite connotation to Hardy's elementary mathematics since it attempts to exclude the calculus and traditionally makes no claims of practical utility. 3 A. H. Beiler, Recreations in the Theory of Numbers (Dover, 1964) describes the attempt to carry this through without computers.

THE MATHEMATICALINTELLIGENCERVOL. 15, NO. 2 9 1993Springer-VerlagNew York 63

However, it is not so common to have large numbers presented in factored form, and factorisation of large numbers is difficult. The first interesting algorithm of the subject is the Euclidean algorithm which makes it possible to find the greatest common divisor of two large numbers efficiently without factorising them, and, as an application, gives the solutions of linear congruences of the form ax =- b mod m. The highlight of the course is the theory of quadratic congruences and, in particular, Gauss's Theorema Aureum, the law of quadratic reciprocity. Suppose p and q are odd prime numbers. The Legendre symbol is defined by

mod i ol ble =

-

if x2 ~ p mod q is insoluble.

The law of quadratic reciprocity asserts that

An efficient process akin to the Euclidean algorithm based on this law can be used to evaluate Legendre symbols and so to decide the solubility of quadratic congruences. Another variant of the Euclidean algorithm leads to continued fractions and thence to the solution of the Pellian equation. Koblitz covers all this in his first 52 pages, with asides and exercises taking in much more. Along the way, he gives us a smattering of group theory and the theory of finite fields. The characteristics of Hardy's real mathematics are here in abundance, from the beauty of quadratic reciprocity to the unexpected intervention of complex exponential sums in the reciprocity proof. However, according to the well-known platitude, the digital computer has affected every part of our lives. The computer has certainly infected mathematics and influenced the questions which are being studied, and it has transformed elementary number theory into computational number theory. Koblitz's account of the elements of number theory gives a good illustration of this trend, emphasising the algorithms and their computational complexity. Incidentally, the approach provides opportunities for estimation and practising the basic techniques of analysis as well as algebra. Even in this well-filled field, the search for efficiency and the stress on computation occasionally produce n e w solutions that improve on the classical constructions. For example, an exercise in Chapter I introduces the greatest common divisor algorithm of Brent and Kung, which is faster than the Euclidean algorithm because it requires only subtractions and no multiplication or division. The basic observation is that (a, b) = 2(a/2, b/2) if a and b are both even (a/2, b) if a is even and b is odd (a - b,b) i f a a n d b a r e b o t h o d d a n d a > b . 64 THE MATHEMATICALINTELLIGENCERVOL. 15, NO. 2, 1993

(Division by 2 does not count as division in binary arithmetic.) The price for the extra speed is that this algorithm does not lead to the solution of linear congruences as does the Euclidean algorithm. The BrentKung algorithm was essential in recent work on the security of some of the modern cryptosystems. 4 At this point Carella, a third party-goer, drifted across to Acacia and gushed conspiratorially: "I just heard this marvellous secret. Promise not to let anyone know and I'll tell it to you. 24 14 20 17 4 26 9 14 10 8 13 6"! Bondi thought that he had about as much information as he usually gleaned from Carella. But Acacia could not bear for someone to keep a secret from her, even if unintentionally, and demanded: "Bondi, here is a chance for you to do something really useful with your mathematics. What is the secret?" "24 14 20 17 4 26 9 14 10 8 13 6," said Bondi. s Obviously, cryptography has something to do with mathematics which goes deeper than its juxtaposition with number theory in the title of a book. However, until recently, cryptography has been held in secret by the military and diplomats. The public uses of cryptography have been mostly in puzzles and recreations and in neither case is there much call for mathematical sophistication. This may account for the variety of opinions held of the value of cryptography. A man is crazy who writes a secret in any other way than one which will conceal it from the vulgar. (Roger Bacon, c. 1250) Few persons can be made to believe that it is not quite an easy thing to invent a method of secret writing which shall baffle investigation. Yet it may be roundly asserted that human ingenuity cannot concoct a cipher which human ingenuity cannot resolve. (Edgar Allan Poe, 1843) Ceux qui se vantent de lire lettres chiffr~es sont de plus grands charlatans que ceux qui se vanteraient d'entendre une langue qu'ils n'ont point apprise. (Voltaire, 1769) Until recently, users of cryptosystems have believed in the invulnerability of their systems, and the systems have been routinely broken with the help of a combination of statistical analysis and coincidence. The codebreakers enjoyed the upper hand. 6 Again, the computer has changed all that. As Gilles Brassard points out, 7 this is ironic because the first electronic computer was built for the specific purpose of analysing the Enigma code. Nowadays, the computer has made it possible to design practical codes which have so far 4 B.-Z. Chor, Two Issues in Public Key Cryptography: RSA Bit Security and a New Knapsack Type System (MIT Press, 1985). s Admittedly, not such a concise reply as "42," the answer to the Ultimate Question of Life, the Universe and Everything. [D. Adams, The Hitch-Hiker's Guide to the Galaxy (Pan, 1979).] 6 D. Kahn, The Codebreakers: the Story of Secret Writing (Macmillan, 1967) describes the great successes of cryptanalysis including the breaking of the German Enigma and the Japanese purple codes in World War II. 7 G. Brassard, Modern Cryptology (Springer-Verlag, 1988).

resisted cryptanalysis. The computer has also brought about a number of interesting cryptographic situations in the public domain and changed the nature of the subject. Traditional cryptosystems are built on transpositions and substitutions. The security depends on a secret key exchanged between the parties. The ultimate example is the one-time pad invented by Veruam in 1926. To encipher a message M --- (M 1, M 2. . . . ), say, already numerically encoded, generate a random sequence Z = (Z 1, Z2. . . . ) of the same length as M and add this key to the message to get the ciphertext C = (C1, C2. . . . ) with C i = M i + Zi. Because the key is random, the ciphertext reveals no information about the message and so the one-time pad is perfectly secure. The code illustrates Shannon's theorem that a perfectly secure system must have at least as many keys as messages. Unfortunately, the one-time pad is not really practical except in very special circumstances. (It is rumoured that it has been used on the Moscow-Washington hotline.) Some of the simpler cryptosystems may not provide much secrecy, but they do lead to interesting mathematical exercises. One such system and among the earliest mathematical accounts of cryptography is Hill's encipherment by matrices. 8 Suppose we have an N-letter alphabet. Split the message into blocks of length k and identify each block with a k-tuple of integers modulo N. The key is a k • k matrix A say, and each block M of the message is enciphered as C = AM. The decipherment leads to nice exercises in congruences and linear algebra. Practical cryptosystems usually cannot rely on very long keys, but the use of fast computers makes it possible to use much more complex algorithms. The most complex, if not the ultimate, example is the Data Encryption Standard (DES). The basis of DES is the Lucifer cryptosystem designed by IBM. It works as follows. Break the message into blocks of length 2n. Let M be a typical block and divide it into two equal parts M = (M0, Ms). The key k is a vector which determines subkeys k 1, . . . , k d_ 2. For 2 ~ i ~ d, define recursively Mi = Mi_ 2 + f(ki- 1, Mi- 1) where f is some fixed nonlinear transformation and the algebra is carried out modulo 2. The ciphertext is C = (Me-1, Me). To recover M from C is just a matter of using the keys in the reverse order: Md-2 = Md + f(kd-1, Md-1), M a - 3 = M a - 1 + f(kd-2, Md-2),

and so on, until the original message M = (M 0, M1) is reached. If, for example, f is determined by polynomial equations, then the problem of determining the key from a knowledge of M and C amounts to solving algebraic equations in the key variables modulo 2. This is a L. S. Hill, Concerning certain linear transformation apparatus of cryptography, Amer. Math, Monthly 38 (1931), 135-154.

an NP-hard problem, which gives some grounds for believing that the code might be hard to break. A modified version of the idea was used as the basis of the Data Encryption Standard certified by the National Bureau of Standards in the United States in 1977 for the protection of valuable and sensitive data. DES had a key with 56 bits and there has been much controversy over its adequacy. DES was nearly decertified in 1988, but is still widely used. Despite all the work done on DES, it seems no one has found a shortcut to make cryptanalysis any easier than conducting an exhaustive search for the key, but the 2SS-word key space no longer seems large enough to stand up to modern computers. The information society and the computer age have given rise to innumerable applications for cryptography besides the original motivation of secure communication. These include authentication, digital signatures, user identification, protection of privacy, and more. 9 Distribution of the secret key between the parties is a problem in all the traditional cryptosystems. In 1976, Diffie and Hellman postulated a trapdoor function to solve this problem with an asymmetrical cryptosystem as follows. A user, Acacia, say, has a public encoding function E A and a secret decoding function D A such that E A and D A are easy to compute, E A and D A are inverse functions, and E A is a trapdoor, that is, knowledge of E A does not reveal information about D A. With such a system, Bondi can send a secret message to Acacia by the cryptogram E A ( M ) using the public function E A without first exchanging secret information, and Acacia can decode M = DAEA(M). Moreover, if Bondi sends the cryptogram EADB(M ) which Acacia can decipher because M = EBDAEADB(M ) and the function EB is public, then Acacia knows that the message must have come from Bondi because only Bondi knows the secret function D B. By now, yet another party-goer, Dorrigo, had joined the group and scented the opportunity to gather some free advice. "I was talking to my stockbroker the other day about a nice little stock, and I discovered he charged me commission for buying some even though they went down and I don't want them now. Could I use your trap door to make sure my broker didn't act without genuine authorisation?" Bondi thought he would make up for the free advice by adding: "And your stockbroker could use the system to prove that you had given him genuine instructions too." The more philosophically inclined Acacia hurried to fill the pause. "These trap doors seem to be logically impossible. After all, if I know h o w to encipher then surely I must know how to decipher. ''1~ 9 G. Brassard, Modern Cryptology. i0 D. Kahn, The Codebreakers, cites a historical precedent for this arg u m e n t from the autobiography of Casanova (1757). THE MATHEMATICALINTELLIGENCERVOL. 15, NO. 2, 1993 65

This is, of course, where number theory returns to the fore. Number theory is full of hard problems which have resisted efficient solution even though in many cases they can be solved in principle by exhaustive search. These problems have been the most promising source of trapdoor functions. Knowledge of the function yields information about its inverse only with an impractical amount of computation. One of the first ideas and still the most popular is the RSA cryptosystem, named for its inventors, Rivest, Shamir, and Adleman (1978). It is based on the fact that it is easy to find very large prime numbers but difficult to factor very large numbers. To illustrate how the functions EA and D A are chosen, Bondi asked Acacia to choose two large secret primes, p and q, of about 100 digits each. The product n = pq is part of the public key. The only known method to recover the secret decoding function, Bondi informed her, is to factor the 200-digit number n. Diffie and H e h m a n noted that raising to a power in a large finite field is an easy operation, but its inverse, the discrete logarithm, is computationally much more difficult. This fact provides the basis for a trapdoor function. Another early candidate was the knapsack problem proposed by Merkle and Hellman (1978). The basic problem is to pack a knapsack of volume V with a selection of objects of volume vl, v2. . . . without wasting space. In general, this is a computationally difficult problem. However, the suggested cryptosystern depended on a special type of knapsack problem, and the code was broken by Shamir in 1987. More complicated versions of the original idea are still alive. Edgar Allan Poe, mentioned above, boasted that he could solve any cryptogram. G. W. Kulp presented Poe with a 43-word ciphertext. After working on it laboriously, Poe proclaimed it to be a jumble of characters having no meaning whatsoever. It turned out that Kulp's cipher contained a mistake and 15 misprints, making it difficult to solve. 11 Turning this tale to good use, McEliece, also in 1978, proposed a trapdoor function based on introducing random errors into the encoding process. The authorised decoder can recover the message by using an error-correcting code. All these systems have two basic flaws. They are slow, perhaps only a thousandth of the speed of DES. Also, there are no proofs of their security. The various number-theoretic systems have generated intense interest in the search for algorithms for primality testing, factorisation, and the computation of discrete logarithms. Koblitz discusses primallty tests based on pseudoprimes. To cite one: if n is a prime number, then

11 S e e M. Gardner, Mathematical Games, Scientific American (August 1977), 120-124.

66 THE MATHEMATICALINTELLIGENCERVOL. 15, NO. 2, 1993

for any integer b. On the other hand, if n is composite, then the above equation fails for at least half of the choices of b. So, if the equation fails for some choice of b, we know that n must be composite. If it holds for 200, say, random choices of b, then we can declare that n is "probably" prime, with probability 2 -20o that we are wrong. 12 The calculations are very efficient. Adleman, Pomerance, Rumely, Cohen, and Lenstra (1984) have developed a deterministic primality test which is not quite so fast, but can still routinely test 100-digit numbers for primality. It is based on an extension of pseudoprimes to number fields. Factorisation of large numbers is another story. Koblitz describes the traditional machinery (Pollard's rho method, factor bases, and continued fractions) with modern improvements and analyses their running times. In all, Koblitz gives 59 pages on cryptosystems and 38 pages on primality testing and factoring. This outline could serve excellently as the basis for a course in applied and computational number theory. There are a number of books which discuss the previously mentioned topics from various points of view and the comparisons are very revealing. Here, gathered from recently published accounts, are some alternative approaches and some conclusions. The algebraic closure: Childs 13 presents coding theory and cryptography as an illustration and motivation for abstract algebra, emphasising the theory of finite fields and the elements of Galois theory. The book is an excellent example of the well-known (antimilitary?) sentiment of "the general embedded in the concrete.'" The exercises teach the reader how to do computations in finite fields. Perhaps for this reason, the algebra and the number theory fit together like Tweedledum and Tweedledee. On the other hand, Childs does not cover the modern algorithms for factorisation and computing discrete logarithms. The discrete completion: Schroeder 14 is much more discursive. For him, the theme behind the applications of number theory seems to be the discrete Fourier transform. Fourier transforms, convolutions, and spectra feature in the analysis of randomness. The author provides an account of signal propagation and spread spectrum communication using quadratic residues. This leads on to self-similarity, fractals, and art. The book has something for everyone, with all the advantages and disadvantages that go with it. The combinatorial explosion: Welsh is gives what might be called the orthodox account of codes and cryptography. He describes Shannon's information theory and 12 Un ph~nom~ne dont la probabilit6 est 10-so ne se produira donc jamais, ou du moins ne sera jamais observe. (E. Borel) 13 L. Childs, A Concrete Introduction to Higher Algebra (SpringerVerlag, 1979). 14 M. R. Schroeder, Number Theory in Science and Communication (Springer-Verlag, 1986). 15 D. Welsh, Codes and Cryptography (Oxford University Press, 1938).

developments in computational complexity with much less attention to number theory. This is certainly appropriate because Shannon's information theory was based on his previous work in cryptography. However, current trends are to replace the question "Does the cryptanalyst have enough information to decipher a cryptograph?" with the complexity question, "Does the cryptanalyst have enough time?" At present, it seems that the computer has tipped the balance in favour of the encoder. The elliptic universe: "It is possible to write endlessly on elliptic curves. ''16 Koblitz's book concludes with 30 pages on elliptic curves. Very recently, the theory of elliptic curves has been applied both in cryptography and in the theory of factorisation. An elliptic curve is defined by an equation y2 = x3 + ax + b, or more general forms, and the particular interest is in the points on these curves which are defined over a finite field. These points can be turned into a finite group analogous to the groups of residues modulo n which appeared earlier. The advantage is that there is much more flexibility in choosing the group in which to work. Thus, there are analogues of the public-key cryptosystems involving elliptic curves, and there is an elliptic factorisation method of Lenstra which imitates an algorithm of Pollard. Koblitz's account includes a full working knowledge of the basic features of arithmetic on elliptic curves, a tour de force in 30 pages, with examples and exercises to make sure that the reader can do the required computations. Indeed, a feature of Koblitz's book is the wealth of imaginative exercises and the careful and revealing answers. Beyond the fringe: Quantum cryptography, using polarised photons, has been proposed as a means to transmit information with perfect security. 17 The idea is that the uncertainty principle can be invoked to postulate a communication channel on which it is impossible to eavesdrop without a high probability of disturbing the transmission and thereby being detected. Such a system would be secure against unlimited computing power. An experimental system which begins to realise this idea has been announced. Did anyone say that "real" physics should be almost wholly useless? The happy ending: Acacia and Bondi were last heard arguing animatedly about secrecy, texts, and binary division as their hosts cleared the tables and the last of the other guests roared off into the night. School of Mathematics, Physics, Computing and Electronics Macquarie University North Ryde New South Wales, 2109 Australia 16 S. Lang, Elliptic Curves: Diophantine Analysis (Springer-Verlag, 1978). Lang has nearly proved the truth of this assertion. 17 G. Brassard, Modern Cryptology, and C. H. Bennett, F. Bessette, G. Brassard, and L. Savail, Experimental quantum cryptography, Advances in Cryptology-Eurocrypt "90 (Springer-Verlag, 1991), pp. 253-265.

Computers, Chess, and Cognition edited by T. Anthony Marsland and Jonathan Schaeffer New York: Springer-Verlag, 1990. xiii + 323 pp. US$35.00 ISBN 0-387-97415-6. Reviewed by Robert Levinson Thirteen years have passed since the classic book on computer chess, Chess Skill in Man and Machine [1], appeared and seven years since a later expanded edition. In this time, the field of computer chess has made remarkable strides forward. In the early 1980s only the best computers were playing at the master level. Now storebought machines are competing with masters and the best computers are occasionally beating top grandmasters. Only a mere 40 years ago Claude Shannon published a paper [9] laying the foundations for computer chess, and already the long sought goal of artificial intelligence, to defeat the World Chess Champion, is in sight.--Ken Thompson, in the foreword.

Introduction Computers, Chess, and Cognition is an important book. It will be welcomed by all interested in the state of the art in computer-chess research and in particular by those pondering the relationship between this specialized field and the general goals of artificial intelligence and cognitive-science research. This collection of 18 papers, written by the top researchers, has 5 parts and 323 pages including a 14-page bibliography and a thorough index. The book provides a useful overview of significant efforts in computer chess research and suggests what progress will be required for computers eventually to defeat the human World Champion. Obvious omissions from the book are descriptions of Soviet projects [7], commercial chess machines (perhaps not present for proprietary reasons), and recent work in machine learning and artificial intelligence that uses chess as a testbed. But the book was not meant to be comprehensive--it grew directly out of the papers presented at the 6th World Computer Chess Championship. In the preface, the authors enthusiastically declare the inevitable: that a computer will defeat the World Champion (perhaps soon), and that this will mark the achievement of the long-sought-after goal of artificial intelligence, "a field of computer science devoted to creating the illusion of machine intelligence." They acknowledge that this will be done through means largely different from modeling " h u m a n thought processes." Although I agree with the premise that computers will soon be champion and with the conclusion that an important milestone will have been reached, I know that I am not alone in declaring that the potential uses of chess in artificial intelligence and cognitivescience research remain largely untapped. Whereas some artificial intelligence researchers may be satisfied with an "illusion of intelligence," I believe that most of them wish to create "true intelligence," (informally defined as the rational and creative soluTHEMATHEMATICAL1NTELLIGENCERVOL.15, NO. 2, 1993 67

tion of the problems of life, regardless of the means), and those w h o call themselves cognitive scientists would like this to be done by cognitive modeling of human problem-solving. Achievements in computer chess demonstrate strikingly the p o w e r of brute-force search in a wellstructured problem space. I claim that this power may n o w be s u c c e s s f u l l y t r a n s f e r r e d to o t h e r wellstructured domains such as organic chemical synthesis

"'How can computers and humans be good at chess, but in such different ways?" and to automatic theorem-proving. Granted, there is much of abstract mathematics that is beyond current brute-force techniques. Still, I believe that there is a host of problems that are currently considered too "deep" for computers that will soon fall to the application of brute-force search, in particular, to the "selective search" techniques that researchers are developing to narrow the search space. Even now, chess computers are solving many complex middlegames. What is it about the human approach to chess that is so different from the way computers currently play? Humans search only a small set of positions in the game-tree (10-100). They base their moves and evaluations on structural/perceptual patterns acquired through experience. Computers, on the other hand, typically examine millions of positions before deciding what to do in the next position. Thus, we employ knowledge to compensate for our inherent lack of searching ability. H o w to supply this human knowledge to a machine is a puzzle that no one has solved. Cognitive scientists would like to understand how the human approach can be modeled computationaUy. AI researchers are interested in resolving the conundrum: how can both computers and humans be good at chess, but in such different ways? Perhaps these methods can be combined synergistically to produce a chessplayer far stronger than we can currently conceive. We quote from S. Skiena [11]: The first psychological study of chess and perception is Binet's, in 1894, but the major work in the field is Thought and Choice in Chess by Adrian de Groot (1965), who analyzed the thought processes for chess players of various levels of ability on difficult positions. The experiments surprisingly show that Grandmasters do not search the game-tree deeper than less talented players, but invariably select only good moves for further study. Performance on blitz games, where the player does not have time to explore the tree to any depth, illustrate this idea. Play is still at a very high level, suggesting that, in human chess, pattern recognition plays a more important role than search. (de Groot confirmed these experiments, as have others [2,6].) Masters explicitly search a tree of about 50 nodes before selecting a move, using perceptual mechanisms to prune the tree to eliminate the need to consider other possibilities. Even with search, 68 THE MATHEMATICALINTELLIGENCERVOL. 15, NO. 2, 1993

players evaluate leaf nodes as a perceptual process, rating them subjectively rather than explicitly assigning a score. Instead of some form of chess aptitude, 1 what distinguishes a player as a master is the accumulation of problem-specific knowledge. A player needs at least five years of intense chess study to become a Grandmaster, and even the cases of child prodigies show the periods of total dedication necessary to achieve excellence. This suggests that learning plays an important role in chess play, and that attempts to codify this knowledge into a few expertsystem rules probably are doomed to failure. Clearly, chess masters perform as well as they do because of an internal library of patterns that they apply to suggest promising moves. Simon and Gilmartin [10] estimated the number of patterns for a Grandmaster at 50,000. When compared to the astronomical number of possible board positions or nodes in the complete game-tree this is a tractable number, and thus suggests an avenue through which computers can improve their play.

Part I. M a n and Machine In the first article, Tony Marsland gives a brief history of computer chess, from those chess-playing machine hoaxes where a chessplayer hid in a box, up through the current leader in computer chess, Deep Thought. He makes the important point that the success of the current chess programs is not due to luck but to "hard work, direct application of simple ideas and substantial public t e s t i n g . . . [and] in hardware/software support systems." I must agree: The competitive framework of computer chess has produced true engineering successes. Should similar effort be put into other areas of AI research, I expect that we shall observe similar progress. Danny Kopec provides a complementary article in which the competition between man and machine is traced over the years, with examples of games from computer nemesis David Levy (who lost to Deep Thought 4-0 in December 1989) to the match between World Champion Garry Kasparov and Deep Thought in October 1989, w o n by Kasparov 2-0. Kopec also describes the Thompson endgame databases and an endgame match against Grandmaster Walter Browne. Jonathan Schaeffer follows this article up with a report on the 1989 World Computer Chess Championship. Deep Thought won: a perfect score of 5-0 in a field of 24. The next World Championship took place in Spain in November of 1992. David Levy ends Part I with a short article, " H o w will chess programs beat Kasparov?'" He claims that the key advances will come in software rather than hardware. He cites the singular extensions algorithm used in Deep Thought as a great advance, comparable to the analysis strong human players use to extend their search along what they deem to be important lines:

1 This s t a t e m e n t s e e m s to be false at t h e h i g h e c h e l o n of play w h e r e exceptional creativity is d e a r l y evident.

9 . . The tournament player will draw on an ever growing ever more analytically detailed body of annotated precedent. He must have in orderly mental reach an inventory of previous end games in which analogous positions, [and] analogous combinations of pieces have turned up. Under time pressure, he cannot hope to find optimal strategies across the board but must rely for his tactical choices on a recognition of previously explored lines and configurations. From the middle game onward, the expert player is projecting, by inner vision, the terminal situations he must aim for or avoid. He knows, without even thinking, that bishops of opposite colors lead to a draw, that the side whose king first reaches the central squares has a large advantage, that split pawns are a nagging weakness, that a posture on the enemy's seventh rank is often paralyzing, that major pieces ought normally to be behind advancing pawns9 He knows in what position a lone king can force a draw against king and pawn. Above all, he remembers the denouements of previous master encounters. [12] Levy argues that "quantum" advances in each major phase of the game can be made by supplying human knowledge to the program: attaching plans, subplans, and moves to each opening; coupling middlegame "'fragments or chunks" with winning plans; and conducting "fuzzy matching" with previously stored endgames and their evaluations. These are all sound suggestions. But one must admit that knowledge of this sort, although possibly sufficient to beat the World Champion, does not provide much more to the AI researcher than a furthering of the "illusion of intelligence." After all, h o w smart can a computer be if we have to tell it everything? I argue that if knowledge structures similar to those Levy suggests storing were instead learned from experience then we would truly be moving forward in AI. That most c o m p u t e r chess researchers have not granted their programs the benefit of experience is not surprising--they have ignored experience in almost all areas of artificial intelligence research, including machine learning, where "supervised" models (in which examples are pre-classified for the program) have dominated. It is like the parent who complains that the child does not mature, while not giving it any responsibility to act (and learn) on its own!

Part II: Chess Programs Part II describes the architecture and design of three of the major chess programs today: Deep Thought (the current World Champion), HiTech, and Cray Blitz. The first two programs use special-purpose hardware. The third uses multiple processors on a supercomputer. Although these are the deepest searching chess programs currently available, the authors also emphasize the type of knowledge used in these programs: Deep Thought's evaluation function was "tuned" to make it more consistent with those moves chosen by grandmasters--an example of the supervised learning paradigm of the previous paragraph. HiTech's design is unique and uses "pattern recognizers," a precursor

to the type of mechanisms envisioned by David Levy. Without improving hardware speed, HiTech could go from a Master to a Senior Master on search algorithm and knowledge improvements alone! The paper on Cray Blitz gives a detailed description of the procedures it uses to evaluate chess positions. It concludes by considering the difficulty of determining whether to "selectively search" or to continue fullwidth searching. Even these top programs combine

"'How s m a r t can a c o m p u t e r be if w e h a v e t o

tell it e v e r y t h i n g ? " chess knowledge with high-speed search. The Deep Thought team points out that the commercial chessmachine designers have made up for slower hardware by giving their programs more knowledge than exists in other research machines. This leaves us with some important unanswered questions: " H o w do they do it?" and "What if such knowledge was combined with Deep Thought's processing power?" Is it possible that

the means already exist for defeating Kasparov, but competition prevents the necessary merger of software and hardware? Part III: Computer Chess Methods Part III contains several exciting papers on new ideas in computer chess. The paper by Hermann Kaindl on "Tree Searching Algorithms" is an excellent introduction to current search-algorithms. The issues involve some sophisticated mathematical ideas and structures: aspiration windows, minimax versus product evaluation, pruning heuristics, selective searching, node ordering heuristics, quiescence, and problems of evaluation. The paper by Gordon Goetsch and Murray Campbell discusses the use of a "null-move" heuristic. The idea is to evaluate a board in which Player A is to move as if it were player B's move. This evaluation gives, in essence, a minimum value of the board from A's perspective: if B cannot inflict any damage, then certainly B will have even less success after A's move. (The exception, of course, is the "zugzwang" type of position in chess, in which all possible moves are disadvantageous.) Thus, by examining null-moves, the machine determines search cutoffs with little effort9 In an original article, Peter Jansen considers problematic positions and speculative play, i.e., when may it be desirable for a computer to make a less than optimal move against a human or computer opponent? Jansen also considers the importance of problematic positions in annotating chess games. This article contains several interesting suggestions ripe for future research. Bob Herschberg, Jaap van den Herik, and Patrick Schoo consider the problem of whether an omniscient endgame database can be recast in terms of strategy THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993 6 9

and rules. They support the claims A. Troitsky made for the two-knights versus rook-pawn ending in 1934. Although the article shows how to verify heuristics, the authors admit that getting a computer (or even sometimes a human) to come up with the necessary heuristics in the first place is much more difficult. This is one of the few articles that considers representation issues surrounding chess knowledge. In an important paper, the programmers of Bebe (world runner-up in 1989) describe how the program can learn from its previous games and significantly improve against a given opponent. The method used involves storing previously reached positions and updating, based on further information provided by the game in question--a form of rote learning akin to what Arthur Samuel [8] did with his checkers program in the 50s. This, obviously, is an important step in getting chess programs to improve with experience. Still, a lot remains to be done: Much more important is getting computers to exploit similarities or analogies, rather than exact matches, with previously seen positions. Achieving this would bring us much closer to the human cognitive model. In the final article of Part III, Tony Marsland describes a comparison of the programs in the 1989 World Championship based on the 24 positions of the well-known Bratko-Kopec test suite. The construction and use of test suites is critical to the evaluation (and therefore improvement) of chess programs circumventing the hundreds of games required for accurately using the standard schemes for rating chess players.

Part IV: Computer Chess and AI Part IV considers directly the relationship between computer chess and artificial intelligence research in three thoughtful papers by John McCarthy, Donald Michie, and Misha Donskoy with Jonathan Schaeffer. McCarthy's article is titled "Chess as the Drosophila of AI." He suggests that computer chess--looked at by many in the scientific community as having little relevance to practical AI problems could be used to explore many basic issues related to machine learning and cognition in general, much in the way that the humble fruitfly has become a standard in genetic research due to the ease of breeding and storing it. A main obstacle to computer chess taking on a more ubiquitous role in AI studies, McCarthy suggests, is that few of its practitioners have been motivated to publish scientific papers explaining how their results were achieved and how others might build upon them. McCarthy lists four "drosophila" test problems for AI. The first problem involves types of chess endgames in which the solution is obvious to humans, due to their ability to search schematically, but is extremely difficult for computers because the winning plan forces them far beyond the search horizon. For example, a 70

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993

c o m p u t e r will naturally centralize a king in the endgame, but it may be a requirement of the position that the king take a long path along the edge of the board to accomplish its objective. The second problem is a problem in non-monotonic logic known as the Yale Shooting Problem. The third problem--an idea McCarthy has been pushing over the years---is that of getting computers to provide relative (better or worse) assessments of chess positions rather than numerical evaluations, because this is closer to what we humans do. For the fourth drosophila, McCarthy suggests the game of go: Due to the enormous branching capabilities required by the game, computers will require sophisticated pattern-matching to make progress. In an entertaining and thought-provoking article, Michie considers the future of computer chess, the human response to the growing strength of computers, and the possibility that brute-force computation may take us beyond the limits of h u m a n understanding, in science as well as chess. (Contrast this with the paper by Herschberg, Herik, and Schoo.) The third paper is titled "Perspectives on Falling from Grace." It examines several of the issues that I have already alluded to; specifically, that the competitive aspects of computer chess, coupled with the resounding success of brute-force techniques, seem to have made the field one of engineering, not of science. Although this characterization may be partially correct, the author could have been far more positive: If computer chess is engineering, it is good engineering. The progress in this area surpasses many other applications of artificial intelligence research. Although computer-chess research has not addressed many of the fundamental questions of AI, it is essential that we adopt from computer chess those methods that have worked and apply them to other well-structured domains.

Part V: A N e w Drosophila for AI? Part V of the book contains two papers on the game of go, which McCarthy and others are suggesting as the new drosophila of AI. That these papers should be included in a book on computer chess is consistent with the book's approach of considering the larger context in which chess research is taking place. The first paper describes a knowledge-based program of AI research: such an approach allows us to measure the progress of projects with remarkable accuracy. The second paper describes a priority-tree-based approach to go in which nodes of the tree are procedures for deciding which move to make next. The result is a highly refined selective search mechanism. The paper concludes by giving eight areas in which further improvements in go programs will occur. Although these two papers cannot be considered representative of ongoing go research, they provide an interesting starting point for someone newly interested in this area.

Conclusions of the Reviewer Though limited by the number of contributing papers, Computers, Chess, and Cognition provides a fine, carefully edited introduction to the state of the art. Computer-chess researchers as well as those w h o simply want to k n o w more about the area will enjoy it. Though many papers brush against the issues of human cognition, it would be false to say that this is a major topic of the book. In this review I have argued for the wide acceptance of computer-chess research, while simultaneously claiming that the potential uses of chess as a testbed in AI and cognitive-science research remain largely untapped. Further, I have argued the importance of giving computers the benefit of experience. In closing, allow me the opportunity to describe one research project in this direction (putting my money where my pen is, as it were): Morph [3,4,5] is a self-learning pattern-oriented chess program whose design was first presented at the 1989 World Chess Championship. In Morph's cognitively inspired learning framework, knowledge must be learned incrementally from experience without pre-classified examples and with little guidance about relevant features. Since 1989, the design has been fully implemented, and Morph has been tested over the last nine months. Starting from little chess knowledge, by playing another chess program, Morph has learned the values of the pieces and many tactical and positional features, and has to its credit over twenty draws and one victory over its trainer. Whether such progress can continue remains to be seen. When will a computer beat the human World Champion? Let me list a few considerations. 9 The charts that attempt to project rating-point increases over time (e.g., see pages 10 and 242 of the book) to conclude that Kasparov's days are numbered are probably not relevant. At the grandmaster level, the difference between one grandmaster and another has more to do with strategic play and endgame expertise than with the tactical play that dominates the lower levels of chess. 9 The World Chess Champion may be considered the "Einstein" of chess and can establish entirely new styles and modes of play. Any player this good can develop an effective anti-computer style, especially for a specific computer. Many consider the current World Champion Gary Kasparov to be the greatest chessplayer of all time. He will not take the challenge from computers lightly. 9 Beating the champion in a single game is far different from defeating the champion in the match required for the human championship. Given the current state of the art, playing in the match would require a tremendous amount of human engineering during the match, thus leaving the question of the computer's superiority in doubt.

Still, being generous to the humans (i.e., the computer-chess researchers), I believe that two more computer world championships will pass before the human champion's first defeat by a computer (about 1995-1996). Not until after the turn of the century will the domination of computers be obvious. This prediction is both more pessimistic and more optimistic than most. Perhaps such a defeat will not come about until we can produce a book entitled Cognition, Computers, and Chess!

References 1. P. W. Frey, editor. Chess Skill in Man and Machine. Springer-Verlag, New York, 1983. 2. E. Hearst. "Man and machine." In P. W. Frey, editor, Chess Skill in Man and Machine. Springer-Verlag, New York, 1983. 3. R. Levinson. "A self-learning pattern-oriented chess program." Intern. Computer Chess Assoc. Journal, 12(4):207215, December, 1989. 4. R. Levinson. "Experience-based creativity." In T. Dartnail, editor, AI and Creativity, Kluwer Academic Press, Amsterdam, 1993. 5. R. Levinson and R. Snyder. "Adaptive pattern oriented chess." In Proceedings of AAAI-91, 601-605. MorganKaufman, 1991. 6. H. Pfleger and G. Treppner. Chess: The Mechanics of the Mind. The Crowood Press, North Pomfret, VT, 1987. 7. A. Reznitsky and M. Chudakoff. "Pioneer: A chess program modelling a chess master's mind." Intern. Computer Chess Assoc. Journal, 13(4):175-195, December, 1990. 8. A. L. Samuel. "Some studies in machine learning using the game of checkers." IBM Journal of Research and Development, 3(3):211-229, 1959. 9. C. E. Shannon. "Programming a computer for playing chess." PhilosophicalMagazine, 41(7):256--275, 1950. 10. H. A. Simon and K. Gilmartin. "A simulation of memory for chess positions." Cognitive Psychology, 5(1):29-46, 1973. 11. S. Skiena. "An overview of machine learning in computer chess." Intern. Computer Chess Assoc. Journal, 9(3): 20--28, 1986. 12. George Steiner. Fields of force: Fischerand Spassky at Reykjavik. Viking Press, New York, 1974.

Department of Computer and Information Sciences Applied Sciences Building University of California Santa Cruz, CA 95064 USA levinson@cis, ucsc.edu

generatingfunctionology by Herbert Wilf N e w York: Academic Press, 1990. viii + 184 pp. US $34.50 (ISBN 0-12-751955-6).

Reviewed by E. Rodney Canfield Recently, a friend spotted a book I was holding while standing in line at the bank. "I can't even pronounce the title of that book," he said. The book in question was not the one under review here; "Combinatorial Enumeration," I replied slowly, syllable by syllable. THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993

71

"What's that," he teased, "a glorified name for addition?" "Worse than that," I confessed, "it's a glorified name for counting." Indeed, I realized later, those of us who concern ourselves with one aspect or another of enumeration may well be driven by the most primordial of mathematical urges. For the practicing enumerator there is no tool more trusted, or more called upon than the generating function. Although the topic has been treated expertly in several modern treatises, the book generatingfunctionology by Herbert S. Wilf offers several distinctive and attractive features. First, it limits its scope to the basics of the subject: What are generating functions? H o w does one manipulate them? What are they useful for? The result is a compact book that one may comfortably recommend to the ambitious undergraduate for selfstudy. Second, Wilf acknowledges that power series possess both a formal and an analytic theory; he develops both, and explores their relationship. Third, although suitable for the beginner, the book includes a taste of more advanced material, like WZ pairs and Hayman's method (see more below). As one has come to expect, Wilf's writing is clear and friendly; his exercises are instructive and plentiful. In the remainder of this article I shall highlight a few results of the subject following Wilf. At the least, the reader will have been introduced to generating functions and, perhaps even better, he will be inspired to study the book and learn more. In his Introductio in Analysin Infinitorum, Euler (1742) studied p(n), the number of partitions of the natural number n and presented several identities which nowadays we regard as belonging to the realm of generating functions. However, the name itself, and the first systematic treatment, are found in Laplace's Th~orie analytique des probabilit& (1812). The idea is simple enough: If we are studying a sequence of numbers an indexed by the natural numbers, we use all the numbers an to build one function, fix) = s n. Then we convert information about fix), the ordinary generating function of the sequence an, into information about the sequence itself. Here is an example arising in the analysis of the Quicksort algorithm [3]. Let Hn be the nth harmonic number, that is, 1

1

1

H,=l+~+g+...+-.n Give a closed, summation-free formula for s Hk. Telling you that there is a closed formula is a big help; once you have guessed the answer, (n + 1)(Hn+ 1 - 1), the proof by mathematical induction follows quickly. But through the use of generating functions, we can eliminate the guesswork and the uncertainty as to existence of a closed formula. The function -log(1 - x) 72

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993

is the ordinary generating function for 1/n, and so h(x) = -log(1 - x ) / ( 1 - x ) a n d -log(1 - x ) / ( 1 - x)2 are the ordinary generating functions for H n and s Hk, respectively. But -log(1 - x) (l-x)

2

1

-h'(x)

(l-x)

2'

and the above-reported closed formula follows on comparing the coefficients of x n on both sides. To be honest, the example loses its appeal of simplicity when we imagine explaining it to a person w h o has never heard of the logarithm function, or of the derivative. The magic and simplicity of generating functions stem, in large part, from the cultural coincidence that no human ever encounters the ring of formal power series without having been taught calculus first. The lessons learned in the latter render obvious many propositions of the former. Let us review a few fundamentals of formal power series, and note some of the unconscious intuition which contributes to our comprehension of the harmonic number example. A formal power series (fps) over a field ~ is an expression s n, in which x is an indeterminate and each coefficient a n belongs to ~. Readers desiring strict rigor may alternatively define an fps to be an infinite sequence a n of field elements, but the use of the indeterminate x makes everything easier to remember. Addition and scalar multiplication of these series is defined coordinatewise, and product is defined by the Cauchy convolution; thus, n

C = AB,

cn = s

akbn-k,

k=0

a n, bn, and cn being the coefficients of A, B, and C, respectively. With these operations, the set of power series forms an algebra over the field ~, frequently denoted ~[[x]]. It is a perfectly well-defined algebraic object, a bit reminiscent of the ring of polynomials ~[x] which it contains as a subalgebra. Please note there is n( n~tion of convergence involved; as Wilf expresses it,-K,ve must learn to manipulate with abandon and not have a guilty conscience over convergence. One of the first propositions is that the units of ~[[x]], the power series which possess reciprocals, are precisely those whose leading coefficient, a o, is not 0. Proving this proposition is a good exercise. If B = s n is an fps, we understand the meaning of B2,B3, indeed Bn for any integer n, as fps. But, in fact, if b0 = 0 and a n is an infinite sequence of field elements, the infinite summation s n, makes perfectly good sense as a fps. To compute the coefficient of x n in the infinite sum, one needs only a finite amount of arithmetic over the field ~ and needs the sequences ao,al. . . . and bl,b2 . . . . only through an and bn, respectively. Thus, our algebra ~[[x]] possesses another

binary operator, called composition, denoted A(B) and defined whenever the leading coefficient of B is 0. There is a childish justice at play here, whereby if you cannot be reciprocated, at least you can be composed, and vice versa. Restricted to those power series A for which a 0 = 0 and a 1 # 0, the composition operator yields a group. For obvious reasons, the identity element can be denoted x. To establish this assertion, one must make peace with several issues, for example, associativity. Having made the point that these constructions are valid over an arbitrary field ~;, we now specialize to the case that ~; has characteristic 0, so that the rational numbers Q are contained in ~. This allows us to define a few special series based on our experience with Taylor expansions in calculus, namely, the exponential function exp(x) = Exn/n!, and the logarithmic function log(1 + x) = x - x2/2 + x3/3 - . . . . As you might guess, the composition exp(log(1 + x)) is the power series 1 + x; but this deserves a proof. If you resort to your knowledge of complex analysis, that is understandable, but inappropriate. If you envision yourself proving a series of identities by mathematical induction, then you are into the spirit of formal power serfes. However, there is a neat w a y to prove that exp(log(1 + x)) is 1 + x: Compute that the leading coefficient is 1, and show that the derivative is 1. Did I say derivative? Yes, if A is a formal power series whose coefficient of x n is a n, then we can define the derivative of A, A', to be the formal power series whose coefficient of x n is (n + Dan + 1. In no time at all, you see that differentiation is a linear operator on the ring of fps which obeys the Leibniz product rule; also that exp(x) is its own derivative and that the derivative of log(1 + x) is 1/(1 + x). Thus, instead of focusing on the exp(log(1 + x)) problem mentioned above, it would be more productive to prove the chain rule: A(B)' = A'(B)B'. Each reader will appreciate the opportunity to exercise her imagination and prove, once again after all these years, the chain rule. The ring Q[[x]] has been the scene of some interesting research, for example the problem of integration in closed form dating back to Liouville and having modern relevance for symbolic manipulation systems. But without going too far afield, we have one final question for the reader about fps. If f is a fps whose leading coefficient is not 0, then the equation w

= xf(w)

determines uniquely a fps w = w(x). What can be said about the coefficients of w? Using [xn] for the linear operator, "extract the coefficient of x TM, the answer is provided by the classical Lagrange inversion theorem

G ;3 Figure 1. The two cycles built on the labels {1, 2, 3}

Lagrange's original proof of (1) is in effect a fps proof, the calculus of residues being unknown at the time. See [1,2] for modern nonanalytic proofs, the first of which is combinatorial. We n o w re-join Wilf in Chapter 3, where the topic is exponential generating functions (egO. The egf of a sequence a, is the fps s The usefulness of this concept for enumeration is seen from the fact that if A and B are the egf's for sequences an and bn, respectively, then the product C = AB is the egf of a sequence c, defined thus: t/

(2) k=O

The latter is e n d o w e d with combinatorial meaning once we understand the idea of building an object on a set of labels. For instance, given the labels 1, 2, and 3, there are two ways to build a cycle (see Figure 1). Or, even simpler, given the labels 1, 2, 3, and 4, there is only one way to build a set on these labels, namely, {1,2,3,4}. In general, there are (n - 1)! ways to build a cycle on n labels, and one w a y to build a set on n elements. We say, in the vernacular, that - l o g ( 1 x) is the egf for cycles, and that exp(x) is the egf for sets, meaning that the coefficient of x" is a,/n!, where a, is the number of objects constructable with n labels. So, as another example, 1/(1 - x) is the egf for permutations. This given, we see from (2) that the product AB of two egf's is the egf for ordered pairs (A-object, B-object), the union of whose labels is the set {1,2 . . . . . n}. In other words, partition the set of labels into two disjoint blocks, build an A-object on the first and a B-object on the second. From this simple observation we conclude that Ak/k! is the egf for k-sets of objects, and hence exp(A) is the egf for sets of A-objects. This is a powerful tool, and the result is called the exponential formula. Utilizing just the three egf's introduced above, the fact that a permutation is a set of cycles is summarized by the identity 1/(1 - x) = exp(-log(1 - x)), whereas we see that the numbers B,, defined by

[ X n ] W = -1 n

[xn_llfn"

(1)

exp(ex - 1) = F~Bnxn/n!,

(3)

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993 7 3

~(nk k)_

(n=0,1,2

. . . . ).

k~>O

Figure 2. A rooted tree viewed as a pair (root, set).

count the number of partitions of an n-element set into nonempty pairwise disjoint blocks. The coefficients B, are known as the Bell numbers, and they possess many beautiful properties. For instance, knowing the coefficient of x ~ in emX/m! suggests Dobinski's formula 0c

1 Bn

=

e

~

mn/m!,

m=l

although justification of the infinite summation is needed. Or again, noting that a rooted tree (see Figure 2) on n labels is really an ordered pair, consisting first of a root r, and second of a set of rooted trees (namely, the subtrees of the original tree, their o w n roots being the points of attachment to the original root r), we see that the egf for rooted trees, T, must satisfy the functional equation

"(a) Identify the free variable, say n, that the sum depends on. Give a name to the sum that you are working on; call it fin)." Let us consider step (a) done; we are ready to read the next three steps: "(b) Let F(x) be the ordinary generating function for the sequence f(n), the sum you would love to evaluate. (c) Multiply the sum by x n and sum on n. Your generating function is now expressed as a double sum over n, and over whatever variable was first used as a d u m m y summation variable. (d) Interchange the order of the two summations that you are now looking at, and perform the inner one in simple closed form. For this purpose it will be helpful to have a catalog of series whose sums are known, such as the list in section 2.5 of this book." After interchanging the order of summation, we find, for our example above,

E E (n k~>0

By the binomial theorem, the inner summation is a slightly disguised version of (1 + x) k, and so we conclude

T = x exp(T). The preceding equation completely determines the coefficients tn/n! of the fps T, and calculating the first five values of tn you will be struck by an unmistakable numerological pattern. If you are lucky enough not to have seen this before, try it, and then see if you can prove your observed pattern with the Lagrange inversion formula (1). We are only skimming the surface of Wilf's Chapter 3 here. He offers a general, but easily visualized, version of the exponential formula, expressed in the language of cards, decks, and hands. These picturesque terms for describing the theory are attributed to A. Garsia, whose many more serious contributions to combinatorics will be appreciated by readers w h o work in the subject. The fourth chapter of generatingfunctionology presents several applications. Perhaps the reader is already familiar with the technique of using an analytical formula for P,(x) = Ea, kxk to get closed formulas for the moments ~kkan,k and ~kk2an,k as functions of n, or with the expression of the principle of inclusion/ exclusion in terms of generating functions. However, it is less likely that he has had the opportunity to use the " 'Snake Oil' method for easier combinatorial identifies," and because Wilf's presentation of this is so nicely done, we will illustrate with one of his examples and quote his five-step method verbatim. Consider the sum 74

THE M A T H E M A T I C A L INTELLIGENCER VOL. 15, NO. 2, 1993

n

1

F(x) = ~ xk(l + x) k - l _ x _ x 2"

(4)

k~>0

This brings us to the fifth and final step of "Snake Oil," "(e) Try to identify the coefficients of the generating function answer, because those coefficients are what you want to find." The uninitiated reader w h o has not had the benefit of everything in Wilf's book preceding this example will just have to believe that the rational function on the right side of (4) is an old friend, and those with some experience know already that this old friend generates the Fibonacci numbers, F n. (The latter sequence begins 1,1,2,3,5 . . . . . and is covered on page 1 of Wilf's book.) Hence,

Z(n k~>0

From these excerpts, the reader can see h o w ideally suited the book is for teaching a beginner h o w to use generating functions effectively. Even issues of limits of summation are discussed carefully and clearly, and indeed this is a stumbling block for many beginners. But I emphasize again that the book is valuable reading for even the best of specialists, and I cannot imagine anyone not learning something n e w from the more advanced sections.

One such advanced section is that dealing with WZ pairs. This is a technique developed relatively recently by Wilf and Zeilberger by means of which a computer can verify an identity, once given a rational function proof certificate. The verification is a mechanical sixstep process, quite suitable for a symbolic manipulation package. There is even an algorithm, due to R. W. Gosper, Jr., which can decide for a given identity whether or not the required rational function proof certificate exists, and it has been implemented in some commercially available packages. The number of deep identities which can be proven by this method is impressive. Finally, Chapter 4 closes with a brief discussion of a few more applications of generating functions, to unimodality, convexity, and congruences. Just when you are beginning to feel comfortable in the newfound freedom to manipulate p o w e r series shamelessly with no guilt about convergence, it is time for the fifth and final chapter of generatingfunctionology: analytic and asymptotic methods. In the words of Rademacher [4], as he prepared to present the H a r d y Ramanujan-Rademacher expansion for p(n), "If we now, however, replace the indeterminate x by a complex variable x, the formal power series become power series in the ordinary sense to which the concept of convergence [absent earlier] applies. The convergent power series are analytic functions, and the formal identities become equations between analytic functions. This step opens the whole theory of analytic tools for the treatment of arithmetical problems in additive number theory." As noted by Wilf, the ability to change gears quickly between formal and analytic methods can be helpful. For instance, which is larger, the number of permutations of an n-set, n!, or the number of partitions of an n-set, Bn? Of course, because each permutation has a unique underlying partition, n! is larger. Do you suppose that the ratio Bn/n! might go to zero? Those with quick mental facilities will see that this is the case, but the mortals among us, especially those w h o are a bit on the lazy side, will just note that the left side of (3) is an entire function, and so, of course, its coefficients B,/n! must tend to zero. In fact, having been reminded of our halcyon days in a complex variables course, we make the crudest imaginable bounds on Cauchy's integral formula, and conelude that

Bn/n! <~ exp(eR - 1)/R"

(5)

a minimum. He would find the appropriate choice of R to be the unique positive root r of the equation ye r =

n.

Substituting this special value of r into (5), our industrious reader will have found a n e w upper bound for Bn/n!, but he may well wonder if the new bound itself even goes to zero as n --* ~? In fact it does, and we are now not far from the complete truth,

Bn/n! = exp(e r -1)r-n(2~rerr(r - 1 ) ) -1/2 (1 + O(n-1)). The preceding is an example of what can be proven using Hayman's method as presented by Wilf in Chapter 5. Again though, it is not Wilf's style to plunge us into a raging sea of complexities without preparing us first. As early as Chapter 2, we are reminded of Abel's theorem that when complex values are assigned to the indeterminate variable of a power series, the result is a sum which converges for [z[ < R, diverges for [z[ > R, and defines an analytic function with a singularity somewhere on the circle [z[ = R. The radius of convergence R is expressed explicitly in terms of the coefficients a n by the Cauchy-Hadamard formula R = (lim sup lanlVn) -1. Thus, we are prepared early for the idea that some information about the coefficients a, can be gleaned from the location of the singularities of their generating function. Wilf presents two theorems, one due to Darboux, the other to Szeg6, which tell h o w to get a world of information about an out of a little knowledge of the g e n e r a t i n g f u n c t i o n ' s singularities. The a b o v e mentioned method of Hayman is the extreme case in which R = ~. In closing, we thank Ira Gessel for historical informarion about Lagrange's inversion theorem, and we wish the reader many happy and successful experiences with generating functions.

References 1. I. M. Gessel, A factorization for formal Laurent series and lattice path enumeration, J. Combinatorial Theory, Series A 28 (1980), 321-327. 2. P. Henrici, An algebraic proof of the Lagrange-B/irmann formula, J. Math. Anal. Appl. 8 (1964), 218-224. 3. D. E. Knuth, The Art of Computer Programming, volume III: Sorting and Searching, Reading, MA: Addison-Wesley (1973). 4. H. Rademacher, Topics in Analytic Number Theory, Berlin: Springer-Veflag (1973).

no matter what positive value may be given to R. [The R here is the radius of a circle about the origin, around which we integrate exp(ez - 1)/z n+l to extract 2-rri times the coefficient BJn!; when Izl = R, the integrand never exceeds exp(e a - 1)/R n +1 in absolute value, and the path of integration has length 2~rR.] Department of Computer Science An industrious person would now think of the idea University of Georgia of choosing R in such a w a y that the right side of (5) is Athens, GA 30602 USA

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993 7 5

Robin Wilson*

Greek Mathematics IVmThe School of Alexandria

Euclid.

Euclid and his students (Raphael).

Hipparchus.

Around 300 B.C., with the rise to power of Ptolemy I, mathematical activity moved to the Egyptian part of the Greek empire. Ptolemy founded a university in Alexandria, which became the intellectual center for Greek scholarship for some 800 years. The first mathematician of importance there was Euclid (ft. c. 300 B.C.), whose Elements was to become the most influential mathematical work of all time. Books V and XII of the Elements, on the theory of proportion and the method of exhaustion, are sometimes ascribed to Eudoxus of Cnidus (c. 400-350 B.C.), a mathematician and astronomer who studied at Plato's Academy. Eudoxus advanced the hypothesis that the Sun, Moon and planets move around the Earth on concentric spheres, a hypothesis later supported by Hipparchus (c. 150 B.C.), the greatest astronomer of antiquity, who discovered the precession of the equinoxes and constructed the first known star catalogue. Hipparchus, the "father of trigonometry", showed how to construct a "table of chords", which gives, essentially, the sine of each angle from 0~ to 90~ in1o creasing by ~ each time. The Earth-centered hypothesis was developed by Ptolemy (2nd century A.D.), and eventually became known as the Ptolemaic system. A different hypothesis, that the Earth rotates and revolves around the Sun, was advanced somewhat earlier by Aristarchus of Samos (ft. 270 B.C.), thereby anticipating by 1700 years the revolutionary work of Copernicus.

Eudoxus' system. Ptolemy (flanked by Aristotle and Copernicus).

Aristarchus.

Column editor's address: Facultyof Mathematics;The Open University,Milton Keynes, MK7 6AA, England. 76

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2 9 1993 Sprlnger-Veflag New York