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

W h a t Sort of P r o b l e m s for t h e 2 1 s t Century?. In The Mathematical InteUigencer vol. 20, no. 2, you present Steve Smale's list of problems for the new century, and ask for counter-proposals. To me, it seems the model is wrong. To pose problems and hope that their solution will be a major part of the mathematics of the coming years is to confine ourselves too much to a schoolboy vision. Solution of already well-defined problems calls for proficiency in existing and well-established theories. Progress lies more and more in the creation of new and effective theories, and major theorems more and more are those which validate the development of those theories. To me, therefore, only two of Smale's problems seem to be on the right track:

Introduction of Dynamics into Economic Theory. Extend the mathematical model of general equilibrium theory to include price adjustments. Limits of Intelligence. What are the limits of intelligence, both artificial and human? These are truly fundamental, and may well call for whole new theories, not just solution of problems. Elemer E. Rosinger Department of Mathematics and Applied Mathematics University of Pretoria Pretoria 0002 South Africa e-mail: [email protected]

P r o b l e m s R e l a t e d to Finite G r o u p s In your Spring 1998 issue you published Steve Smale's list of problems for next century, inviting readers to submit their own lists. Here is a short

one, consisting of problems related to fmite groups, an area not covered by Smale's list. 1. S i m p l e Groups The classification of the finite simple groups is one of the great achievements of twentieth-century mathematics. It is the result of the joint efforts of dozens of mathematicians, who have written hundreds of papers and thousands of pages. Nobody in their right mind believes that such a huge amount of material can be free of errors and gaps. It is an article of faith among group theorists that the gaps can all be filled. But even the people who worked at the classification are not unanimous on the question whether the proof in its present shape is complete or not. I would like to state two questions. l a . T i d y i n g up. That means, first, fill all the gaps, and second, write the proof in an intelligible form, unifying the styles of the many papers involved, eliminating steps that turned out to be unnecessary, etc. The late Daniel Gorenstein initiated such a programme of rewriting the proof, and his collaborators continue the project, while other people work on redoing large parts of the proof by different methods (this is known as "revisionism"). These efforts automatically include filling the gaps, and there are still others who concentrate on just such repairs and completion of the proof. It is to be hoped that this task will be accomplished soon. This would still leave open the problem of finding a more satisfactory proof. lb. A Grand Unified Theory. The classification of the fmite simple groups was preceded by the classification of the simple Lie groups, by Killing and E. Cartan at the end of last century, and by the classification of the simple algebraic groups by Chevalley in the middle of this century. The lists of simple Lie groups and simple algebraic

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g r o u p s are identical, consisting o f four large families of "classical" groups, and five families of "exceptional" ones. In the finite case w e have the s a m e groups, this time d e f m e d over finite fields, together with the "twisted" forms that s o m e of t h e m can take, a n d w e also have the alternating g r o u p s and 26 "sporadics." Thus it can be said that the a n s w e r is virtually the s a m e in all t h r e e cases, b u t the three p r o o f s a r e v e r y diff e r e n t from each other. We c a n a d d a further classification t h e o r e m , n o t yet proved. Model-theorists have a p r o j e c t to classify simple groups o f so-called "f'mite Morley rank," and again the exp e c t e d a n s w e r is the a b o v e list of classical a n d e x c e p t i o n a l groups, and again the m e t h o d s are different. My question is:

Can we f i n d a unified proof of all three (or four) classification theorems, or at least proofs that proceed along similar lines? 2. Galois Groups Galois s h o w e d h o w to a s s o c i a t e to e a c h p o l y n o m i a l equation (in one unk n o w n ) a fmite group, its Galois group, t h e r e b y creating a b s t r a c t algebra. The following is k n o w n as the in-

verse problem of Galois theory: Can every f i n i t e group occur as the Galois group of some equation with rational coefficients? The p r o b l e m s e e m s to have been originated b y Hilbert, w h o also prov i d e d w h a t is still a m a j o r tool, the

Hilbert Irreducibility Theorem, and u s e d it to s h o w that the s y m m e t r i c groups can so occur. Shafarevich proved that all finite soluble groups a r e Galois g r o u p s over the rationals. Following the classification o f the finite simple groups, efforts c o n c e n t r a t e d on realizing simple, or nearly simple, g r o u p s as rational Galois groups. Even if this w e r e achieved, it w o u l d still b e a nontrivial s t e p to p a s s from this to the realization o f all t'mite groups.

3. Groups o f finite e x p o n e n t . A group has e x p o n e n t n ff all its el-

e m e n t s satisfy x *~= 1. The Burnside P r o b l e m is w h e t h e r finitely g e n e r a t e d groups o f finite e x p o n e n t are finite. This w a s d i s p r o v e d by Novikov a n d Adjan, w h o s e results, s u p p l e m e n t e d b y m o r e r e c e n t ones of Ivanov and Lyseniok, establish that for all large e n o u g h n t h e r e exist infinite groups o f e x p o n e n t n g e n e r a t e d b y t w o elements. On t h e o t h e r hand, ff t h e exponent is 2, 3, 4, o r 6, then the groups are l'mite. Thus t h e first o p e n c a s e is:

Are f i n i t e l y generated groups of exponent 5 f i n i t e ? 4. D i v i s i o n algebras A division ring D is a ring in w h i c h all non-zero e l e m e n t s have inverses. Its centre, the s e t o f all e l e m e n t s t h a t c o m m u t e with all o t h e r elements, is a field Z, a n d D is a division algebra if it is finite-dimensional as a v e c t o r s p a c e over Z. Let F be a maximal subfield o f D. Then F c o n t a i n s Z, and it is k n o w n that the dimension, s a y n, of D o v e r F is equal to the d i m e n s i o n of F o v e r Z, so that the d i m e n s i o n o f D over Z is n 2. The n u m b e r n is called the degree o f D. Generally t h e r e exist m a n y non-isom o r p h i c m a x i m a l subfields. S u p p o s e that one o f t h e m can be c h o s e n to be a Galois e x t e n s i o n of Z. This is the s a m e a s saying that F is o b t a i n e d from Z by adding all r o o t s of s o m e equation. Then the s t r u c t u r e of D is c o m p l e t e l y d e t e r m i n e d by F, the Galois group G o f F over Z, a n d a certain function, called a 2-cocycle, from G x G to F. In that case the division algebra D is t e r m e d a crossed product of F and G. This notion w a s i n t r o d u c e d by E m m y Noether, and the question a r o s e w h e t h e r all division algebras are c r o s s e d p r o d u c t s . This is easy for n = 2, w a s p r o v e d b y W e d d e r b u r n for n = 3, and b y A l b e r t for n = 4. The AlbertBrauer-Hasse-Noether T h e o r e m o f 1932 s t a t e s that if Z is the field o f rational n u m b e r s , o r a finite e x t e n s i o n o f it, then D is a c r o s s e d product, with a cyclic Galois group. F o r t y y e a r s l a t e r Amitsur s h o w e d that n o n - c r o s s e d p r o d u c t s o f d i m e n s i o n n exist whene v e r n is divisible by 8 o r b y the square

of an o d d prime. So again the first o p e n case is

Are division algebras of degree 5 crossed products? Avinoam Mann Institute of Mathematics Hebrew University Givat Ram, Jerusalem 91904 Israel e-mail: [email protected]

Twentieth-Century Mathematics: A Methodological Question I found Malet's p a p e r o n F e r r a n Sunyer i Balaguer (The Mathematical Intelligencer, Vol. 20, no. 2, pp. 23-30) m o s t interesting. I m u s t a d m i t I k n e w nothing a b o u t Sunyer a n d I was deeply imp r e s s e d by several a s p e c t s of his life: the disabled p e r s o n ' s fight to live a n o r m a l life, the help o f his family, his m a t h e m a t i c a l a c h i e v e m e n t s , . . . However, t h e r e is a m e t h o d o l o g i c a l question in the p a p e r I w o u l d like to discuss. It is n o t j u s t a scholarly matter, b u t has h u m a n implications. The last p a r t of Malet's p a p e r is devoted to the m a t h e m a t i c a l c o m m u n i t y in F r a n c o ' s Spain (the forties until the early seventies). Malet m a k e s a serious and h o n e s t a t t e m p t to f m d an objective m e t h o d to evaluate the nature (quality?) of the m a t h e m a t i c a l r e s e a r c h in this period. He p r o v i d e s two tables giving the n u m b e r of publications by Spanish m a t h e m a t i c i a n s of this period, in Spanish j o u r n a l s (Table 1) and in foreign j o u r n a l s (Table 2). F r o m these tables he suggests s o m e conclusions. To begin with, it is s t r e s s e d that "many m a t h e m a t i c i a n s w h o w e r e highly productive or p r o d u c t i v e in Spanish published nothing or marginally in any foreign language." This is clearly seen as anomalous. However, let us reflect for a while. Was this actually a n o m a l o u s at that time? If so, I believe the a u t h o r should have clearly e x p l a i n e d why. 1 I am afraid that b e h i n d all this w e find clearly s u g g e s t e d the a s s u m p t i o n that, at least in a c o u n t r y such as Spain, the quality of the r e s e a r c h essentially d e p e n d s on the n u m b e r of p a p e r s a n d

1Malet makes some allusion to the way scientific productivity is developed today in small countries. But the point is how things were then.

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b o o k s p u b l i s h e d abroad: "the m o r e y o u publish a b r o a d the b e t t e r you are" (of course, this is an oversimplification). This has b e e n the p o l i c y in m a n y countries. This is very often seen as "progress." I w o u l d s a y it is j u s t a fashion (which I try to follow, b e c a u s e I have b e e n e d u c a t e d in this atmosphere). Of course, this issue has b e e n c o n s i d e r e d several times; 2 I do n o t wish to go into these discussions, b u t to p o i n t out that w e can n o t t a k e for g r a n t e d it w a s the same, let us say, in the fifties. On the contrary, I think that then, all over the world, things w e r e c o m p l e t e l y different. This i m p r e s s i o n c o m e s from informal talks with old m a t h e m a t i c i a n s , from leafing through old issues o f j o u r n a l s (one s e e s t h a t m o s t a u t h o r s w e r e locals and w r o t e in their o w n language), from biographical n o t e s on o u r eldest c o l l e a g u e s , . . , a n d from s o m e i n s t a n c e s I will give below. Malet u s e s again the s a m e underlying i d e a to s e l e c t the group of the m o s t active Spanish m a t h e m a t i c i a n s at that time (they w o u l d be the ones w h o published a b r o a d the most). He t a k e s the list he has u s e d to m a k e his tables, a n d selects the 8 m a t h e m a t i c i a n s in his list w h o p u b l i s h e d m o r e t h a n 5 items a b r o a d ("in non-Iberian languages" to b e p r e c i s e ) b e t w e e n 1940 a n d 1972. He a s s u m e s "they all w e r e i n c l u d e d a m o n g the d e e p e s t and m o s t researcho r i e n t e d Spanish mathematicians." We learn that 4 o f t h e m left the country: 3 in the fifties, and one m o r e in 1960. This s o u n d s hard: one has the impression that "the b e s t ones left." We r e a c h n o w the h u m a n point: What a b o u t the others, the ones who r e m a i n e d in the country? And also, w h a t about the ones who left the country but usually published in their own language? (I would say this w a s typical among those who went, for instance, to Argentina.) Were m o s t of t h e m not so good? I do not wish to m a k e comparisons or p e r s o n a l judgements. We j u s t m a y w o n d e r w h e t h e r going a b r o a d was a conse-

quence of the quality of the research (supposedly equivalent to the n u m b e r of foreign publications) or w h e t h e r they published abroad j u s t because they were abroad. So, let us have a l o o k at the four m a t h e m a t i c i a n s in Malet's selection who left the country: Gil Azpeitia (A. G. Azpeitia in Mathematical Reviews), M. Balanzat, E. Corominas, a n d F. Gaeta. We f'md3: (a) In the first pap e r published a b r o a d b y Azpeitia, in the U.S.A., one can read he was already in the U.S.A. Co) E. Corominas published papers exactly in the country where he was, with the exception of at most two papers in France (after having completed his Ph.D. thesis there). (c) During the time under consideration F. Gaeta visited many countries and w o r k e d in Spain only for a b o u t five years. (d) In the case of Balanzat, it s e e m s there is a m i s t a k e in Malet's paper: he did not leave Spain in the fifties but during the Civil War (1936-39). Since he w a s never in Spain in the p e r i o d u n d e r consideration, I do n o t k n o w w h e t h e r Malet w o u l d have i n c l u d e d him in his study o r not, b u t in any case, I w o u l d say that he follows a similar pattern. It is n o t even c l e a r to m e w h e t h e r he really r e a c h e d Malet's s t a n d a r d of foreign publications: he certainly p u b l i s h e d m o r e t h a n 5 p a p e r s in non-Iberian languages (he p u b l i s h e d 6 papers, in French), but not in non-Iberian journals (one of the 6 papers in French was published in a Portuguese journal, and another one in a Brazilian journal). One final remark. A casual r e a d e r could think that the list of Spanish m a t h e m a t i c i a n s of 1940-1972 u s e d b y Malet to m a k e his t a b l e s (see footnote 7 of his p a p e r ) is m o r e o r less exhaustive. It is not. Let me mention two mathematicians o f m y specialty (Analysis) w h o are n o t in the list: A. de Castro Brzezicky and M. Valdivia. The first one w o u l d be a m o n g the m o s t productive in foreign j o u r n a l s according to Malet's s t a n d a r d s ( m o r e than five

items). The s e c o n d one w o u l d not be, although he has p u b l i s h e d (since 1971) an impressive n u m b e r o f p a p e r s in j o u r n a l s of i r r e p r o a c h a b l e pedigree, a n d has h a d (and has) a v e r y s t r o n g influence in Spanish mathematics. I t h i n k that Malet should have m a d e c l e a r h o w he got his sample. Jose Mendoza Departamento de An&lisis Matematico Universidad Complutense de Madrid 28040 Madrid Spain e-mail: mendoza@sunaml .mat.ucm.es

Antoni Malet responds: I very m u c h a p p r e c i a t e P r o f e s s o r Mendoza's c o m m e n t s on m y article on Sunyer i Balaguer and the Spanish m a t h e m a t i c a l c o m m u n i t y during the F r a n c o regime. This is an o p p o r t u n i t y to clarify s o m e of the p o i n t s I made. To begin with, I did not attempt to c o m p a r e the value of individual mathematicians. Actually, m y b o o k on Ferran Sunyer (mentioned in m y article) explicitly warns the r e a d e r that "any quantitative evaluation of published articles and b o o k s always is a dubious exercise. A single good article can be incommensurably more important than 100 mediocre articles. Moreover, it is hard to find an objective w a y to decide what should be counted and h o w to count it. All this m a k e s it impossible to qualitatively compare the importance of two or m o r e individual authors. Nonetheless it does seem to be possible to a p p r o a c h quantitatively the production of a mathematical community." (p. 256-7) In the s a m e place I provided examples of difficulties found when counting the publications of individual mathematicians. Accordingly, my article does not say anything about the value of the production of individual mathematicians. One of the main a s s u m p t i o n s of m y article is the p e r i p h e r a l c h a r a c t e r o f the Spanish m a t h e m a t i c a l c o m m u n i t y - t h a t is to say, I a s s u m e that in the

2See for instance, "The 'Indexed' Theorem" by Alfredo Octavio, The Mathematical Intelligencer 18, no. 4, 1996, pp. 9-11, and some letters in connection with it in Vol. 19, no. 3. 3Concerning the papers, my source is, of course, the Mathematical Reviews. Concerning the places where each of these mathematicians were working my sources are: (a) For Azpeitia, his own papers. (b) For Balanzat, the book Julio Rey Pastor matematico by Sixto Rios, Luis A. Santal6 and Manuel Balanzat, Instituto de Espaha, Madrid 1979 (pp. 124-125)--an excellent source on Spanish mathematics of this century. (c) For Corominas, the book just mentioned and Malet's paper. (d) For Gaeta, I just asked him. He has been Professor at my University since the early eighties. Now he is Emeritus.

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period here under consideration the Spanish journals of mathematics were not world-class journals. They were not among the journals carrying cutting-edge results, nor were they widely read in leading departments around the world, nor were they very demanding as regards the quality of the articles. The suggestion that in the fifties there were no journals drawing contributions from an international community and catering to an international audience is just e r r o n e o u s - - i t is enough to have to look at the issues of Acta Mathematica. In Spain, the obsession with the differences between

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the standards of the Spanish scientific community and those of leading European centers is not a recent one. It comes at least from the turn of the century, when under the direction of Nobel prize winner Ramon y Cajal a program of fellowships was implemented that sent young promising scientists abroad. The program was kept alive until the Civil War. After the war the Franco regime introduced a similar program, now as part of the Consejo Superior de Investigaciones Cientfficas (CSIC). The main point of my article and one which seems to have eluded Prof.

Mendoza, is that mathematical communities have values of their own, shaped by local cultures and conditions, which determine the dynamics and production of their individuals. The Spanish authorities in the Ministry of Education and the CSIC valued relationships with foreign centers and foreign scientific communities in the 1950s and 1960s as much as or perhaps even more than today. My article shows the tensions, and indeed the contradictions between the official values and discourses of these authorities and the realities of who (and how and why) was actually supported and promoted in those days.

MIKLC)S REDEI

"Unsolved Problems in Mathematics J, von Neumann's Address to the International Congress of Mathematicians, Amsterdam, September 2-9, 1 954

The Invitation On b e h a l f o f the p r o g r a m c o m m i t t e e o f the 1954 International Congress of Mathematicians, H.D. Kloosterman, c h a i r m a n of t h e committee, w r o t e a l e t t e r [13] to J o h n von N e u m a n n at the Institute for A d v a n c e d Study. In this letter, K l o o s t e r m a n informs von N e u m a n n that a p r o p o s a l had b e e n m a d e in the c o m m i t t e e to c o n s i d e r an a d d r e s s to the Congress, an a d d r e s s similar to Hilbert's f a m o u s lecture in 1900 in Paris a b o u t u n s o l v e d p r o b l e m s in mathematics. K l o o s t e r m a n also points out the c o m m i t t e e ' s a w a r e n e s s of the increasing specialization in m a t h e m a t i c s , w h i c h might p r o h i b i t one p e r s o n from being able to p r e p a r e such an address. The c o m m i t t e e t h e r e f o r e c o n s i d e r e d the following t h r e e options:

1. A talk p r e p a r e d and delivered b y one mathematician. 2. A small t e a m of m a t h e m a t i c i a n s p r e p a r e s and one mathematician delivers an address. 3. A small t e a m p r e p a r e s the a d d r e s s a n d the m e m b e r s of the t e a m r e p o r t individually. K l o o s t e r m a n writes: As I m e n t i o n e d a l r e a d y the p r o g r a m c o m m i t t e e has a p r e f e r e n c e for the first of t h e s e suggestions. On the o t h e r h a n d the c o m m i t t e e ' s opinion is that you are prob-

H

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ably the only active m a t h e m a t i c i a n in the w o r l d who is m a s t e r of the w h o l e m a t h e m a t i c s to such a degree as to be able to deliver an a d d r e s s o f the c h a r a c t e r e x p r e s s e d above. F o r this r e a s o n y o u will oblige m e v e r y m u c h to comm u n i c a t e to m e if y o u w o u l d kindly a c c e p t an invitation to deliver b e f o r e the International Congress of Mathematicians in A m s t e r d a m an a d d r e s s on u n s o l v e d problems in m a t h e m a t i c s . In any case y o u r opinion a b o u t t h e s e suggestions s t a t e d w o u l d be m o s t valuable to our committee. Apparently the N o v e m b e r 27 letter of Kloosterman never reached von Neumann. Kloosterman contacted von Neumann again in a letter d a t e d March 20, 1953, a n d he also e n c l o s e d a c o p y of the N o v e m b e r 27 letter, renewing the invitation. Von N e u m a n n r e c e i v e d this s e c o n d l e t t e r on March 25 a n d replied i m m e d i a t e l y - - b u t cautiously [24]. E x p r e s s i n g his deep appreciation, and indicating that in view of the exceptional confidence that the invitation e x p r e s s e s , he is inclined to accept, he a s k e d for s o m e m o r e time to c o n s i d e r the m a t t e r carefully b e f o r e making a final commitment. Considering von N e u m a n n ' s famous mental speed, he hesitated r a t h e r long: it t o o k him 2 w e e k s to r e a c h a decision, a n d the d e c i s i o n w a s that "If this is the p r e f e r e n c e of y o u r Committee, and if it is o t h e r w i s e a c c e p t a b l e to you, I will

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give an a d d r e s s on the basis of alternative 1 that y o u ment i o n e d - t h a t is, an individual a d d r e s s 'On Unsolved Probl e m s in Mathematics 9 [25]. With characteristic precision and modesty, von Neumann also defmes, however, his t a s k m o r e restrictively b y adding, The total subject of m a t h e m a t i c s is clearly t o o b r o a d for any one of us. I do n o t think that any m a t h e m a t i c i a n since Gauss has c o v e r e d it uniformly and fully, even Hilbert did not, and all o f us are of c o n s i d e r a b l y l e s s e r w i d t h (quite a p a r t from the question of d e p t h ) t h a n Hilbert. It would, therefore, b e quite unrealistic n o t to admit, that any a d d r e s s I c o u l d possibly give w o u l d n o t be b i a s e d t o w a r d s s o m e a r e a s in m a t h e m a t i c s in w h i c h I have had s o m e experience, to the d e t r i m e n t o f o t h e r s w h i c h m a y be equally o r m o r e important. To be specific, I c o u l d not avoid a bias t o w a r d s those p a r t s o f analysis, logics, and certain b o r d e r a r e a s of the a p p l i c a t i o n s o f m a t h e m a t i c s to o t h e r sciences, in which I have w o r k e d . If y o u r Committee feels t h a t an a d d r e s s w h i c h is aff e c t e d b y such i m p e r f e c t i o n s still fits into the p r o g r a m of the Congress, and if the v e r y generous c o n f i d e n c e in m y ability to deliver continues, I shall be glad to und e r t a k e it. The t a s k r e p r e s e n t s a very interesting a n d inspiring challenge, a n d I w o u l d certainly try to m a k e the limitations that I have d e s c r i b e d above as p a l a t a b l e to the a u d i e n c e as I can. [25] The c o m m i t t e e m u s t have felt satisfied with this reply, a n d in the afternoon s e s s i o n of the first d a y o f the Congress, von N e u m a n n d e l i v e r e d the address 9 Von Neumann's Address The Plan

As far as one can tell on the b a s i s of published and archival material, von N e u m a n n did n o t p r e p a r e a t e x t for his talk: the 1-hour invited lectures w e r e p u b l i s h e d in Volume I o f the three-volume p r o c e e d i n g s o f the Congress [10]; however, "no m a n u s c r i p t w a s available" in von N e u m a n n ' s case. 1 There are two d o c u m e n t s in the Von N e u m a n n Archive in the Library of Congress that are r e l a t e d to his talk: one is a h a n d w r i t t e n s k e t c h of the topics he p l a n n e d to d i s c u s s [26], the o t h e r is a t y p e s c r i p t of the talk [27]. As his s k e t c h reveals, v o n N e u m a n n ' s plan w a s to talk about 9 p r o b l e m s in a p a r t i c u l a r a r e a of m a t h e m a t i c s - - o p e r a t o r theory, v i e w e d in its c o n n e c t i o n with certain o t h e r subjects, specifically in its algebralcal aspects, a n d its relationship to q u a n t u m theory, and t h r o u g h this to logic and to the t h e o r y o f probability. The unsolved p r o b l e m s to which attention is called fall into three groups.

1. P r o b l e m s involving the algebraic structure of rings of operators. 2. The role a n d meaning of t h e s e in v i e w of the p r e s e n t difficulties a n d uncertainties in q u a n t u m theory. 3. P r o b l e m s of reformulation a n d unification in logics and p r o b a b i l i t y theory b a s e d on this approach 9 [26] Von N e u m a n n ' s h a n d w r i t t e n s k e t c h [26] gives a list of 24 m o r e specific i s s u e s he i n t e n d e d to d i s c u s s in the talk. A c o m p a r i s o n of this list with the t y p e s c r i p t s h o w s that in the address, he did not, in fact, bring up s o m e of the issues and p r o b l e m s that he originally h a d p l a n n e d to discuss. This is not surprising in view of the fact that von N e u m a n n ' s s k e t c h e s t i m a t e s the time to be 89 m i n u t e s (he w a s supp o s e d to give a 1-hour lecture), a n d in his sketch, he had allocated only 5 m i n u t e s to state a n d d i s c u s s the isomorp h i s m p r o b l e m o f finite von N e u m a n n algebras, 2 b a r e l y 4 minutes for detailing the c h a r a c t e r i z a t i o n (i.e., classification) p r o b l e m of infinite von N e u m a n n algebras, and j u s t 3 minutes to deal with the infinite d i r e c t product. These are technically involved, deep issues that p r o v e d to be extremely challenging for m a n y t a l e n t e d m a t h e m a t i c i a n s to come. Von N e u m a n n must have realized t h a t he w o u l d not be able to do j u s t i c e to the c o m p l e x i t y of t h e s e p r o b l e m s in the time he h a d and he s k i p p e d t h e m in the lecture. ( F o r a review of the legacy of von N e u m a n n in the t h e o r y of ope r a t o r algebras, s e e [12, 15].) What von N e u m a n n decided to do in the lecture was to concentrate on the s e c o n d and third groups of problems: He gave a general motivation for the theory of operator rings and, in particular, he discussed the possible conceptual significance of a particular ring, the fmite, continuous ring (the type II1 von N e u m a n n algebra), both for the mathematical theory of o p e r a t o r s and for a better understanding of quant u m mechanics, quantum logic, and (quantum) probability. Main Points in t h e Talk

In the first p a r t o f t h e talk, von N e u m a n n m a k e s clear that a satisfactory t h e o r y of u n b o u n d e d o p e r a t o r s is absolutely i n d i s p e n s a b l e b e c a u s e u n b o u n d e d o p e r a t o r s are forced u p o n us b y q u a n t u m mechanics: the H e i s e n b e r g canonical c o m m u t a t i o n relation cannot b e satisfied b y b o u n d e d operators " . . . a n d t h e r e could not be t w o w a y s a b o u t it" [27, p. 5]. This obliges us to use infinite-dimensional Hilbert s p a c e s to m o d e l quantum systems, and l e a d s to the problem that the o p e r a t o r s are not e v e r y w h e r e defined in general. "As s o o n as one o b s e r v e s that n o n - b o u n d e d o p e r a t o r s are n e c e s s a r i l y n o t e v e r y w h e r e defined operators, one imm e d i a t e l y gets into a host of difficulties a n d one immediately runs into a n u m b e r of wide o p e n p r o b l e m s . . . " [27, p. 7]. The c o r e o f t h e s e o p e n p r o b l e m s is the p r o b l e m of finding s o m e r e a s o n a b l e set of r e q u i r e m e n t s that p e r m i t forming an a l g e b r a of u n b o u n d e d o p e r a t o r s :

~Von Neumann was not the only invited speaker not submitting a paper; for instance, A. Tarski did not submit one either. 2In 1954, yon Neumann algebras were called "rings of operators"; the name "von Neumann algebra" was proposed by Dieudonne in 1954--just about the time of von Neumann's talk.

8

THE MATHEMATICALINTELLIGENCER

it is quite clear that the interesting applications, quite p a r t i c u l a r l y in quantum theory, a b s o l u t e l y call for a sett l e m e n t o f these questions: to tell h o w to o p e r a t e on n o n - b o u n d e d operators, h o w to f o r m sums a n d products, w h a t to do if their d o m a i n s have nothing in common, i.e., if there is no p o i n t w h e r e b o t h are defined, w h a t to do if the points w h e r e t h e y are both defined exist a n d are dense, but one s o m e h o w s u s p e c t s that the set is not large enough (and this c a n be given a s h a r p e r meaning), generally speaking, h o w to introduce an algebra. [27, p. 10] 9

This, von N e u m a n n p o i n t s out, is the chief motivation to l o o k for suitable s u b s e t s o f b o u n d e d o p e r a t o r s that determine a well-behaved set of u n b o u n d e d operators 9 After an informal discussion o f w h a t is n o w k n o w n as von N e u m a n n ' s "double c o m m u t a n t t h e o r e m " (a self-adjoint set S of b o u n d e d Hilbert s p a c e o p e r a t o r s is strongly d e n s e in its s e c o n d c o m m u t a n t S"), von N e u m a n n briefly s k e t c h e s the 1935 M u r r a y - v o n N e u m a n n classification t h e o r y of factors, emphasizing the analogy b e t w e e n relative d i m e n s i o n and the cardinals: 9 the w h o l e algorithm o f Cantor is such that it goes over on this case. One can p r o v e various t h e o r e m s on the additivity of equivalence and the transitivity of equivalence, w h i c h one w o u l d n o r m a l l y expect, so that one can i n t r o d u c e a t h e o r y o f alephs here, j u s t as in set theo r y . . . . [27, p. 15] But the really exciting c o n s e q u e n c e of the d i m e n s i o n t h e o r y is in von N e u m a n n ' s view that One can, however, go a great deal further. Specifically in the c a s e o f Hilbert space, one can define fmiteness a n d infiniteness in the s a m e w a y as Cantor did b y equivalence to a p r o p e r subset 9 One c a n prove m o s t of the Cantoreal p r o p e r t i e s of finite and infinite, and, finally, one can p r o v e that given a Hilbert s p a c e and a ring in it, a simple ring in it, either all linear sets e x c e p t the null set are infinite (in which c a s e this c o n c e p t of alephs gives y o u nothing new), or else the dimensions, the equivalence classes, b e h a v e e x a c t l y like n u m b e r s and t h e r e are t w o qualitatively different cases. The dimensions either b e h a v e like integers, o r else t h e y b e h a v e like all real numbers. There are t w o subcases, n a m e l y t h e r e is either a finite top or t h e r e is not. So, w h e n t h e y b e h a v e like integers, t h e y either b e h a v e like all integers from one to a finite n, or like all integers to infmity plus a s y m b o l infinity. When they are continuous, they either b e h a v e like all real n u m b e r s from null to a fmite numb e r a, inclusive, or else like all real n u m b e r s up to infinity with a symbolic top at infinity. In total t h e r e are t h e r e f o r e five classes, I m e a n like the integers which m a y have a finite top or not, like all real n u m b e r s which m a y have a finite top o r not, and,

finally, the case w h e r e only the infinite d i m e n s i o n s exist, a p a r t from the d i m e n s i o n null. The case which is entirely finite, w h e r e all you have are the d i m e n s i o n s w h i c h are integers that have a finite ceiling, is always i s o m o r p h i c to the m a t r i c e s of the Euclidean space 9 The case w h e r e y o u have integers going to infinity, is i s o m o r p h i c to all m a t r i c e s o f Hilbert space, so there nothing is gained. A b o u t the infinite cases very little is known. [27, pp. 15-16] The upshot of this classification theory, in von Neumann's view, is to turn o u r attention to the c a s e " . . . w h e r e the dimensionality is like real n u m b e r s with a finite ceiling." [27, p. 16], which can be c h o s e n to b e 1. This case, k n o w n as the type I I l case, f a s c i n a t e d von N e u m a n n from the mom e n t of its d i s c o v e r y [14], and he m a d e clear m o r e than once that in his view this structure might b e m o r e suitable for quantum t h e o r y t h a n ordinary Hilbert s p a c e theory. 3 It has not e s c a p e d the attention of r e v i e w e r s that von N e u m a n n p l a c e d high h o p e s on this structure (see, e.g., [1, 4, 8, 12]); however, t h e r e is general a g r e e m e n t that physics has n o t d e v e l o p e d in the direction von N e u m a n n s e e m s to have envisaged. Araki even s e e s in von N e u m a n n ' s preference of the t y p e II1 v o n N e u m a n n algebras a "mathematical Utopia for q u a n t u m calculus" [1, p. 119]9 But w h a t w a s the rationale b e h i n d von N e u m a n n ' s conviction that the type II1 structure might b e m o r e suitable for quantum m e c h a n i c s than the o t h e r types, in p a r t i c u l a r type I, w h i c h c o r r e s p o n d s to the s t a n d a r d Hilbert s p a c e quantum m e c h a n i c s ? Surely, von N e u m a n n m u s t have h a d g o o d r e a s o n s w h e n he s u g g e s t e d that the Hilbert s p a c e formalism, w h i c h in its p r e c i s e m a t h e m a t i c a l form w a s largely his own creation, is n o t entirely a p p r o p r i a t e after all. What w e r e then t h e s e r e a s o n s ? Unfortunately, von N e u m a n n n e v e r p u b l i s h e d a p a p e r d e v o t e d to a s y s t e m a t i c analysis of the c o n c e p t u a l significance of the t y p e II1 case. His 1954 lecture is thus a m a j o r s o u r c e of information in this regard, for von N e u m a n n c o n s c i o u s l y a d d r e s s e s this issue in it. One r e a s o n von N e u m a n n gives for the privileged status of the t y p e II1 a l g e b r a is that the u n b o u n d e d o p e r a t o r s affiliated with this a l g e b r a are a very w e l l - b e h a v e d set: " . . . one can s h o w that any finite n u m b e r of them, in fact any c o u n t a b l e n u m b e r o f them, are s i m u l t a n e o u s l y defined on an e v e r y w h e r e d e n s e set; one can p r o v e that one can indulge in o p e r a t i o n s like adding and multiplying o p e r a t o r s and one n e v e r gets into any difficulty whatever. The whole symbolic calculus goes through" [27, p. 16]9 So, quantum s y s t e m s m o d e l e d b y a t y p e II1 a l g e b r a are p e r f e c t l y wellbehaved, including their u n b o u n d e d o p e r a t o r s . What are the quantum s y s t e m s that are m o d e l e d b y this structure? Von N e u m a n n claims that One can further s h o w that such s y s t e m s of o p e r a t o r s are in m a n y w a y s v e r y similar to certain o p e r a t o r syst e m s u s e d in q u a n t u m theory. I will n o t a t t e m p t to go into detail at this occasion, b u t it is true t h a t actually

3See the Introduction in [14] and footnote 33 in [5].

VOLUME21, NUMBER4, 1999 9

the so-called method of second quantization, which introduces the operators of quantum theory depending on certain processes of counting of states, permits a very plausible generalization which leads exactly into this kind of operator ring, and which is therefore immune to the usual pathology of operator rings. [27, p. 16] It is not clear what von Neumann is referring to in the above passage. He had developed a quantum theory of infinite quantum systems in a manuscript dated 1937 [23]; he never published that theory, however. It might be that the procedure in [23] of creating the algebra of observables leads to the type II1 case--an analysis of this manuscript is yet to be done. Today, the typical example of type IIt structure is the infinite tensor product of 2 • 2 (complex) matrices in the representation given by the state whose restriction to any finite tensor product is the product of the 2 • 2 traces. This structure describes a lattice gas in the infinite temperature state [6]. Von Neumann's main motivation in preferring type II1 is, however, related to the third group of problems he set out to discuss: the relation of logic and probability in quantum mechanics. To appreciate fully von Neumann's position in his address, one would have to reconstruct the development of his ideas from 1927 on, when he published his three "fundamental papers" [18-20], through his 1932 book [21] and his joint paper with Birkhoff on quantum logic in 1936 [5]. This cannot be done here (see [16; 17, Chap. 7] for the details of this intellectual history). Skipping over this historical development, one can formulate the main points needed to understand his view of quantum probability in his address as follows. The closed linear subspaces of the (infinite-dimensional) Hilbert space describing a quantum system form an orthocomplemented lattice ("Hilbert lattice") with respect to set-theoretical intersection as A, orthogonal complement as complementation A ~-~A--, and V defined by A V B = (A~- /~ B~-)--. In quantum logic, one interprets the elements of this lattice as quantum propositions and the lattice operations A, V, and - as the logical connectives corresponding to and, or, and not, respectively. On this "quantum logic interpretation," the Hilbert lattice is the quantum analog of the Boolean algebra that represents the propositional system of a classical propositional logic. A Boolean algebra also appears in another role, however: it represents the algebra of random events in classical probability theory, with probability being an additive measure /z on the algebra. The measure/z has the following property: /z(A) +/z(B) =/~(A fq B) +/z(A U B),

(1)

where the interpretation of A rq B is that it represents the joint occurrence of the events A and B. Property (1) is crucial if the probabilities/z(A) and/z(B) are to be viewed as relative frequencies: if N/z(X) (with

X = A, B, A r B, A U B) are the numbers of occurrences of the events in a fixed ensemble of N events, then (1) obviously holds. Von Neumann wanted also to interpret quantum logic as representing random events of a noncommutative probability theory; furthermore, he, too, viewed probabilities as relative frequencies in the years 1927-1936. The trouble is that noncommutative probability measures (i.e., the normalized maps & defined on the Hilbert lattice which take values in [0, 1] and which are additive on orthogonal elements) violate (1) in general: the following equation cannot hold for all A and B: &(A) + ~b(B) = (#(A A B) + ~b(A V B). So von Neumann faced the following options: (i) Give up the frequency interpretation of probability in favor of an interpretation that is capable, in principle, of handling infinite probabilities. (ii) Give up the interpretation of quantum logic as random event structure. (iii) Give up the Hilbert lattice (of an infinite-dimensional Hilbert space) as quantum logic. None of these options is particularly attractive. In his 1936 paper published with Birkhoff [5], von Neumann chose option (iii): quantum logic is postulated to be a lattice that admits a normalized noncommutative probability measure 4 satisfying (2). What made such a choice possible was that by the time of the paper with Birkhoff in 1936, Murray and von Neumann had already discovered type II1 von Neumann algebras [14], which are distinguished by the fact that on their projection lattice there exists a unique (up to constant) finite, noncommutative probability measure ~-that satisfies (2). Soon, it also became known that T can be extended from the projection lattice to a trace on the algebra, where "trace" means that for all X and Y 9( x r ) = ~(YX).

THE MATHEMATICALINTELLIGENCER

(3)

In short, the projection lattice of a type II1 von Neumarm algebra with the trace 9 giving the probabilities is a noncommutative probability structure, the probabilities of which could be viewed as relative frequencies. Thus, it would seem that, remaining within the mathematical framework of type II1 von Neumann algebras, one can restore the harmonious classical picture: random events can be identified with propositions stating that the event happens, and probabilities can be viewed as relative frequencies of occurrence of the events. But this restored harmony is deceiving since of all the noncommutative probability measures definable on type II1 algebra, only the trace ~-satisfies condition (2), which is necessary for a frequency interpretation--and the trace is exactly the functional which is insensitive [in the sense of (3)] to the noncommutativity of the algebra. In other words, there are no

4In that paper, this measure is called "the apriori thermodynamic weight of states." For an explanation of this terminology, see [17].

10

(2)

"properly n o n c o m m u t a t i v e " p r o b a b i l i t y s p a c e s - - a s long as one insists on the frequency i n t e r p r e t a t i o n of probability; hence, if one w a n t s to entertain the i d e a of n o n c o m m u t a tive p r o b a b i l i t y spaces, the frequency view has to go. A n d it did: von N e u m a n n a b a n d o n e d the frequency int e r p r e t a t i o n after 1936. In an unfinished m a n u s c r i p t writt e n a b o u t 1937 and entitled "Quantum logic (strict- a n d p r o b a b i l i t y logics)," he writes: "This view, the so-called 'frequency t h e o r y o f probability,' h a s b e e n very brilliantly upheld and e x p o u n d e d b y R. von Mises. This view, however, is n o t a c c e p t a b l e to us, at least n o t in the p r e s e n t 'logical' context" [22; 30, p. 196]. Instead, von N e u m a n n e m b r a c e s in this unfinished n o t e a "logical t h e o r y of probability," which he a s s o c i a t e s with J. M. Keynes, but which he d o e s n o t spell out in detail. It is in his 1954 a d d r e s s t h a t he is m o r e explicit and explains the logical interpretation at length w i t h o u t ever mentioning relative frequencies: Essentially if a state of a s y s t e m is given b y one vector, the transition p r o b a b i l i t y in a n o t h e r state is the inner p r o d u c t of the two w h i c h is the square of the cosine of the angle b e t w e e n them. In o t h e r words, probability corr e s p o n d s precisely to introducing t h e angles geometrically. F u r t h e r m o r e , there is only one w a y to i n t r o d u c e it. The m o r e so b e c a u s e in the q u a n t u m m e c h a n i c a l machinery the negation of a statement, so the negation of a s t a t e m e n t which is r e p r e s e n t e d b y a linear set of vectors, c o r r e s p o n d s to the o r t h o g o n a l c o m p l e m e n t of this linear space. And therefore, as s o o n as you have introd u c e d into the projective g e o m e t r y the ordinary machinery of logics, you m u s t have i n t r o d u c e d the c o n c e p t o f orthogonality. This actually is rigorously true and any a x i o m a t i c e l a b o r a t i o n of the s u b j e c t b e a r s it out. So in o r d e r to have logics you n e e d in this set a projective g e o m e t r y with a c o n c e p t o f orthogonality in it. In o r d e r to have p r o b a b i l i t y all y o u n e e d is a c o n c e p t of all angles, I m e a n angles o t h e r than 90 ~ Now it is perfectly quite true that in geometry, as s o o n as you can define the right angle, y o u can define all angles. A n o t h e r w a y to p u t it is that if you t a k e the c a s e of an orthogonal space, t h o s e m a p p i n g s o f this s p a c e on itself, which leave orthogonality intact, leave all angles intact, in o t h e r words, in t h o s e s y s t e m s w h i c h can be u s e d as m o d e l s o f the logical b a c k g r o u n d for quantum theory, it is true that as s o o n as all the o r d i n a r y c o n c e p t s of logic are fixed u n d e r s o m e i s o m o r p h i c transformation, all o f p r o b a b i l i t y t h e o r y is a l r e a d y fixed . . . . This means, however, t h a t one has a formal m e c h a n i s m , in w h i c h logics and p r o b a b i l i t y t h e o r y arise s i m u l t a n e o u s l y and are d e r i v e d simultaneously. [27, pp. 21-22] It was the s i m u l t a n e o u s e m e r g e n c e a n d mutual determ i n a t i o n of p r o b a b i l i t y and logic that von N e u m a n n found intriguing a n d n o t at all well u n d e r s t o o d . He very m u c h w a n t e d to have a detailed a x i o m a t i c s t u d y of this phen o m e n o n b e c a u s e he h o p e d t h a t it w o u l d s h e d " . . . a great deal of n e w light on logics and p r o b a b l y alter the whole formal s t r u c t u r e of logics considerably, if one s u c c e e d s in

deriving this s y s t e m from first principles, in o t h e r w o r d s from a suitable set of axioms" [27, p. 22]. He e m p h a s i z e d - and this w a s his last t h o u g h t in his a d d r e s s - - t h a t it w a s an entirely o p e n p r o b l e m w h e t h e r / h o w such an axiomatic derivation c a n b e c a r r i e d out. Von Neumann's call s e e m s to have r e m a i n e d u n a n s w e r e d so far. And if one looks at all seriously at the e n o r m o u s philosophical and foundational literature on quantum mechanics, one has to conclude that we are in the peculiar situation where noncommutative m e a s u r e theory has been developed into a rich mathematical discipline, yet a satisfactory interpretation of noncommutative m e a s u r e as probability and of the relation of this n o n c o m m u t a t i v e (quantum) probability to (quantum) logic is still lacking. It s h o w s von Neumann's deep interest in philosophical and conceptual issues that he had chosen this particular topic for a p l e n a r y lecture at the world Congress of Mathematicians in 1954. ACKNOWLEDGMENT

I wish to t h a n k Marina von Neumann-Whitman for h e r granting me p e r m i s s i o n to quote from the u n p u b l i s h e d material held in the von N e u m a n n Archive in the Library of Congress. This w o r k w a s s u p p o r t e d by AKP, by OTKA (cont r a c t n u m b e r s T 025841 and F 023447), a n d by t h e Dibner Institute MIT, w h e r e I w a s staying during the 1997-1998 a c a d e m i c y e a r as a Resident Fellow. REFERENCES

[1] H. Araki, "Some of the legacy of John von Neumann in physics: theory of measurement, quantum logic, and von Neumann algebras in physics," in [11], pp. 119-136. [2] E.G. Beltrametti and G. Cassinelli, The Logic of Quantum Mechanics, Addison Wesley, Reading, MA, 1981. [3] E. Beltrametti and B.C. van Fraassen (eds.), Current Issues in Quantum Logic, Plenum Press, New York, 1981. [4] G. Birkhoff, "Lattices in applied mathematics," in [9], pp. 155-184. [5] G. Birkhoff and J. von Neumann, The logic of quantum mechanics, Ann. Math. 37 (1936), 823-843; also in [30], pp. 105-125. [6] O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol. II, Springer-Verlag, New York, 1981. [7] F. Brody and T. V~mos (eds.), The Neumann Compendium. World Scientific Series of 20th Century Mathematics, Vol. I. World Scientific, Singapore, 1995, pp. 163-181. [8] J. Bub, "What does quantum logic explain?" in [3], pp. 89-100. [9] R.P. Dilworth (ed.), Lattice Theory (Proceedings of the Second Symposium in Pure Mathematics of the American Mathematical Society, April, 1959, American Mathematical Society, Providence, RI, 1961. [10] J.C.H. Gerretsen and J. De Groot (eds.), Proceedings of the lntemational Congress of Mathematicians, Amsterdam, September2-9, 1954, Vols. 1,2, and 3, North-Holland, Amsterdam, 1957. [11] J. Glimm, J. Impagliazzo, and I. Singer (eds.), The Legacy of John yon Neumann, American Mathematical Society, Providence, RI, 1990. [12] R.V. Kadison, "Operator algebras--an overview," in [11], pp. 61-89. [13] H.D. Kloosterman, "Letter to John von Neumann, November 27, 1952," John von Neumann Archive, Library of Congress, Washington, DC.

VOLUME 21, NUMBER 4, 1999

11

[14] F.J. Murray and J. von Neumann, "On rings of operators," Ann. Math. 37 (1936), 6-119; also in [29], pp. 6-119. [15] D. Petz and M. Redei, "John von Neumann and the theory of operator algebras," in [7], pp. 163-181. [16] M R6dei, Quantum Logic in Algebraic Approach, Kluwer Academic, Dordrecht, 1998. [17] M. Redei, "Why John von Neumann did not like the Hilbert space formalism of quantum mechanics (and what he liked instead)," Studies Hist. Phil. Mod. Phys. 27 (1996), 493-510. [18] J. von Neumann, "Mathematische Begr0ndung der Quantenmechanik, Gdttinger Nachrichten (1927), 1-57; also in [28], pp. 151-207. [19] J. von Neumann, "Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik," Gdttinger Nachrichten (1927), 245-272; also in [28] pp. 208-235. [20] J. von Neumann, "Thermodynamik quantenmechanischerGesamtheiten," G6ttinger Nachrichten (1927), 245-272; also in [28], pp. 236--254. [21] J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Dover, New York, 1943, first American edition; first edition: Springer-Verlag, Heidelberg, 1932. [22] J. von Neumann, "Quantum logics (strict- and probability logics)," unfinished manuscript, John von Neumann Archive, Library of Congress, Washington, DC, reviewed by A.H. Taub in [30], pp. 195-197.

[23] J. von Neumann, "Quantum mechanics of infinite systems," Mimeographed version of a lecture given at Pauli's seminar held at the Institute for Advanced Study in 1937, unpublished, John von Neumann Archive, Library of Congress, Washington, DO. [24] J. von Neumann, "Letter to H.D. Kloosterman, March 25, 1953," unpublished, John yon Neumann Archive, Library of Congress, Washington, DC. [25] J. von Neumann, "Letter to H.D. Kloosterman, April 10, 1953," unpublished, John von Neumann Archive, Library of Congress, Washington, DC. [26] J. von Neumann, "Amsterdam talk about 'Problems in Mathematics,' September 2, 1954," handwritten sketch of talk, unpublished, John von Neumann Archive, Library of Congress, Washington, DC. [27] J. von Neumann, "Unsolved problems in mathematics," typescript of the address to the International Congress of Mathematicians, Amsterdam, September 2-9, 1954, unpublished,John von Neumann Archive, Library of Congress, Washington, DC. [28] J. von Neumann, Collected Works Vol. I. Logic, Theory of Sets and Quantum Mechanics, edited by A.H. Taub, Pergamon Press, London, 1962. [29] J. von Neumann, Collected Works Vol. III. Rings of Operators, edited by A.H. Taub, Pergamon Press, London, 1961. [30] J. von Neumann, Collected Works Vol. IV. Continuous Geometry and Other Topics, edited by A.H. Taub, Pergamon Press, London, 1961.

12

]-HEMATHEMATICALINTELLIGENCER

Iii[~L'Al-I~Ii[:-]i*[':-II[r

B0rr0me0 Revisited

Dirk H u y l e b r o u c k ,

Editor I

The paper on Borromean rings by P. Cromwell, E. Beltrami, and M. Rampichini in The Mathematical Intelligencer 20 (1998), inspired several

readers, yielding three contributions interlaced by a Borromean link.

I. Another Geometrical Object Associated with a Borromeo Cathy Kessel

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafd where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? 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 Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende, Belgium e-mail: [email protected]

After reading Cromwell, Beltrami, and Rampichini's article "The B o r r o m e a n rings," I w o n d e r e d if s o m e of the drawings below were m e a n t as an allusion to Borromean rings. It's a bit of a s t r e t c h - but the rings can be distorted into the 3leafed rose on page 14 in Figure 22, and the intertwining in Figures 22, 24, and 27 is reminiscent of s o m e depictions of the rings. Moreover, the figures come from a b o o k dedicated to a Borromeo: Countess Clelia Grillo Borromeo. The book,

Flores geometrici (Geometric Flowers), was p u b l i s h e d in 1728, and, like m a n y b o o k s of that era, w a s w r i t t e n in Latin. Unlike Borromeo, I d o n ' t r e a d Latin, b u t the c o n j e c t u r e t h a t r o s e curves w e r e an allusion to B o r r o m e a n rings s e e m s plausible if one l o o k s at the b o o k ' s figures and dedication. A c o p y of [7ores geometrici has traveled from 18th-century F l o r e n c e to the Bancroft Library, the rare b o o k collection at the University o f California. Reading a b o o k at this library is a little like going to a n o t h e r country: the lib r a r y c u s t o m s are s o m e w h a t different from those of o t h e r libraries on camp u s and, of course, s o m e o f the b o o k s t h e m s e l v e s are strange and wonderful. Like m a n y i n h a b i t a n t s of foreign countries, the b o o k s of the Bancroft do n o t go h o m e with tourists. TTores geometrici w a s written by Guido Grandi, a m o n k of the Camoldolese order who became a professor

of mathematics at the University of Pisa. The b o o k describes rose curves, that is, curves of the form r = R sin nO, e. g., Figures 22-25, and c l e l i a s I c u r v e s inscribed in a sphere w h o s e projections yield rose curves (presumably the boundaries of the regions in Figures 32 and 33). Grandi had written to Leibniz a b o u t these curves in 1713. Ten years later, he described t h e m in a m e m o i r p r e s e n t e d to the Royal Society of London [1]. The c o m p l e t e treatment was Flores geometrici, which was dedicated to Clelia Borromeo, who would be "able to smell the s c e n t of this geometric bouquet." Grandi m a y have k n o w n B o r r o m e o b e c a u s e of h e r interest in m a t h e m a t i c s and science o r b e c a u s e she f o u n d e d the A c a d e m y for Natural Science in Milan. The A c c a d e m i a dei Vigilanti (also k n o w n as A c a d e m i a Claelia Vigilantium) s u p p o r t e d e x p e r i m e n t a l physics, anatomy, and m a t h e m a t i c s . It w a s frequented b y T o m a s s o Ceva, w h o d e d i c a t e d a w o r k to T e r e s a B o r r o m e o (see [2]; p e r h a p s she is a relative of Clelia, but I have f o u n d no further information a b o u t her). Giovanni Girolamo Saccheri, a f o r m e r s t u d e n t of Ceva and n o w p e r h a p s b e s t k n o w n for his w o r k on n o n - E u c l i d e a n geometry, w a s also w e l c o m e d at the A c a d e m y [3]. The little information that I have f o u n d a b o u t Clelia B o r r o m e o herself is i n t r i g u i n g - - a n d will p e r h a p s p r o v o k e m o r e p r o f o u n d historical research. She w a s b o r n in G e n o a in 1684, m a r r i e d Giovanni B e n e d e t t o B o r r o m e o , and

The Bancroft Library at the University of California, Berkeley. Photo by Ben Ailes.

9 1999 SPRINGER VERLAG NEW YORK, VOLUME 21, NUMBER 4, 1999

13

Figure 2.

Facing page: Drawings from Flores geometrici,

dedicated

to

Clelia

Borromeo.

QA567.G7, The Bancroft Library.

II. The 3-ring Symbol of Ballantine Beer Ned Glick

died in Milan 1777 (see [4]), so she w o u l d have b e e n a b o u t 44 w h e n TTores geometrici w a s published. She r e a d Latin, Italian, Spanish, French, English, German, a n d Arabic, and w a s so l e a r n e d that a m e d a l with the inscription "Gloria Genuensium" ("Glory of the Genoese") w a s s t r u c k in h e r h o n o r [5, 6]. She c o r r e s p o n d e d with p e o p l e in m a n y E u r o p e a n countries, a n d h e r h o m e w a s frequented b y cultivated citizens of Milan a n d illustrious travelers [5, p. 75]. In Woman in Science, Mozans d e s c r i b e d her as follows: In addition to a special talent for languages, she p o s s e s s e d so great a c a p a c i t y for m a t h e m a t i c s and mechanics that no p r o b l e m in these s c i e n c e s s e e m e d b e y o n d h e r comprehension. Surely a p e r s o n w h o w o u l d a p p r e c i a t e t r a n s f o r m i n g B o r r o m e a n rings to 3leafed r o s e s to clelias!

In their p a p e r on the B o r r o m e a n rings, P. Cromwell, E. Beltrami, and M. Rampichini note that the B o r r o m e a n design is k n o w n in North A m e r i c a as the Ballantine rings, a n d that this t e r m is used in a t e x t b o o k (cf. page 60). In fact, the rings w e r e widely k n o w n in the 1950s w h e n Ballantine beer, " 'crisp' as a line-drive double," w a s a principal b r o a d c a s t s p o n s o r for New York Yankees b a s e b a l l games. F r o m a 1958 b a s e b a l l game t h a t I a t t e n d e d at the Yankee S t a d i u m "Home of Champions" (again in 1998!) I still have a souvenir "Official P r o g r a m and S c o r e c a r d - - 1 5 cents" in w h i c h there is a full-page a d v e r t i s e m e n t for the b e e r p r o d u c e d by "P. Ballantine & Sons, Newark, New Jersey," as the radio alw a y s announced. This b r a n d w a s s o l d in and a r o u n d New York City. I do not k n o w h o w o r w h e n it c a m e to b e a s s o c i a t e d with the Falstaff Brewing Corporation, as s t a t e d b y Cromwell, Beltrami, and Rampichini. (Falstaff has a complic a t e d history. At v a r i o u s times it con-

REFERENCES

1. A. Nantucci, Guido Grand/, in Dictionary of Scientific Biography, ed. C. Gillispie, Scribner, New York, 1970, 498-500. 2. R. Westfall, Ceva, http://es.rice.edu/ES/ humsoc/Galileo/Catalog/Files/ceva_tom. html. 3. R. Westfall, Saccheri, http://es.rice.edu/ES/ humsoc/Galileo/Catalog/Files/saccheri.html. 4. Dizionario Enciclopedica Italiana, vol. 2, 1955. Thanks to Gary Tee at the University of Auckland for providing this information. 5. G. Maugain, E-tude sur I'E-volution Intel-

Cathy Kessel University of California at Berkeley Graduate School of Education Berkeley, California 94720-1670 USA e-mail: [email protected]

trolled b r e w e r i e s in St. Louis, Omaha, New Orleans, San Francisco, and elsewhere, as well as in Cranston, Rhode Island, w h e r e Ballantine b e v e r a g e s m a y have b e e n p r o d u c e d for s o m e period.) The a u t h o r s refer to B o r r o m e a n ring s y m b o l i s m in m e d i e v a l Christian i c o n o g r a p h y (pages 58-59). But the Ballantine rings w e r e l a b e l e d Purity, Body, and Flavor; a n d these values m o s t p r o b a b l y did n o t refer to the myst e r y o f the Christian Trinity. Dr. Tim Higgs and I have p r o d u c e d from the a d v e r t i s e m e n t s e p a r a t e good quality enlargements o f the 3-ring symb o l a n d of the b e e r can on w h i c h it was featured. Other images of Ballantine cans can be found on the World Wide Web; see w w w . o n e i m a g e . c o m / - a m -

mie/FAM.html and w w w . breweriana. com/beercans.html--and t h e r e m a y be

lectuelle de I'ltalie: De 1657 a 1750 Environ,

Librarie Hachette et c/e, Paris, 1909. 6. H. J. Mozans [pseudonym of John Augustine Zahm], Woman in Science, MIT Press, Cambridge, MA, 1974 (first published 1913).

The Ballantine rings: Purity, Body, Flavor.

allan'1ine beer the "crisp" refresher...

Ballantine beer advertisement, 1958.

o t h e r sites offering Ballantine images. One site indicates that novelist E r n e s t H e m i n g w a y w a s a Ballantine ale enthusiast, having b e e n i n t r o d u c e d to this b r e w by Robert Benchley, the humorist. The Web, however, s e e m s to have m o r e sites d e v o t e d to B o r r o m e a n rings t h a n to Ballantine. Many refer to The Mathematical InteUigencer or r e p e a t information from it. I offer one further context, however. The 1986 World E x p o s i t i o n in Vancouver, British Columbia, Canada, u s e d three interlocking rings to c r e a t e a stylized 86 within its E x p o 86 symbol, which a p p e a r e d t h r o u g h o u t British Columbia, e.g., on t e l e p h o n e directories. In D e c e m b e r 1998 this image

VOLUME 21, NUMBER 4, 1999

15

EXPO

The 1986 World Exposition Vancouver British Columbia, Canada May 2 - October 13, 1986 Exposition internationale de 1986 Vancouver Colombie-Britannique, Canada Du 2 mai au 13 octobre 1986

A Provincial Crown Corporation

O Sanctioned by the International Bureau of Expositions @1985, Expo 86 Corporation. EXPO 86 name and Iogos are registered trademarks. The logo with Borromeanrings used in the 1986 World Exposition in Vancouver(a registered trademark). with interlocking rings was reversed for a one-day music-oriented Expo 98. Yet, the 1950s advertising symbol of Ballantine beer made the strongest impression on the present contributor. I remember the brand up to today, although I was then too young to drink the product. Maybe the Borromean rings enchanted a future mathematician long before I even knew what the subject was about. Ned Glick Department of Statistics and Department of Health Care & Epidemiology University of British Columbia Vancouver, B.C. V6T 1Z2 Canada e-mail: [email protected]

III. If One of the Borromean Links Fails, so Do the Liberal Arts

esting mathematical significance with plenty of history behind them, as shown in the paper on Borromean rings by P. Cromwell, E. Beltrami, and M. Rampichini. Their topological properties are an excellent way to represent my belief that the Sciences, Humanities, and Social Sciences only fmd their full strength when linked together; if one of the links falls, so do the Liberal Arts. Since last fall the logo (see figure) is used on the web page www.cas.gmv.edu and on the letterhead of the College of Arts and Sciences at George Mason University. Daniele C. Struppa Dean, College of Arts and Sciences George Mason University Fairfax, VA 22030 U.S.A.

Daniele C. S t ~ p p a As new Dean of the College of Arts and Sciences at George Mason University, I was tasked with graphically representing the interrelationship between the different disciplinary clusters, the Sciences, the Humanities, and the Social Sciences, which make up the m o d e m Liberal Arts. Being a mathematician, and being from Milan (only an hour from the Isole Borromee), I thought that the Borromean tings were the perfect symbol. The Borromean tings have an inter-

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16

THE MATHEMATICAL INTELLIGENCER

Figure 2.

K.H. KIM, F.W. ROUSH, AND J.B. WAGONER

The Shift Eq u iva ence Prob em

ubshifts of f i n i t e type have appeared naturally in subjects ranging f r o m dynamical systems, to ergodic theory, to statistical mechanics, to C*-algebras, to coding and information theory. A key method in studying them uses equivalence theory.

This grew out of R.F. WiUiams's fundamental work [Wil], in

which he formulated an algebraic approach to the topological classification problem for subshifts of finite type by introducing the concepts of strong shift equivalence and shift equivalence, the latter being much more accessible. Since 1974, the question of whether shift equivalence implies strong shift equivalence has been a well-known and tantalizing problem, called the Shift Equivalence Problem or Williams's Conjecture. This is a marvelous problem for at least three reasons. First, it comes from the fundamental and still open conjugacy problem in dynamics. Second, it is easy to state in very elementary terms. Third, it is unexpectedly and closely related to other branches of mathematics outside dynamics, such as algebraic K-theory and topological quantum field theory. We will give an account of the Conjecture, its recent disproof, and where this leaves the subject. The Shift Equivalence Problem came from the 1960s and early 1970s, which were tremendously exciting years for dynamics. Simultaneously, there was an explosion of activity in differential topology and algebraic K-theory. Taken

18

strong shift

THE MATHEMATICAL INTELLIGENCER 9 1999 SPRINGER-VERLAG NEW YORK

together, the papers [Fr, Mi2, S1] established a close connection between zeta functions, which count periodic points of dynamical systems, and Alexander-Reidemeister torsion, which gives invariants of knots in R 3 and is related to the algebraic K-theory group K1. In [C, HW] a connection is made between diffeomorphisms of manifolds and the algebraic K-theory group K2. So it was natural to wonder whether there might be a connection between symmetry groups of dynamical systems and K2. This theme was developed during the 1980s and 1990s, and it ultimately produced machinery which was used in [KR5, KR6] to find the first counterexamples to the Shift Equivalence Problem for primitive matrices in 1997. We will explain how a certain cohomology class sgc~, constructed by studying how symmetries of dynamical systems act on periodic points, is cruciai for the counterexamples. Another approach was subsequently found in [W7] using an invariant r in the algebraic K-theory group K2 of a truncated polynomial ring. The construction of ~h2mcomes very directly from the analogy with K2 invariants for diffeomorphisms and is quite dif-

ferent from t h e c o n s t r u c t i o n of sgc2. The first t w o a u t h o r s have r e c e n t l y s h o w n that, surprisingly, 62 = sgc2. Even t h o u g h the w o r k on shift equivalence and the w o r k on d i f f e o m o r p h i s m s and K2 w e r e d o n e a r o u n d the s a m e time in the early 1970s b y p e o p l e w h o w e r e friends, a connection w a s n o t m a d e b e t w e e n the t w o areas at that time. Only now, a h n o s t a quarter of a c e n t u r y later, t h e s e ideas turn out to b e closely related. This article primarily d i s c u s s e s the a p p r o a c h to the c o u n t e r e x a m p l e s using sgc2. See [G, W8] for m o r e details on the relationship a m o n g strong shift equivalence, algebraic K-theory, and topological q u a n t u m field theory. The n e x t section explains subshifts of finite t y p e and gives the p r e c i s e s t a t e m e n t of the Shift Equivalence Problem. The following two s e c t i o n s develop a general strategy for analyzing the difference b e t w e e n shift equivalence and s t r o n g shift equivalence. Then, w e introduce sgc2 and p r o c e e d to give an explicit c o u n t e r e x a m p t e to the Shift Equivalence P r o b l e m using it. The final section briefly disc u s s e s further p r o b l e m s a n d topics. G o o d general b a c k g r o u n d r e f e r e n c e s are [K, LM, R]. A sampling o f articles is [A, BH2, CK2, E, M, $1, $2].

Subshifts of Finite Type and the Classification Problem Messages can b e r e p r e s e n t e d as s t r e a m s of s y m b o l s moving from right to left, and it is r e a s o n a b l e to c o n s i d e r t w o long s e q u e n c e s of s y m b o l s to be close to one a n o t h e r o r a l m o s t the s a m e if t h e y agree e x c e p t p o s s i b l y for a few s y m b o l s at the beginning a n d at the end. This can b e m a d e p r e c i s e b y defining thefuU n-shift Xn to be the set of biinfinite s e q u e n c e s x = {Xk}, w h e r e each symbol Xk is drawn from an n-letter alphabet { 0 , . . . , n - 1}. It is e q u i p p e d with the p r o d u c t t o p o l o g y making it a Cantor set. The shift h o m e o m o r p h i s m o-n : Xn -"->Z n is defined b y o-n(X)k = Xk+l (i.e., shift one s p a c e to the left). A subshift of f i n i t e type arises by fLXing a finite set F of w o r d s (of f'mite length) in the s y m b o l s { 0 , . . . , n - 1} and then excluding from Xn all x which contain a w o r d in F. Very p r a c t i c a l e x a m p l e s are the run-length-limited subshifts u s e d in d a t a storage. Consider the subshiff L(2, 7) of X2, w h i c h consists of all those x which have infmitely m a n y ones to the right and left and which have at least two b u t not m o r e than seven zeros b e t w e e n any two successive ones. This m o d e l s a magnetic tape zipping by, with 1 being the s y m b o l for reversal of magnetic fields and 0 being the s y m b o l for nonreversal. Insisting upo n at least two zeros helps p r e v e n t intersymbol interference o r confusion of magnetic fields. The b o u n d of seven zeros helps maintain accuracy of a clock, which is u p d a t e d at each reversal. Alternatively, and equivalently, w e c a n obtain a subshift of finite type b y considering an m • m z e r o - o n e transition m a t r i x A. Given such an A, define [LM] the subshift of finite t y p e {XA, O-A} of the Bernoulli m-shift (Xm, o-m) b y letting XA b e the s u b s e t of s e q u e n c e s satisfying A(xk, Xk+t) = 1 for all - w < k < ~. In o t h e r words, XA is obtained by excluding t h o s e x in Xm that contain 2-blocks [ij] w h e r e A(i, j ) = 0. By definition, O-A : O-m~(A9

Example: The m • m matrix of all l ' s gives the full m-shift (i.e., any s y m b o l is a l l o w e d to follow any other one). The finite set of e x c l u d e d b l o c k s is empty. A n o t h e r e x a m p l e is the golden m e a n shift Xgm, w h i c h is a subshift of )(2 and arises from the m a t r i x

m=(ll 10) Since Agm (1,1) = 0, the s y m b o l 1 c a n n o t b e followed by itself. Thus, Xgm consists of all s e q u e n c e s o f O's and l ' s w h e r e each 1 m u s t have a 0 to its i m m e d i a t e right. The only excluded b l o c k is [11]. The transition m a t r i x is always in ZO, the set of z e r o - o n e matrices. Here is a n o t h e r p r o c e d u r e , called the edge shift c o n s t r u c t i o n [LM], which yields a subshift of finite type (XA, O-A) from any m • m non-negative integral m a t r i x A = {Aij}. The m a t r i x A can be v i e w e d as a d i r e c t e d graph which has m vertices and w h i c h h a s Aij edges going from the v e r t e x i to the v e r t e x j . O r d e r the set S # of edges of the graph A, and define the z e r o - o n e m a t r i x A # : S # • S # ~ {0, 1 } by letting A # (a, fl) = 1 iff the end vert e x of the edge a is the start v e r t e x of the edge/3. Then, one defmes (XA, O-A)

=

{XA#, O-A#}"

(1)

This is a subshift of the full shift (Xm#, o-m#), w h e r e m # is the n u m b e r of edges o f A. If A is a z e r o - o n e matrix, there are two c o n s t r u c t i o n s of a subshift of finite type associa t e d to A; n a m e l y {XA, O-A}and (XA, O-A) = {XA#, O-A#}.These are well k n o w n to be canonically equivalent. See [W4]. A subshift of finite type is an e x a m p l e of a discretetime d y n a m i c a l s y s t e m (X, f ) (i.e., a h o m e o m o r p h i s m f : X ~ X of, say, a c o m p a c t s p a c e X). F o r any (X, f), one s e e k s its intrinsic p r o p e r t i e s or invariants; this is m a d e precise by the i d e a of topological conjugacy. We say that f : X---> X and g : Y---~ Y are topologically conjugate provided there is a h o m e o m o r p h i s m a: X---~ Ysuch that a f = gc~. One of the m o s t i m p o r t a n t a n d basic invariants of topological c o n j u g a c y is the topological entropy h ( f ) defined b y Adler, et al. [AKM]. This is a t o p o l o g i c a l version of ent r o p y in ergodic t h e o r y [Kol] and channel c a p a c i t y in inf o r m a t i o n t h e o r y [Sh]. In our ease, X = XA a n d f = O-A.The topological e n t r o p y can be c o m p u t e d b y the beautiful formula

h(o-A)

=

lim 1 log

n ---~~ n

Bn

=

log /~A,

:IGURE

A and A #

a#=

|111

/ooo \lll

VOLUME 21, NUMBER 4, 1999

19

where Bn is the number of blocks of length n appearing in sequences in XA and where hA is the Perron-Frobenius eigenvalue, the largest real root of det(tI - A). See [LM, Sect. 4]. For example, the Perron-Frobenius eigenvalue AgE for the golden mean shift is the largest root . F ~the equation det(tI - A g m ) - - t 2 - t - 1 = 0. So AgE = (s Another important invariant is the Artin-Mazur zeta function, which elegantly combines information about the n u m b e r of periodic points. For a subshift of finite type, it is given by the formula ~A(t) = exp ( ~n=t Pn(~

tn)

where pn(erA) is the n u m b e r of points x in XA such that O~AA(X)= X. In fact, pn(O-A)= Tr (An), and the B o w e n Lanford formula states that 1

~(t) -

d e t ( I - tA)

As an example, consider the golden mean shift. Induction shows

n

(rn+t

fn

AgE= \ Fn

Fn-g '

where, as usual, the Fibonacci numbers are defined byF0 = 0, Ft = 1, F2 = 1, and Fn = Fn-1 + Fn-2 for n -> 3. Therefore, pn((Tgm) : Fn+l + Fn-1 and (gin(t) = exp ( ~n=l Fn+t+Fn-ttn)n 1 d e t ( I - tAgm) 1 1 -

t-

t2 "

Another way in which subshifts of fmite type arise is through the use of topological Markov partitions, which were a key feature in the development of dynamical system theory in the 1960s. Smale's influential article [S1] summarized h o w topological Markov partitions could be used to study the periodic and recurrent behavior of discretetime dynamical systems f : X - - ~ X and continuous flows. See also [A; K, Sect. 1.2; LM, Sect. 6.5; $2] for some beautiful examples. Typically, X might be the nonwandering set of a diffeomorphism on a high-dimensional manifold; that is, X might be the set of points which keep coming arbitrarily close to themselves under iterates o f f rather than eventually moving away to another part of the manifold. When X is zerodimensional, a topological Markov partition U = {U1, 9 9 9 Un} is a way of writing it as a disjoint union

X = U1U U2 U ... U Un

(2)

of sets Ui which are open and closed as subsets of X and which satisfy certain axioms [K, LM, PT, Wl]. The standard Markov partition of a subshift of finite type XA inside Xn is given by

X=U1AU...UUn, 20

THE MATHEMATICAL INTELLIGENCER

where UA consists of all sequences x = {Xk} in XA with x0 = i. Given (2), define the zero-one matrix M = M(U) = {Mij} by the condition

Mij = 1 iff U/N f - l U j is not empty.

(3)

Strictly speaking, the order of the sets Ui of a topological Markov partition is not specified in advance. To get the n • n matrix M, we choose an order for the Ui and then apply (3). A different choice of ordering produces a matrix conjugate to M b y a permutation. As expected, M(UA) = A. And any topological Markov partition (2) arises in this way! Indeed, we can define a topological conjugacy 77 : X ---> X M

(4)

by the condition that 7r(X)k = i ifffk(x) is in Ui. This just says that ~r(x) is the sequence of symbols chosen from the set { 1 , . . . , n} obtained by reading off where the point x lands after applying the p r o c e d u r e f to it k times. Topological Markov partitions are the topological counterparts of Markov partitions which o c c u r in statistics and ergodic theory, where the transition matrices have nonintegral entries specifying transition probabilities from one state to another. It is a well-known fact [PT] that the nonnegative integral transition matrix for a subshift of finite type determines a stochastic transition matrix, which, in turn, gives rise to a unique measure of maximal entropy preserved by topological conjugacy. We often k n o w that a topological Markov partition exists [Bow]. If its structure is clear enough to read off the transition matrix [A, K, LM, S1], it can be used to compute such information as the entropy and the zeta function. However, these two invariants are certainly not complete. Moreover, if one Markov partition exists f o r f : X ~ X, then there are infinitely many. In other words, there are typically infinitely m a n y ways to present dynamical systems that have equivalent behavior. So there is the fundamental and still open Classification Problem. Given non-negative integral matrices A and B, when are the corresponding subshifts of finite type (XA, (rA) and (XB, (rB) topologically conjugate?

Williams formulated an algebraic approach to this question, which we n o w review. An elementary strong shift equivalence (R, S) : A ---)B over the non-negative integers Z + consists of two non-negative integral matrices R and S of finite size satisfying the strong shift equivalence equations

A = RS,

SR = B.

(SSE)

Williams defined the matrices A and B to be strong shift equivalent over Z + iff there is a chain of elementary strong shift equivalences over Z + between them, and in [Wil] he proved T h e o r e m 1. (XA, O'A) and (XB, O'B) are topologically conjugate iff A and B are strong shift equivalent over Z +

Also see [LM, Fr, Wl]. Here is an example. Let

z_-(i1 '1)'

aXA) and (XB, O~B) are topologically conjugate for k - N. Williams [Wil] a n d the first two a u t h o r s [KR1] p r o v e d

01, 1 1

T h e o r e m 2. (XA, O-A) and (XB, O'B) are eventually conjugate i f f A and B are shift equivalent over Z +.

/10/

0

1

1

C h e c k that A = R S and SR = B. Moreover, {XA, O-A} = (X2, 0"2) and {XB, O'B} is a subshift of 0(3, 03). W h e n e v e r w e have A = R S and SR = B in ZO, the strong shift equivalence (R, S) : A --~ B p r o d u c e s a topological conjugacy

c(R, S) : {XA, O-A}---> {XB, ~

1 = A(Xk, Xk+l) = ~

R(Xk, i)S(i, Xk+l). i B e c a u s e A, R, and S are z e r o - o n e matrices, there is e x a c t l y one i for w h i c h R(xk, i ) = S(i, X k + l ) = 1. By definition, Yk = i. We have c(S, R)c(R, S) = c(A, 1) = O-A a n d c(R, S)c(8, R) = c(B, 1) = O-BThis generalizes to m a t r i c e s over Z + using the #-construction; namely, as e x p l a i n e d in [LM, 7.2] and [W4, Sect. 2], the equations A = R S and SR = B over Z + yield

GA = {vl V E R A and yAP ~ Z n for s o m e integer p > 0} G~ = {v I v E GA and vAP >_ 0 for some integer p > 0} (9) SA = the i s o m o r p h i s m of GA i n d u c e d b y A. E x a m p l e . A = {2}. Then GA = Z[89 G~ = Z[89 +, a n d SA is multiplication b y 2.

c(R, S) : (XA, O-A) --> (XB, O-B).

(6)

A m o r e algebraically t r a c t a b l e c o n c e p t is shift equivalence. The m a t r i c e s A and B are said to be shift equivalent over Z + iff there are non-negative m a t r i c e s R a n d S a n d a positive integer k called the lag satisfying the shift equivalence equations

B S = SA,

Shift equivalence over Z + is m u c h e a s i e r to d e t e r m i n e than strong shift equivalence over Z +. Shift equivalence is decidable [KR1, KR2], and Krieger c h a r a c t e r i z e d shift equivalence in t e r m s o f the d i m e n s i o n group triple (GA, G~, SA). See [LM, 7.5]. The m o s t c o n c r e t e definition of GA uses the eventual range RA. Suppose A is an n • n matrix, and c o n s i d e r it as a linear t r a n s f o r m a t i o n from Qn to itself. We let RA = Am(Qn), w h e r e m is sufficiently large that A : Am(Q n) ~ Am+ I(Q n) is an isomorphism. Then

S#R # = / 3 #

o v e r ZO, w h i c h then give a c o n j u g a c y

A R = RB,

Shift Equivalence Problem. Does SE over Z + i m p l y SSE over Z + ?

(5)

as follows. Let x = {xk} and y = {Yk} w h e r e y = c(R,S)(x). Then

A # = R#S ~,

See [LM, 7.5.15]. A s t e p in the p r o o f of this result is the observation that (XA, O~A)is topologically conjugate to (XA k, OAk) 9 Obviously, c o n j u g a c y implies eventual c o n j u g a c y and strong shift equivalence implies shift equivalence. Williams's influential w o r k [Wil] brought forth the following question.

A k = RS,

SR = B k.

(SE)

An e l e m e n t a r y strong shift equivalence occurs w h e n k = 1. Two subshifts of finite type (XA, O-A) and (XB, O-B) are eventually conjugate if t h e r e is an integer N such that (XA,

E x a m p l e . A = Agm. Since det(Agm) = - 1, w e have GA = Z 2. The v e c t o r ( s 1) is a left eigenvector c o r r e s p o n d i n g to the P e r r o n - F r o b e n i u s eigenvalue A = Agm, which satisfies A2 = A + 1. The c o r r e s p o n d e n c e sending the p a i r of integers (a, b) to aA + b induces an i s o m o r p h i s m b e t w e e n GA a n d Z[A], c o n s i d e r e d as a subgroup of the real n u m b e r s R equipped with its usual ordering. Via this isomorphism, SA c o r r e s p o n d s to multiplication b y A a n d G~ b e c o m e s Z[A] +. The n a m e "dimension group" c o m e s from o p e r a t o r alg e b r a theory, w h e r e a d i m e n s i o n function on the projec-

:IGURE

Strong shift equivalence A(Xk, Xk+l) = 1 Xk

J

•R•Xk • Yk) =

1/ /

Xk+1

Xk+2

A stream

'R(Xk+l, Yk+l) = 1 / / / /

S. \

r B stream Yk

Yk+l B(yk, Yk+l) = 1

VOLUME 21, NUMBER 4, 1999

21

IGURE '~

:IGURE 3

Two representations of (2, 3): 6

111 111 111 lll lll lll

A = (6) = B A#=

6 loops

111 111 111 111 111 111

Edge-shift

S#

R#

A #=

01001 001000 10010 01001 00100

11100~ 11100 00011 00011 00011

S#

B#=

Transition matrix

A= [212]

=B #

(1000 11100 ) )

R -- (2), S = (3)

6=3.2

Elementary SSE in two representations

Transition matrix

Edge shift

6=2.3

~ 6

"--

A#=

%"':

A = [2 O] I111

A#:

R#

"- l] 0]

0100 0010 1001 0100

c(2, 3) = 0"2 X I on (X6, 0"6) ~ (X2 X X3, 0"2 X 0"3) tions in an infinite-dimensional algebra m a y t a k e on nonintegral real n u m b e r values. In particular, a non-negative integral matrix A gives rise to a certain C*-algebra via a Bratelli diagram, and the range of the d i m e n s i o n function is GA. See [E, W6] for e x p o s i t i o n s of these ideas. Krieger's t h e o r e m [LM, 7.5.8] states T h e o r e m 3. Non-negative integral matrices A and B are shift equivalent over Z + iff (GA, G~, SA) is isomorphic to (GB, G~, SB). The m o s t i m p o r t a n t c a s e is when A and B are primitive. This m e a n s there is a positive integer m such that b o t h A m a n d B m have all entries positive. The d y n a m i c a l signifi c a n c e is that the subshifts (XA, O-A)and (XB, O'B) are topo-

llOO1 ,1~176 OOLO/

I i00l[ 0i; oo,o/

.,__

0101 0000 0000

GA = Z ~,

SA = the isomorphism of GA given by A.

B3 arc in graph of B

22

THE MATHEMATICAL INTELLIGENCER

(10)

Thus, Krieger's T h e o r e m and the a b o v e discussion yield C o r o l l a r y 4. Suppose det(A) = _+1 and det(B) = - 1. Then A and B are shift equivalent over Z + iff the matrices A and B are conjugate in GI,~(Z). An old result o f Effros and Williams says that SE over Z implies SSE over Z (without primitivity). See [W3].

Shift equivalence for k = 3 A3

110 001 001 000

logically mixing. It turns out [LM, 7.5] that primitive matrices are shift equivalent over Z + iff t h e y are shift equivalent over Z. This implies that primitive A and B are shift equivalent over Z + iff the d i m e n s i o n group pairs (GA, SA) and (GB, SB) are isomorphic. In this article, we will mainly c o n s i d e r A s u c h that det(A) = _ 1, a n d t h e n there is a very c o n c r e t e description:

:IGURE

arc in graph of A \

ooo

0001J

oo17

111000 11100 00011 00011 00011

11110 0000 0000 0000

A3

B3

-

A 3 stream

-

B 3 stream

Hence, a reformulation of the Shift Equivalence Problem over Z + is S t r o n g Shift E q u i v a l e n c e P r o b l e m . If A and B are primitive non-negative integral matrices, does SSE over Z imply SSE over Z+?

The answer is sometimes "yes" for 2 x 2 matrices [B, CK1, Wi2], and yet in the following well-known example the question is still open. Let An=(ll

n(nl-1))'

Rn=(nl

B n = ( n 1- 1 1 ) ' 1 1)"

(11)

Then det(Rn) = - 1 and AnRn = RnBn. So A n and Bn are SSE over Z, because (Rn, Rn 1 An) : A~ --~ Bn. Observe that A2 = B2. Kirby Baker s h o w e d As is SSE to B3 over Z +. There is a chain of seven elementary SSEs between them involving 4 • 4 matrices. See [LM, Chap. 7]. Is A n SSE to Bn over Z + for n -> 4? It was shown in [KR7, KR8] that An is SSE to Bn over Q+ for n -> 2. Forgetting positivity momentarily, a question coming from the Effros-Williams result is

various c(R, S) which would correspond to deformations or homotopies between paths connecting A and B. Put slightly differently, is there a natural notion of deformation or homotopy between paths connecting A and B such that all homotopic paths give the same T? And does this notion of homotopy capture all relations between t h e c(R, S)? ~The answer to these questions is "yes" and involves the CW complex SSE(ZO). Start by letting the vertices of SSE(ZO) be finite, square matrices A with z e r o - o n e entries. Then, edges from A to B are labeled by elementary strong shift equivalences (R, S) : A ---) B over ZO. It is possible that A = B, in which case a loop is created at the vertex A. This occurs, for example, when we have a permutation P such that A P = PA, for then (P, P-tA) : A ~ A. The 2-cells come from triangles ( R ~ / ( B ~ A

(R3, $3)

2)

(13)

~C

where the following Triangle Identities hold:

R1R2 = R3, R283 = St,

S3Rt =

(14)

s2.

A l g e b r a i c Shift E q u i v a l e n c e P r o b l e m . For what rings A does SE imply SSE?

One goes on to define n-cells of SSE(ZO) as in [W2] or in Number 8 of the last section of this article.

Boyle and Handelman [BH1] have shown SE implies SSE if A is a Dedekind ring.

L e m m a 5. If the Triangle Identities hold, then

c(R3, $3) = c(R1, S1)c(R2, S2).

Strong Shift Equivalence Spaces A general program for studying the difference between SSE over Z + and SSE over Z involves the strong shift equivalence spaces SSE(Z +) and SSE(Z). These are CW complexes arising in the study of topological conjugacies and symmetries of subshifts of finite type, and they were motivated by an analogy with pseudo-isotopy theory and algebraic K-theory as explained in [Wl]. See [BaW, WI-W5, KRWl] also. Previously, they were denoted by RS(Z +) and RS(Z) in the literature. The SSE notation is more appropriate. Consider the following version of Williams's Classification Theorem. T h e o r e m 4. Let T : {XA, qA}---) {XB, ~rB} be a topological

conjugacy, where A and B are in ZO. Then there is a chain of elementary strong shift equivalences (R1, $1), 9 9 9 (Rm, Sm) between A and B in ZO such that T = H c(Ri, Si)%

(12)

i=1

where ei = +1 i f (Re, Si) : Ai-t--) A i and ei = - 1 (Ri, Si) : Ai --->Ai- 1.

if

There are many ways in which T can be written as a product like (12) corresponding to different "paths" of elementary strong shift equivalences connecting A and B. It is natural to ask if there are some general relations between the

This is the basic ingredient for Theorem 6 below which is a precise formulation of how to describe topological conjugacies and symmetries of subshifts of finite type in terms of homotopies of paths and relations among the c(R, S). The definitions of SSE(Z +) and SSE(Z) are exactly the same with ZO replaced by Z + or Z. The same is true for SSE(A +) and SSE(A) for more general rings A. It follows from the definitions and Williams's paper [Will that 9ro(SSE(ZO)) = strong shift equivalence classes over ZO

I1 ~-0(SSE(Z+)) = strong shift equivalence classes over Z +, (15) 7r0(SSE(Z)) = strong shift equivalence classes over Z. See [W4, 2.1] also. The group Aut((rA) of automorphisms of the shift (XA, ~rA) consists of those h o m e o m o r p h i s m s f : XA -->XA which commute with ~rA. It was fwst extensively studied by Hedlund and co-workers at IDA in the 1960s in connection with coding theory. See [H]. The one-to-one and onto m a p f c a n be viewed as a way to encode and decode symbol sequences uniquely. Continuity means that if two sequences agree on a long block, then so will their coded images. Commuting with ~rA means that a block is coded the same way no matter when it appears (i.e., the coding is independent of time).

VOLUME 21, NUMBER 4, 1999

23

In the 1980s, t h e r e w a s a r e n e w a l of i n t e r e s t in Aut(o-A). Clearly, o-A itself b e l o n g s to Aut(~A). Any p e r m u t a t i o n on n s y m b o l s gives an e l e m e n t of Aut(qn) b y s i m p l y perm u t i n g each symbol; t h a t is, a(X)k = a(Xk) for x = {Xk}. Aut(o-n) contains the d i r e c t s u m of any c o u n t a b l e set of finite g r o u p s as well as t h e d i r e c t sum o f c o u n t a b l y m a n y c o p i e s of Z. In general, Aut(o'A) is a huge n o n c o m m u t a t i v e c o u n t a b l e group w h e n A is primitive. See [BLR] a n d [K, Chap. 3]. Other w o r k on Aut(crA) m a y be f o u n d in [BF, BK1, BK2, F, KRWl-KRW3, N, W l - W 5 ] . An i m p o r t a n t normal s u b g r o u p of Aut(o-A) is the group Simp(~rA) of s i m p l e a u t o m o r p h i s m s [N], w h i c h arise from a u t o m o r p h i s m s of d i r e c t e d graphs that k e e p the v e r t i c e s fixed. We let Aut(sA) b e t h e group o f a u t o m o r p h i s m s of the d i m e n s i o n p a i r (GA, SA). By definition, this c o n s i s t s of t h o s e i s o m o r p h i s m s of the group GA w h i c h c o m m u t e with the i s o m o r p h i s m SA. As m e n t i o n e d in (10), the c a s e relevant to this article is w h e r e det(A) = _+1. We t h e n have the v e r y c o n c r e t e description Aut(sA) = the elements of Gln(Z) w h i c h c o m m u t e with A.

(17)

w h e r e O is the ring o f integers in the n u m b e r field Q[A] -~ Q[A] g e n e r a t e d b y the P e r r o n - F r o b e n i u s eigenvalue A of A. See [LM, BLR, BMT]. In particular, using (17), it is possible in m a n y c a s e s to c o m p u t e generators for Aut(sA) explicitly using the c o m p u t a t i o n a l algebra p r o g r a m PARI. T h e o r e m 6. There are i s o m o r p h i s m s (A)

~rl(SSE(ZO), A) = Aut((rA), ~ri(SSE(ZO), A ) = 0 f o r i >-- 2;

(B)

~rl(SSE(Z+), A) = Aut(~A)/Simp((rA);

(C)

~-I(SSE(Z), A) = Aut(sA), ~-i(SSE(Z), A ) = 0 f o r i >- 2.

The proofs of T h e o r e m s 4 a n d 6 uses the analogy with p s e u d o - i s o t o p y theory, considering the set of t o p o l o g i c a l M a r k o v partitions to b e like the contractible s p a c e of realv a l u e d functions on a manifold. In fact, the set PA of all t o p o l o g i c a l Markov p a r t i t i o n s of a subshift of finite t y p e (XA, ~rA) can be given the s t r u c t u r e of a simplicial c o m p l e x w h i c h turns out to be contractible [BaW, Wl]. The m a i n p o i n t in Williams's p r o o f o f T h e o r e m 4 really a m o u n t s to s h o w i n g that PA is connected. The p r o o f of T h e o r e m 6 u s e s simple connectivity of PA for the first p a r t of (A) a n d full contractibility Of PA for the s e c o n d p a r t of (A). The Triangle Identities (14) c a m e from triangles in PA. See [W8].

SSE Strategy We are n o w r e a d y to d e s c r i b e a strategy, n o t i c e d independ e n t l y by the first t w o a u t h o r s and by the third author, for p r o d u c i n g c o u n t e r e x a m p l e s to the Strong Shift Equival e n c e Problem. It c o m e s from the e x a c t h o m o t o p y se-

24

THE MATHEMATICALINTELLIGENCER

A :r

S +) --~ G

(18)

w h e r e G is, say, an abelian group, a n d w h e r e A has the properties

(16)

If the characteristic p o l y n o m i a l of d e t ( t I - A) is irreducible, w e have Z[A]* C hut(sA) C O* C Q[A]*,

quence o f the p a i r (SSE(A), SSE(A§ w h e r e A is a ring containing i a n d having a set of non-negative elements A + which c o n t a i n s 0 and 1 and w h i c h is c l o s e d u n d e r addition and multiplication. We will a s s u m e A + satisfies the condition that if a and b are in A + and a + b = 0 or ab = 0, t h e n either a = O o r b = 0. Typical e x a m p l e s o f A are the integers, a subring o f the real numbers, a ring of p o l y n o m i a l s in c o m m u t i n g o r n o n c o m m u t i n g v a r i a b l e s with integer coefficients, a ring o f Lanrent p o l y n o m i a l s with integer coefficients, a n d the integral group ring o f a group. Let S + d e n o t e a union of c o m p o n e n t s in SSE(A+). F o r example, S + c o u l d be SSEn(A+), w h i c h consists of t h o s e c o m p o n e n t s o f SSE(A § containing vertices A satisfying Tr(A) = Tr(A 2) . . . . . Tr(A n) = O. Let ~rl (SSE(A), S +) denote h o m o t o p y classes of p a t h s in SSE(A) with e n d p o i n t s in S +. We w o u l d like to find a function

h ( a f i ) = A(a) + h(fl), A ( a ) = 0 w h e n e v e r a lies in S +.

(19)

Let A and B be vertices in S +, and c h o o s e a path a from A to B in SSE(A). If fi is a n o t h e r p a t h from A to B, w e have A(Oz) = A(~) -{- A(O/~-I).

Consequently, t h e r e is an invariant A(A~B) = A ( a )

in

G m o d A(~-i(SSE(A), A)),

(20)

which v a n i s h e s if there is a p a t h from A to B in S +. In particular, a c o u n t e r e x a m p l e to the Strong Shift Equivalence P r o b l e m can b e o b t a i n e d b y finding a function A t o g e t h e r with m a t r i c e s A a n d B such that A(A,B) = A(fl) r 0 for s o m e fi A ( a ) = 0 w h e n e v e r a is in ~-I(SSE(A), A).

(21)

The Sign-Gyration-Compatibility Condition Consider an edge (R, S) : P--~ Q in SSE(Z). Let sgc2(R, S) = Z

RikSkiRjlSlj +

i<j,k>l

i <j,k>-t

RikSkjRjtSli + ~ Rik(Rik -- 1) S2i. i,k 2

(22)

Let A a n d B b e vertices in SSE2(Z+). C o n s i d e r a p a t h ~/ from A to B in SSE(Z). Write ~/as a c o n c a t e n a t i o n

]/=

fi

k=l

Y(Pk, Qk) Ek

where ek = + 1 if (Pk, Qk) : A k - 1 --~ Ak and ek = -- 1 if (Pk, Qk) : Ak --) Ak 1. T h e o r e m 7. The f o r m u l a sgc2(y) = Z k=l

ekSgC2(Pk, Qk)

mod 2

g(R) : (Gp, Sp) --) (GQ, SQ)

:IGURE (

just comes from the isomorphism of the eventual range Rp to the eventual range RQ induced by R. Moreover, if (R, S) : P--~ Q over ZO or Z +, then g(R) takes G~ isomorphicaUy to G~. In view of the isomorphisms in Theorem6, 6A may also be described as the sequence of fundamental groups 6A : Irl(SSE(ZO), A) --> rrl(SSE(Z+), A) --) 7rl(SSE(Z), A). Lying inside Aut(~A) is the important normal subgroup of inert automorphisms defined by the exact sequence 1 --) Inert((rA) --) hut((rA) ~ Aut(sA)

sgcz(fi) v 0 sgc2(c0 = 0 for all loops cr

As observed in (17), Aut(sA) is typically a finitely generated abelian group, and therefore Inert((rA) contains most of the complexity and richness of Aut(~A). For example, if p is prime, then Aut(sp) = Z(~{__I}. The generator of the infinite cyclic summand is Sp = ~ p ( O ' p ) , and Aut(~p) = Z@Inert((rp).

defines a function sgc2 : ~'I(SSE(Z), SSE2(Z+))--~ Z/2Z

satisfying (19). Of course, the rather complicated formula (22) and Theorem 7 did not just appear from thin air. They have a history going back more than 10 years and a conceptual background which comes from studying Aut(o-A). The two main ingredients are the sign and gyration numbers and the dimension group representation. The first homomorphisms are representations of Aut((rA) to finite cyclic groups. The second is essentially a matrix group representation. What connects them are the sign-gyration-compatibility conditions, which will be the lever by which we will implement the strategy we outlined above. In the influential paper [BK1], Boyle and Krieger defined the sign and gyration number homomorphisms OSm : Aut(vrA) --->Z/2Z for m >-- 1 GYm : Aut(c~A) ---)Z/mZ for m --> 2. OSm(a) is the sign of the permutation a induces on the orbits of length m and GYm(a) is the average measure of how a moves orbits of length m parallel to themselves. To define GYm(a), list the orbits of length m and choose a point bi on the ith orbit. Write O'A(bi) = (y~i (bj). Then GYm(a) = ~ ri

mod m.

i

This is independent of the choice of base points bi. Boyle and Krieger proved that certain sign-gyration-compatibility conditions (see below) hold between GYm(a) and various OSm/2r(a) for involutions a in Aut(~n). This was subsequently proved for simple automorphisms by Nasu [N]. The dimension group representation 6A : Aut((rA) --~ Aut(sA) was first defined dynamically by Krieger. This approach [BLR, LM] shows that if (R, S) : P--> Q over Z, then the induced isomorphism of dimension groups

In particular, any product of finite-order automorphisms in Aut(qp) is inert. Fiebig [F] extended the work of Boyle and Krieger by showing that the sign-gyration-compatibility conditions hold for finite-order inert automorphisms. Finally, the authors [KR3, KRWl] proved the sign-gyration-compatibility conditions hold for all inert automorphisms. These results may be interpreted as vanishing of the so-called sign-

gyration-compatibility condition homomorphism SGCCk : Aut(o-A) --->Z/kZ on various types of elements of Aut(o-A). See [KRWl]. By definition, SGCCk = GYk + Z OSk/2~, i>0

where O S k / 2 i = O if k/2 i is not integral and where Z/2Z is identified with the subgroup {0, k/2} of Z/kZ when k is even. For example, when k -- 2, we have SGCC2 = GY2 + OS1. The expression (22) is simply the explicit algebraic formula [KRWl] for the dynamically defined SGCC2 on the conjugacy c(R, S) : (XA, O'A)--* (XB, ~rB) arising from an elementary strong shift equivalence (R, S) : A ~ B over Z § computed with respect to the lexicographical ordering of fixed points and points of period 2. In [KRWl], it was shown that there is a commutative diagram tut((rA) ~-~ Aut(sA)

Z/kZ when A is primitive. It was this construction that led directly to Theorem 7 and its proof in the primitive case. The main point is that the algebraic expression (22) makes sense for any (R, S) : A ~ B over Z, not just over Z +, and that homotopy invariance is satisfied; namely

VOLUME 21, NUMBER 4, 1999 2 5

P r o p o s i t i o n 8. The cocycle condition

X=

sgc2(R3, $3) = sgc2(R1, $1) + sgc2(R2, $2)

holds f o r a triangle in SSE(Z). This was first verified in [KRW1] u n d e r the condition that the vertices in the triangle lie in a c o m p o n e n t of SSE(Z), which contains a primitive non-negative integral matrix. Boyle subsequently f o u n d a proof, presented in [KR6, W8], which eliminates the primitivity condition and vastly simplifies the original a r g u m e n t in [KRWl].

0 0 0

I! ~

0 0 0 0

(0 1 0 0 0 1 1

A=RS=

The cohomology class sgc2 can be used to give very explicit c o u n t e r e x a m p l e s to the Shift Equivalence P r o b l e m for primitive matrices A a n d B. The method for f'mding A a n d B comes from [KR5, KR6]. In the example below, we use the above strategy with A = sgc2 and formula (22). Let M be the 4 • 4 matrix 00 1 0

01 0 1

01 0 0

from [KRW1, 4.1] with characteristic polynomial t4 - t - 1. It satisfies the equation ( M - / ) ( M 4 + M 3 + M 2) = M.

- 10 1

01 -1

0

F

1

0 0 -1

[

~2

2

2

1

1 1 ,1

2 1 1

2 2 1

1 1 1

so that letting A = R S and B = SR, and letting fl be the path (R, S) : A -o B, we will have the following properties: (I) A and B are primitive, non-negative integral matrices with det(A) = det(B) = _ 1 Tr(A) = Tr(B) = Tr(A 2) = Tr(B 2) = 0. (I0 • r 0. (III) A(a) = 0 for all a in ~'1 (SSE(Z), A) = Aut(sd).

26

THE MATHEMATICAL INTELLIGENCER

0 3 0

01 0 0

0 0 0 0 O 0 1

0 0 0 0 1 0 0

10

I,

0 0 1 0 0 12 12

1 0 0 1 0

12

00 0 0 00 00 13 10

0 0 0 0 0 0 1

0 ~ 0 0 0 , 1 0 0; 0 1 0 0 O 0 0

0 0 1 0 O 1 1

1 0 0 1 O 0 0

1 0 0 0 O 0 0

2 2 2 2 O 3 0

Direct c o m p u t a t i o n shows Property I holds. You can check Property II by substituting R and S into (22). No properties of X, Y, a n d Z are used here except that Z has diagonal entries which are zero. Now for Property III: The characteristic polynomial for A is t 7 - 6 t 4 - 5 t 3 - 6 t 2 - 3 t + 1 . Putting this into the c o m m a n d "buchgenfu( )" of PARI gives

R0 = - 1,

Direct computation using (22) when A = sgc2 shows A(~/) r 0. Next, we w a n t to extend E and F to matrices of the form

Trial and error and luck p r o d u c e s

, Z=

in (17) b e c a u s e {1, A, h 2, . . . , h 6} is a basis for O. The Dirichlet Unit T h e o r e m shows the rank of O* is 4, and PARI computes a set of generators for O* to be

'

= M 4 § M 3 § M 2 --

1 1

Z[A]* = Aut(sA) = O* = Z[A]*

1

-i

Y IY

B =SR =

Let 3/be the loop (E, F) : M--~ M, where

E=M-I=

'

o)

0

12 12

and, therefore,

The Counterexample

i

0

R1 R2 R3 R4

= = = =

A, 2A6 - h 5 + A4 - 13h 3 - 4A2 - 13A - 1, h 6 + h 4 - 7A3 - 6A2 - 8A - 6, 2A6 - 4A5 - 2A3 + 3A.

Substitute A for A in these expressions for each i = 0, 1, 2, 3, 4 to get five loops ai = (Ri, Ri -1 A) : A --> A generating 7rl(SSE(Z), A). A(a0) = 0 b e c a u s e R0 is diagonal and the diagonal entries of A are zero. Direct c o m p u t a t i o n using (22) shows A(ai) = 0 for each i = 1, 2, 3, 4. It was done by computer.

Future Problems and Projects N u m b e r 1. Let A be primitive. Are there only finitely m a n y SSE classes over Z + within the SSE class of A over Z? This q u e s t i o n was asked by D e n n i s Sullivan in the 1970s a n d again recently by Doug Lind. If this is n o t true in general, finding a n example to the c o n t r a r y requires n e w i n v a r i a n t s a n d n e w m e t h o d s for p r o d u c i n g matrices A and B which are c a n d i d a t e s to be SSE over Z b u t n o t SSE over Z +. N u m b e r 2. Are the matrices A equivalent over Z + for n -> 4?

n

and Bn of (11) strong shift

N u m b e r 3. Study the algebraic shift equivalence problem. Does SE imply SSE over a commutative regular ring A? See [BH1]. N u m b e r 4. Study SSE over other subrings of the real numbers. Are the 7 x 7 counterexamples above and in [KR5, KR6] strong shift equivalent over Q+ or R+? N u m b e r 5. For a primitive matrix A, does Image(tiA) always have finite index in Aut(sA)? This is similar to Number 1. For simplicity, assume Aut(sA) is a finitely generated abelian group, which is the case when det(tI - A ) is irreducible as explained in [BLR]. N u m b e r 6. Study strong shift equivalence over more general rings A. For example, the case when A is a ring of integral Laurent polynomials arises from Markov chains. The set A § consists of those Laurent polynomials with non-negative coefficients. See [Bo, MT]. The case when A is the group ring Z[G] of a finite group G arises from subshifts of finite type which are principal G-bundles over base spaces which are subshifts of finite type. The analogy between pseudo-isotopy theory and strong shift equivalence theory is used in [W7] to construct functions CP2m : ~-t(SSE(A), SSE2m(A+)) --->K2(A[t]/(tm+l), (t)) satisfying (19). Back in 1971, van der Kallen [vdK] proved that K2(Z[t]/(t2), (t)) = Z / 2 Z

and, as mentioned previously, the first two authors have shown q)2 -- sgc2. The proof is based on the algorithm in [W7] for evaluating q)2(R, S) derived from [vdK]. The construction of (IL)2m works for fairly general rings such as those above. Moreover, the machinery of algebraic K-theory may be useful in computing the indeterminacy which occurs in (20) and (21) in considering the Strong Shift Equivalence Problem over A. N u m b e r 7. Looking beyond the classification of subshifts of finite type, there is the general problem of studying the relation between the spaces SSE(A +) and SSE(A). For example, when T r ( A ) = Tr(A 2) = 0, a conjectural connection is proposed in [W7, W8] between Inert(o-A)/Simp((~d) = 1r2(SSE(Z), SSE2(Z+); A), counterexamples to the so-called FOG Conjecture produced in [KRW2, KRW3], and the cyclic homology Chern character on K3(Z[t]/(t2), (t)) discussed in [L]. Explore this further. N u m b e r 8. There is another connection between strong shift equivalence theory and cyclic homology theory; namely, the SSE spaces are themselves cyclic spaces [L, Chap. 7]. An n-cell of SSE is given by an n-tuple (A0, 9 9 9 An) of morphisms [e.g., matrices in the case of SSE(A +) and SSE(A)] together with elementary strong shift equivalences (Rij, Sji) : A i ~ A j whenever i < j which satisfy the Triangle Identities

Rik

=

RijRjk,

Sji = RjkSk.~,

Sky ~- SkiRij

for i < j < k. Let R0 = S.~,0 and Ri = R i - l , i for i = 1 , . . . , n. The Triangle Identities show that the simplex is completely determined by the (n 4- 1)-tuple (R0, 9 9 9 Rn). The cyclic operator t~ on n-simplices is given by tn(Ro . . . . . Rn) = (Rn, Ro, 9 9 : ; Rn-1).

It is certainly premature to comment on the significance or ramifications of this observation. However, it is interesting that the SSE spaces, which appeared in a totally different context, do turn out to have cyclic structures. N u m b e r 9. Gilmer [G] has recently shown how the idea of strong shift equivalence over Z + arises in topological quantum field theory invariants (e.g., for knots). Related work has been done by Silver and Williams [SW]. Explore this area further. ACKNOWLEDGMENTS

The first two authors were partially supported by NSF Grants DMS 9024813 and DMS 9405004. The third author was supported in part by NSF Grant DMS 9322498. REFERENCES

[A] R.L. Adler, "The torus and the disk," IBM J. Res. Dev. 31 (2) (1987), 224-234. [AKM] R.L. Adler, A. Konheim, and M. McAndrew, "Topological entropy," TAMS 114 (1g65), 309-319. [B] K. Baker, "Strong shift equivalence of 2 x 2 matrices of non-negative integers," ETDS, 3 (1983), 501-508. [Bow] R. Bowen, "Markov partitions for Axiom A diffeomorphisms," Amer. J. Math. 92 (1970), 725-747. [Bo] M. Boyle, "The stochastic shift equivalence conjecture is false," Contemp. Math. 135 (1992), 107-110. [BF] M. Boyle and U. Fiebig, "The action of inert finite order automorphisms on finite subsystems," ETDS 11 (1991), 413-425. [BH1] M. Boyle and D. Handelman, "Algebraic shift equivalence and primitive matrices," Trans. AMS 336(1) (1993), 121-149. [BH2] - - , "The spectra of nonnegative matrices via symbolic dynamics," Ann. Math. 133 (1991), 249-316. [BK1] M. Boyle and W. Krieger, "Periodic points and automorphisms of the shift," Trans. AMS 302 (1987), 125-149. [BK2] - - , "Automorphisms and subsystems of the shift," J. Reine Angew. Math. 437 (1993), 13-28. [BLR] M. Boyle, D. Lind, and D. Rudolph, "The automorphism group of a shift of finite type," Trans. AMS 306 (1988), 71-114. [BMTJ M. Boyle, B. Marcus, and P. Trow, Resolving Maps and the Dimension Group for Shifts of Finite Type, Mem. Amer. Math. Soc. Voh 70 3 (1987), Number 377, American Mathematical Society, Providence, RI. [BW] M. Boyle and J.B. Wagoner, Nonnegative algebraic K-theory and subshifts of finite type, in preparation. [BaW] L. Badoian and J.B. Wagoner, "Simple connectivity of the Markov partition space," preprint, UC Berkeley (1998), Pac. J. Math. (in press). [C] J. Cerf, "La stratification naturelle des espaces de fonctions differentiables r6elles et le theoreme de la pseudo-isotopie," PubL Math. IHES, 39 (1970), 5-173.

VOLUME 21, NUMBER 4, 1999

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[CK1 ] J. Cuntz and W. Krieger, "Topological Markov chains and dycyclic dimension groups," J. Reine Angew. Math. 320 (1980), 44-51. [CK2] - - , "A class of C*-Algebras and topological Markov chains," Inventi. Math. 56 (1980), 251-268. [E] E.G. Effros, Dimensions and C*-Algebras, CBMS No. 46, American Mathematical Society, Providence, RI 1981. [F] U. Fiebig, "Gyration numbers for involutions of subshifts of finite type I," Forum Math. 4, (1992), 77-108; "Gyration numbers for involutions of subshifts of finite type//, Forum Math. 4 (1992), 183-211. [Fr] J. Franks, Homology and Dynamical Systems, CBMS No. 49, American Mathematical Society, Providence, RI, 1982. [G] P.M. Gilmer, "Topological quantum field theory and strong shift equivalence," preprint, Louisiana State University (1997), Bull. Canadian Math. Soc. (in press). [H] G. Hedlund, "Endomorphisms and automorphisms of shift dynamical systems," Math. Syst. Theory 3 (1969), 320-375. [HW] A. Hatcher and J.B. Wagoner, Pseudo Isotopies of Compact Manifolds, Asterisque No. 6, Societ6 Mathematique de France, 1973. [K] B. Kitchens, Symbolic Dynamics, Springer-Verlag, New York, 1997. [Kol] A.N. Kolomogorov, "New metric invariants of transitive dynamical systems and automorphisms of Lebesgue spaces," Dokl. Akad. Nauk. SSSR 119 (1958), 861-864.

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THE MATHEMATICAL INTELLIGENCER

[Kr] W. Krieger, "On dimension functions and topological Markov chains," Invent. Math. 56 (1980), 239-250. [KR1] K.H. Kim and F.W. Roush, "Some results on decidability of shift equivalence," J. Combin. Inform. Syst. 4 (1979), 123-146. [KR2] - - , "Decidability of shift equivalence," in Dynamical Systems, edited by J.W. Alexander Lecture Notes in Mathematics Vol. 1342, Springer-Verlag, Heidelberg, 1988. [KR3] - "On the structure of inert automorphisms of subshifts," Pure Math. Applic. 2 (1991), 3-22. [KR4] - "Williams's conjecture is false for reducible subshifts," J. AMS 5 (1992), 213-215. [KR5] - "Williams's conjecture is false for irreducible subshifts," ERA AMS 3 (1997), 105-109. [KR6] - "Williams's conjecture is false for irreducible subshifts," Ann. Math. 149 (1999), 545-558. [KR7] - "Path components of matrices and strong shift equivalence over Q+," LinearAIg. Applic. 145 (1991), 177-186. [KR8] - "Strong shift equivalence of Boolean and positive rational matrices," LinearAIg. Appl. 161 (1992), 153-164. [KRW1] K.H. Kim, F.W. Roush, and J. Wagoner, "Automorphisms of the dimension group and gyration numbers," J. AMS 5 (1992), 191-212. [KRW2] - - , "Characterization of inert actions on periodic points,

Part I and Part I1," preprint, UC Berkeley, 1997; Forum Math. (in press). [KRW3] - - , "Inert Actions on Periodic Points," ERA AMS 3 (1997), 55-62. [L] J.-L. Loday, Cyclic Homology, Springer-Verlag, New York. [LM] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995. [M] B. Marcus, "Symbolic dynamics and connections to coding theory, automata theory and system theory," AMS Proc. Symp. Appl. Math. 50 (1995), 95-108. [Mil] J. Milnor, Algebraic K-Theory, Annals of Mathematics Studies Number 72, Princeton University Press, Princeton, NJ, 1971. [Mi2] - - , "Infinite cyclic coverings," Conference on Manifolds, Vol. 13, edited by J.G. Hocking, Prindle, Weber, and Schmidt, Inc., 1968, pp 115-133. [MB] S. MacLane and G. Birkhoff, Algebra, 6th ed., MacMillan, London, 1971. [MH] M. Morse and G.A. Hedlund, "Symbolic dynamics," Amer. J. Math. 60 (1938), 815-866. [MT] B. Marcus and S. Truncel, "Matrices of polynomials, positivity, and finite equivalence of Markov chains," AMS J. 6 (1993), 131-147. [N] M. Nasu, Topological Conjugacy for Sofic Systems and Extensions of Automorphisms of Finite Subsystems of Topological Markov Shifts, Springer Lecture Notes in Mathematics Vol. 1342, SpringerVerlag, Berlin, 1988. [PT] W. Parry and S. Tuncel, Classification problems in ergodic theory, LMS Lecture Notes 67, Cambridge University Press, 1982. [PW] W. Parry and R.F. Williams, "Block coding and a zeta function for Markov chains," Proc. LMS 35 (1977), 483-495. [R] J. Rosenberg, Algebraic K-Theory and Its Applications, SpringerVerlag, New York, 1994. [$1] S. Smale, "Differential dynamical systems," BAMS, 73(6) (1967), 747-847. [S2] - - , "Finding a horseshoe on the beaches of Rio," Math. Intelligencer, 20, no. 1 (1998). [Sh] C. Shannon, "A mathematical theory of communication," Bell Syst. Technol. J. 27 (1948), 379-423, 623-656. [SW] D. Silver and S. Williams, "Knot invariants from symbolic dynamical systems," Report No. 4/96, University of South Alabama. [vdK] W. van der Kallen, "Le K2 des nombres duaux," C.R. Acad. ScL Paris, S~rieA 273 (1971), 1204-1207. [W1] J.B. Wagoner, "Markov partitions and K2," Publ. Math. IHES, 65 (1987), 91-129. [W2] - - , "Triangle identities and symmetries of a subshift of finite type," Pacific J. Math. 144 (1990), 331-350. [W3] - - , "Higher dimensional shift equivalence is the same as strong shift equivalence over the integers," Proc. AMS 109 (1990), 527-536.

[W4] - - , "Eventual finite order generation for the kernel of the dimension group representation," Trans. AMS (1990), 331-350. [W5]--, "Classification of subshifts of finite type revisited," Contemp. Math. 135 (1992), 423-436. [W6] - - , "Markov chains, C*-algebras, and K2," Adv. Math. 71 (2) (1988), 133-185. [W7] - - , "Strong shift equivalence and K2 of the dual numbers," preprint, UC Berkeley, 1998; J. Reine Angew. Math. (in press). [W8] - - , "Strong shift equivalence theory and the shift equivalence problem," Bull. Amer. Math. Soc. 36 (3) (1999), 271-296. [Wil] R.F. Williams, "Classification of subshifts of finite type," Ann. Math. (2) 98 (1973), 120-153; errata, 99 (1974), 380-381. [Wi2] , "Shift equivalence of matrices in GL(2,Z)," Contemp. Math. 135 (1992) 445-451.

VOLUME 21, NUMBER 4, 1999

29

HELAMAN FERGUSON, CLAIRE FERGUSON, TAMAS NEMETH, DAN SCHWALBE, AND STAN WAGON

Invisible I landshake The Event: The ninth annual Breckenridge International S n o w Sculpture Championships, Jan 19-23, 1999. T e a m Sponsor: Wolfram Research, Inc. Sculpture Dedication: To the m e m o r y of A1 Gray, w h o w a s to b e on the t e a m b u t p a s s e d a w a y several w e e k s b e f o r e the event. SW: I n e v e r thought h e ' d agree. H e l a m a n Ferguson, the w e l l - k n o w n s c u l p t o r w h o u s e s s t o n e and bronze to imm o r t a l i z e m a t h e m a t i c a l ideas, p r e f e r s r a w m a t e r i a l s that last centuries, not days. Still, it n e v e r hurts to ask. I h a d j u s t s e e n the s p e c t a c u l a r s n o w s c u l p t u r e s in the 1998 edition of the Breckenridge, Colorado, event, and I a s k e d H e l a m a n if he would c o n s i d e r submitting a p r o p o s a l for 1999. There was s o m e initial reluctance, but after w e began discussing which of his m a n y p i e c e s w o u l d l o o k b e s t in snow, he was hooked. We eventually settled on a Costa surface, w h i c h w e titled Invisible Handshake b e c a u s e it is the c o m p l e m e n t a r y s p a c e o f t w o h a n d s coming together, b u t n o t touching. The c o m p e t i t i o n is b y invitation only: one submits a prop o s a l and h o p e s to land in the roughly half of the submissions t h a t are accepted. We w e r e the only a c c e p t e d t e a m with no p r i o r s n o w sculpting experience. But H e l a m a n is a t a l e n t e d sculptor, and w e m a d e sure our p r o p o s a l inc l u d e d a description of his p a s t work. The Breckenridge event is remarkable. First, it a t t r a c t s t e a m s from all a r o u n d the world: this y e a r saw t e a m s from Lithuania, England, France, Italy, Switzerland, Russia, Mexico, the Netherlands, Canada, and the United States. Second, the t o w n really p u t s out a lot to m a k e the event possible. The giant s n o w b l o c k s are c r e a t e d out of m a c h i n e - m a d e snow. That p r o c e s s starts m o n t h s b e f o r e for

THE MATHEMATICAL INTELLPGENCER 9 1999 SPRINGER VERLAG NEW YORK

the J a n u a r y event, as the ski a r e a s t o c k p i l e s huge m o u n d s of snow. In early J a n u a r y they are t r u c k e d d o w n to the site and b l o w n into large forms (10 feet square b y 12 feet high). At intervals, t o w n s p e o p l e c o m e out to s t o m p on the s n o w so that its c o n s i s t e n c y is firm. A few days b e f o r e the event the forms are r e m o v e d and the b l o c k s s e t up in the cold temperatures. They end up being very firm indeed: not ice, but with a b o u t 50% the density o f ice. E a c h b l o c k weighs 20 tons a n d one m u s t have very sharp tools to w o r k it efficiently. The t o w n also p r o v i d e s r o o m and b o a r d for the entire week. In short, all the teams n e e d do is eat, sleep, a n d sculpt. F o u r a n d a half days are allotted for sculpting. H e l a m a n h a d e x p r e s s e d s o m e c o n c e r n a b o u t the elevation, 10,000 feet a b o v e s e a level. And, worse, I had cont r a c t e d a b a d cold and was s o m e w h a t ill for the first day. But our t e a m m a t e s w e r e strong in t h o s e first few days, and we s o o n w e r e all going full steam. It w a s a real thrill to learn the r u d i m e n t s of sculpting as w e progressed. Our piece w a s ugly at first, slowly b e c a m e less and less ugly, and then e m e r g e d all at once in all its glory. HF: Start Wagon is v e r y persistent. He h a d b e e n interested in m y m a t h e m a t i c a l sculpture for s o m e time, and w e coll a b o r a t e d on a small granite p i e c e illustrating a dissection p r o o f of the pizza t h e o r e m [K]; it is u s e d as the a w a r d for a collegiate c o m p e t i t i o n in Minnesota. W h e n he invited m e to sculpt s n o w I replied, "I do granite; I d o n ' t do snow." The r e a s o n for this s n o b b i s h r e s p o n s e w a s that I didn't k n o w s n o w as a carving medium, h a d n ' t thought a b o u t it, and didn't have time for frivolous ski resorts. Besides, m y idea of m a t h e m a t i c a l sculpture c o m b i n e s timeless conceptual c o n t e n t with timeless-as-possible materials, such

as billion-year-old granite. Material that old will surely last a n o t h e r million years. Snow: Here today, e v a p o r a t e d tomorrow.

But then Stan a s k e d w h i c h of m y m a t h e m a t i c a l sculptures w o u l d b e b e s t in snow. Like he is going to get someone else to do one of m y sculptures? NOT! So that s t a r t e d m e thinking seriously a b o u t snow. Does s n o w have any materiai p r o p e r t i e s like stone? Stone has g o o d c o m p r e s s i v e strength, less tensile strength. With s t o n e you can m a k e an arch, corbel, o r keystone. Snow has u s a b l e c o m p r e s s i v e strength but n o t m u c h tensile strength. But you can m a k e a s n o w arch: an igloo is a r o t a t e d s y s t e m of arches. So, which of m y s c u l p t u r e s [F] have the a r c h as a structural feature? Hmmm, Hnnn, Huuu, I thought, m m m, n n n, u u u (arches). It h a p p e n e d that I h a d d o n e a series of m a t h e m a t i c a l s c u l p t u r e s on the t h e m e of minimal surfaces. The m a i n focus o f the series was the nontrivial t o p o l o g y minimal surface d i s c o v e r e d by Celso Costa [C] in 1983 and the subsequent m a t h e m a t i c a l d e v e l o p m e n t o f Hoffman, Hoffman, and Meeks [H]. A minimal surface belongs to a larger class of s u r f a c e s with nonpositive or, usually, negative Gaussian curvature. This m e a n s that the n e i g h b o r h o o d of e a c h p o i n t is a s a d d l e with a positive principal curvature a n d a negative principal curvature. The p r o d u c t o f these two curvatures is called the G a u s s i a n curvature; half the s u m is called the m e a n curvature. A minimal surface has m e a n curvature zero. What is especially sculpturaily interesting for s n o w s e e m e d to be the negative Gaussian curvature: a r c h e s in m o s t every direction. Indeed, every point s e e m e d to be the k e y s t o n e of a cluster o f arches. I h a d a l r e a d y carved negative G a u s s i a n curvature forms in stone as p a r t of m y minimal surface o r Costa series. Minimal s u r f a c e s first c a m e to m y sculptural attention b y w a y of the late Alfred Gray. Alfred, a differential geometer, faithfully a t t e n d e d m y lectures on m a t h e m a t i c a l sculpture at the University of Maryland and each time encouraged m e to give his differential g e o m e t r y s o m e s c u l p t u r e time with m y NIST SP-2. B e t w e e n m y studio a n d the Intelligent S y s t e m s Group at the National Institute of S t a n d a r d s a n d Technology is a CRADA, a c o o p e r a t i v e res e a r c h and d e v e l o p m e n t agreement, which has gone through t w o g e n e r a t i o n s of a Stewart p l a t f o r m s y s t e m being t e s t e d in m y stone-carving studio. My SP-2 s y s t e m is a w a y of "projecting" equations into a b l o c k of s t o n e - i n d e e d this is a s y s t e m for quantitative direct carving o f stone. (It is a kind of c o m p u t e r - a s s i s t e d pointing m a c h i n e with equations as a virtual o b j e c t to p o i n t from, cf. [F1].) With this exotic studio apparatus, I can learn n e w surfaces, m a t h e m a t i c a l forms that have never b e f o r e seen the light of day, n e v e r b e f o r e b e e n t o u c h e d b y h u m a n hands. Al Gray h a d a M a t h e m a t i c a p r o g r a m for doing the graphics of the Costa minimal surface, b u t it was t o o slow to b e p r a c t i c a l in m y studio environment. I was p e r s i s t e n t in encouraging him to i m p r o v e Costa's p a r a m e t r i z a t i o n b y eliminating the time-consuming n u m e r i c a l integration, which he finally did in a w a y that led to the i m p r o v e m e n t of M a t h e m a t i c a ' s elliptic function routines. This a l l o w e d

Without form and void. (All photos by Stan Wagon.)

m e to do several m i n i m a l surfaces in s t o n e and o t h e r materiais. The i m p o r t a n t p o i n t here was that t h e s e w e r e negative Gaussian curvature forms which I h a d learnt in stone. The c o m p r e s s i v e s t r e n g t h of stone c o u p l e d with the form of saddle n e i g h b o r h o o d s at every p o i n t h a d s o m e remarkable properties. The marble, for example, could b e carved fairly thin, y e t rang quite clearly. Would c o m p a c t e d s n o w b e similar? Well, it w a s a w a r m May in Maryland. Where w a s I to get s n o w for a small e x p e r i m e n t ? Some friends recomm e n d e d that I find a Zamboni: it turns out that j u s t a b o u t every ice rink c o n s t a n t l y piles Zamboni s n o w out back. The Zamboni is a s p e c i a l - p u r p o s e d u m p t r u c k t h a t p l a n e s o f f t h e ice every several hours, a n d d u m p s a l o a d of high-consist e n c y s n o w in big piles on the grass n e a r the p a r k i n g lot. I l i b e r a t e d a few cubic feet of this stuff a n d b r o u g h t it h o m e in m y p i c k u p truck. It w a s a w a r m afternoon. With a giant kitchen s p o o n and s p a t u l a I p r o c e e d e d to carve a form with negative Gaussian curvature everywhere. It h a d lots of holes and I carved it fairly thin, a b o u t an inch thick. Why? B e c a u s e I found I could. Then, in shirtsleeves and u n d e r the late afternoon Maryland sun I w a t c h e d it melt. The form activated the couple o f cubic feet w h e r e m o s t of the s n o w h a d been, and as it m e l t e d the form m a i n t a i n e d its structural integrity; the wails j u s t got t h i n n e r and thinner. Occasionally in a n a r r o w i n g of the wail a hole w o u l d o p e n up, b u t the s t r u c t u r e w a s maintained. I felt I h a d seen enough w h e n the n e i g h b o r kids d i s c o v e r e d I h a d snow, and everything e n d e d with a m a r v e l o u s slushy s n o w b a l l fight. It s e e m s that a C o s t a form could be c a r v e d in snow, and without any special equipment. But h o w w o u l d a t h e o r e m in s n o w fly with the t o w n p o w e r s o f B r e c k e n r i d g e ? Start a n d I m a d e our p r o p o s a l , a i d e d b y lots of c o m p u t e r graphics. By this time, A1 Gray even had a Costa c h a p t e r in his differential g e o m e t r y b o o k [G] and h a d C o s t a images (cut off in a cube) to spare. And Sam F e r g u s o n h a d c o m p u t e d m a n y spherical cutoff i m a g e s . There w e r e 35 p r o p o s a l s to

VOLUME 21, NUMBER 4, 1999

31

The tools.

the event and 16 were chosen, so one can c o n c l u d e that m a t h e m a t i c s in Breckenridge occupies an interest at least above the 50th percentile. I got permission from the m a n a g e r of the ice a r e n a where the Washington Capitals hockey team practices, a n d w o r k e d out in Zamboni s n o w at various times b e t w e e n May and January. Part of the challenge was testing out various sorts of clothing which w o u l d keep me warm through a long day of carving snow. And it was easy to fend tools for working the Zamboni s n o w in local shops. Most useful was a long-handled post-hole digger; b u t in the event this was benched, because Dan had b r o u g h t a magnificent ice auger. Next useful was a crinkle-edged n a r r o w shovel, the special edge being good for making straight tangent cuts in the relatively soft snow. Our raw material consisted of compacted, artificial, nucleated snow, whereas the Zamboni snow is really shaved ice. Our s n o w has a density of a b o u t 30-35 lbs/cu ft, whereas the Zamboni snow is 40-45 lbs/cu ft; for comparison, liquid water is 62.5 lbs/cu ft, ice is a b o u t 60 lbs/cu ft, glacier ice is 50-55 lbs/cu ft, and natural s n o w is 3-8 lbs/cu ft. There are m a n y variables that affect the s n o w in the course of a working day, mostly the temperature and sun conditions. At 20 ~ Fahrenheit the Breckeuridge snow carved crisp and clean a n d could be sanded smooth, very m u c h like wood. At 30 ~ with less than a n h o u r of s u n exposure, the surface could b e c o m e soft and mushy. The week after the event, the daytime temperatures w e n t up to 45 ~ and the traditional sculptures all lost detail, with at least one imploding entirely. It was swept off the site as if it were a dead body o n a lifeboat. But Invisible Handshake didn't change except to b e c o m e a little thinner, d o w n to 2.5-3 inches. This was what I had e x p e r i e n c e d before with my small example, a n d it was gratifying to see it hold on a m u c h larger scale. I attribute this u n u s u a l structural stability to the arched structure of negative Gaussian curvature t h r o u g h o u t the work. After the shape emerged from the snow, we set out to find all the positive Gaussian curvatures. The rule was to eliminate the positive curvatures and replace them with negative curvatures; we did this persis-

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THE MATHEMATICALINTELLIGENCER

tently and thoroughly even though a bit of positive curvature may not affect the structure very much. Team c o h e r e n c e was a matter I was c o n c e r n e d about. The four of us had four days to transform 20 tons of snow, 12' x 10' x 10', a n d I had not m e t two of the team members. To prepare myself for working with others I invited a friend, Neil Coletti, to do a s n o w Costa with me. We did a five-foot cube a n d I saw that things w e n t a lot faster with two working with one mind. The o n e - m i n d part involves c o m m u n i c a t i o n of technical matters a n d having a c o m m o n mathematical language; this makes a great deal of difference. The w e e k before our full team of four met for the first time, Start had stomped his o w n cubic m e t e r of natural s n o w in Colorado. The Monday night we all arrived we started to do a Costa on this block. It b e c a m e clear that the ice auger was more effective than the post-hole digger for making initial holes. The logging saw was helpful for chamfering the initial rough sphere. Some of us were adjusting to the high altitude, so we did n o t finish this off, but I had a very clear idea of the abilities a n d strengths of the three team members. The key sculptural technique I had to teach the team, after the chamfering a n d hole penetration, was the use of tangent cuts for carving negative Gaussian curvature. The fact that we were speaking the same visual m a t h e m a t i c a l language was what provided the team an u n u s u a l degree of coherence. Pacing the team was important, especially since we were acclimating to high altitude. Our p l a n n e d schedule, which we easily maintained, was spheroid Tuesday (a half day), topology Wednesday, differential geometry Thursday

Verifying the connectness of the complement.

and Friday. Saturday morning w a s b a s e p r e p a r a t i o n and surface cosmetics. More explicitly, Tuesday we c h a m f e r e d out a rough sphere. Wednesday, w e did the t o p o l o g y of the holes or passages, t w o w y e s coming t o w a r d each other. Thursday and Friday was differential geometry, b y w h i c h I mean, of course, attaining negative Gaussian curvature throughout. By T h u r s d a y w e h a d the general t h i c k n e s s of eight to twelve What had lain hidden in the block. inches, w h i c h w a s as far as any o f the o t h e r s n o w s c u l p t o r s generally c a r e d to go for standing walls. By the end of F r i d a y w e had the outside wall t h i c k n e s s d o w n to four inches, w o r k i n g m o s t l y with the truss gusset r a s p s m a d e on the s p o t for us b y Bob Salzar, a local friend. This thinness was very impressive and c o n t r i b u t e d to the overall esthetic of the piece. A n d it was gratifying to see that the p i e c e did not d e g r a d e a n d collapse in the 45 ~ weather, b u t j u s t b e c a m e thinner. DS: The elegant simplicity o f the sculpting plan w a s a surprise. After a v e r y s h o r t time with H e l a m a n the entire t e a m k n e w w h a t w a s going to h a p p e n e a c h s t e p along the way. I h a d a n t i c i p a t e d a certain a m o u n t of rough work, at w h i c h p o i n t the real s c u l p t o r w o u l d t a k e over. This w a s n o t the c a s e at all, as H e l a m a n had crafted a careful plan n o t j u s t to sculpt the Costa surface, b u t also to t e a c h us novices h o w to w o r k t o g e t h e r efficiently. In short, the m a s t e r was sculpting the t e a m as m u c h as he w a s the snow. The tools w e r e a focus o f the t e a m t h r o u g h o u t and p r o v e d critical at each stage. As in w o o d w o r k i n g , removing s n o w t o o slowly w a s a l m o s t as b a d as removing s n o w t o o quickly. Start and H e l a m a n p u t in a lot of effort p r i o r to the c o n t e s t choosing tools that p r o v e d especially imp o r t a n t for an i n e x p e r i e n c e d team, s u c h as a four-foot logging saw, small h a t c h e t s and shovels, m a s o n r y tools, and a small w o o d chisel. But w e h a d n o t anticipated the n e e d for rasps. We s a w w h a t the o t h e r t e a m s w e r e u s i n g - - t r u s s gussets m o u n t e d on w o o d e n p l a t f o r m s with a t t a c h e d hand l e s - a n d s o m e local friends quickly m a d e four of t h e m with varying curvatures suitable for rasping negative G a u s s i a n curvature. Ice fishing tools were a natural for easily removing lots of s n o w quickly in the beginning. An ice auger is u s e d for drilling holes t h r o u g h t h r e e feet o f ice. The razor-sharp b l a d e s easily bit through the s n o w and were u s e d s o m e w h a t similarly to the w a y H e l a m a n u s e s d i a m o n d drill bits to w o r k with stone. Indeed, the o t h e r t e a m s envied the s p e e d o f the ice-fishing tool a n d w e f o u n d ourselves lending it out several times!

C F : I k n o w the Costa surface from the beautiful s c u l p t u r e s o f it that Helaman creates, a n d the Breckenridge u n d e r t a k i n g s e e m e d j u s t s h o r t o f impossible to m e w h e n he first d e s c r i b e d it. So the m o m e n t which symbolizes the spirit of the e x p e r i e n c e p i c t u r e s four m e n standing in front of a 12foot high b l o c k o f snow, a r m e d with only shovels and saws. Their implements s e e m outlandishly crude m e a s u r e d against the b e a u t y they i n t e n d to create. The blindingly white b l o c k l o o m s a b o v e t h e m a n d t h e y s h o w no tension w h a t e v e r from the s h o r t four-day time limit. Even m a s k e d b y sunglasses, their e a g e r o p t i m i s m s h o w s in their b o d y language. Of course, e x c e p t for one these are not sculptors. They are mathematicians, a b o u t to carve a surface only r e c e n t l y visualized, and with w h i c h t h e y were n o t very familiar. Moreover, they h a d n o t w o r k e d t o g e t h e r e x c e p t for a few h o u r s on the night b e f o r e the event's start. The first day, they carved a sphere out of the block using ladders and scaffolding provided by the town. On the second, they m a r k e d the outer curves and the holes, then augured through, one turning the h a n d drill while another directed it. They then w i d e n e d the holes and f o r m e d the outer curves. By the end of the third day, the rough Costa shape had appeared. Almost simultaneously, a class of kindergarten children arrived to slide through the sculpture, a highlight of the week. The next day brought another snow~ storm (any w e a t h e r is b e t t e r than w a r m sun) and the team continued to w o r k making the surfaces s m o o t h and refining

The Costa surface, generated by Mathematica and AI Gray's equations [ S W ] .

VOLUME 21, NUMBER4, 1999

the curves. They planed the foot-thick walls first to eight, then to four inches. At that point, the Costa b e c a m e breathtakingly b e a u t i f u l - - b u t this w a s a c c o m p a n i e d b y the threat that it might melt and collapse before the Saturday m i d d a y deadline, should that day turn out to be warm. The day was i n d e e d warm; all b u t one o f the 15 structures survived it. As p h o t o g r a p h e r a n d gofer, I enjoyed watching the prog r e s s i o n of the Costa. This w a s an amazing p i e c e that w a s fun to photograph: an art w o r k of great aesthetic pleasure. TN: When I agreed to j o i n the team, I w a s not really sure w h a t I was getting into. I b e c a m e very e x c i t e d a b o u t the intricate m a t h e m a t i c a l surface, which r e m i n d e d m e o f a giant d i n o s a u r n e c k b o n e , b u t I did not forsee h o w w e w e r e going to carve it out of a giant b l o c k of snow. Surprisingly enough, it w e n t quite smoothly, thanks to the e x p e r t i s e and eye o f our captain. There w a s an intimidation f a c t o r on that first d a y caused by the size o f the raw s n o w block, b u t as w e progressed, o u r relationship with it b e c a m e m o r e a n d m o r e intimate. By Saturday, I h a d grown to like the surface v e r y much: a shiny, warm, and softly curving b e a u t y that e m e r g e d from the coldness. The m a n y h o u r s w e r e w o r t h it. One of the m e m o r a b l e events was an entire class of y o u n g s c h o o l c h i l d r e n that w e allowed to slide d o w n a n d

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THE MATHEMATICAL INTELLIGENCER

crawl t h r o u g h o u r twisted shape: t h e i r fear going in a n d smiles u p o n emerging. REFERENCES

[P] I. Peterson, The song in the stone, Science News, 149:7, Feb. 17, 1996, 110-111. [C] C. Costa, Example of a complete minimal immersion in [~3 of genus one and three embedded ends, BoL Soc. Brasil Math. 15 (1984), 47-54. [F] C. Ferguson, Helaman Ferguson, Mathematics in Stone and Bronze, Meridian Creative Group, Erie, Pa., 1994. [G] A. Gray, Modern Differential Geometry of Curves and Surfaces, 2nd ed., CRC Press, Ann Arbor, Mich. 1998. [K] J. Konhauser, D. Velleman, and S. Wagon, Which Way Did the Bicycle Go? . . . and Other Intriguing Mathematical Mysteries,

Mathematical Association of America, Washington, 1996. [H] D. Hoffman, The computer-aided discovery of new embedded minimal surfaces, The Mathematical Intelligencer 9:3 (1987), 8-21. [F1] H. Ferguson, Stone sculpture from equations, in Proceedings of the 1998 IEEE International Symposium on Intelligent Control,

Section VI, Autonomy, 765-770. [SVV] D. Schwalbe and S. Wagon, The Costa surface, in snow and in Mathematica, Mathematica in Education and Research 8 (1999), 56-63.

VOLUME 21, NUMBER 4, 1999

35

ILV~|,[aii~-lI[C-II:fi1(~ir

Alexander

Shen,

Editor

I

Let m e begin this month w i t h a sto,?] received f r o m Chandler Davis.

Misadventure with the Checkerboard This column is devoted to mathematics for fun. What better purpose is there for mathematics? To appear here,

a r k o ' s friendship with D a r k o w a s m a r r e d b y only one thing: D a r k o always w o n at checkers. They p l a y e d on the usual b o a r d with the usual rules.

M

a theorem or problem or remark does not need to be profound (but it is allowed to be); it may not be directed

To this the Column Editor comments:

only at specialists; it must attract and fascinate. We welcome, encourage, and frequently publish contributions from readers--either new notes, or replies to past columns.

Please send all submissions to the Mathematical Entertainments Editor, Alexander Shen, Institute for Problems of Information Transmission, Ermolovoi 19, K-51 Moscow GSP-4, 101447 Russia; e-mail:[email protected]

~6

s t e a d of diagonally. Darko a c c e p t e d the challenge. Poor Marko! S o m e h o w Darko s e e m e d to m a i n t a i n j u s t the s a m e advantage over him as before!

It m u s t have b e e n j u s t that D a r k o h a d m e m o r i z e d a lot of openings and stand a r d positions. "I d o n ' t have to be b e t t e r t h a n he is," Marko e x c l a i m e d to his w i f e - - " I j u s t ought to win sometimes! . . . . If y o u change the rules," his wife suggested, "then all the s t u d y he's done on the usual g a m e w o n ' t give him any advantage." All right. G o o d idea. Marko designed a n e w b o a r d and p u t the checkers d o w n in the m o s t convenient way. The rules, he p r o p o s e d , s h o u l d b e the s a m e - - e x c e p t , of course, for moving along the r o w s and c o l u m n s in-

THE MATHEMATICAL INTELLIGENCER 9 1999 SPRINGER-VERLAG NEW YORK

This story p o s e s a question to psychologists: to w h a t e x t e n t the "vision" that Darko (like any g o o d player) has w h e n he l o o k s at the p o s i t i o n r e m a i n s intact for the n e w (isomorphic) game. There is a n o t h e r nice e x a m p l e of i s o m o r p h i c g a m e s ( m o s t p r o b a b l y well known, I j u s t d o n ' t r e m e m b e r w h e r e I've r e a d a b o u t it). At the s t a r t t h e r e are nine p i e c e s n u m b e r e d 1,2,3,... ,8,9. Players' m o v e s alternate. At e a c h s t e p one of the players t a k e s one o f the pieces. If you are able to m a k e the s u m 15 with three o f the p i e c e s y o u have taken, you win. (If no one can a n d t h e r e are no p i e c e s left, it is a draw.) It turns out that this game is isomorphic to the s t a n d a r d tic-tac-toe game on 3 x 3 b o a r d w h e r e two players alternate putting their respective signs (X's and O's) until one gets a vertical, horizontal, o r diagonal line a n d wins. (Try to figure out w h y it is isom o r p h i c if you h a v e n ' t seen the p r o b lem before: it is well w o r t h the effort!) Now it is well k n o w n that in tic-tact o e no p l a y e r has a winning strategy,

either can guarantee a draw. Therefore, the s a m e is true for the game of finding triples adding to 15. But please, psychologists, tell us w h e t h e r real h u m a n s play this game differently if t h e y have n o t n o t i c e d that

it is i s o m o r p h i c to tic-tac-toe! They would surely p l a y m o r e carefully, but is there s o m e w a y that t h e y w o u l d play less geometrically? I have the s a m e question a b o u t Darko: for his advantage over the i n e x p e r t p l a y e r to c a r r y

over to the altered c h e c k e r s game, was it n e c e s s a r y that he c o n s c i o u s l y notice the i s o m o r p h i s m - - a n d w o u l d his strategy b e the s a m e if he w e r e using his p r e v i o u s k n o w l e d g e w i t h o u t knowing he did so?

VOLUME 21, NUMBER 4, 1999

37

either can guarantee a draw. Therefore, the s a m e is true for the game of finding triples adding to 15. But please, psychologists, tell us w h e t h e r real h u m a n s play this game differently if t h e y have n o t n o t i c e d that

it is i s o m o r p h i c to tic-tac-toe! They would surely p l a y m o r e carefully, but is there s o m e w a y that t h e y w o u l d play less geometrically? I have the s a m e question a b o u t Darko: for his advantage over the i n e x p e r t p l a y e r to c a r r y

over to the altered c h e c k e r s game, was it n e c e s s a r y that he c o n s c i o u s l y notice the i s o m o r p h i s m - - a n d w o u l d his strategy b e the s a m e if he w e r e using his p r e v i o u s k n o w l e d g e w i t h o u t knowing he did so?

VOLUME 21, NUMBER 4, 1999

37

SOLOMON W. GOLOMB

Mathematics After Forty Years of thc Space Age*

~

hen I was a graduate student at Harvard, in the early 1950s, the question of whether anything that was taught or studied in the Mathematics Department had any practical applications could not even be asked, let alone discussed. This was not unique to Harvard.

matics had to be pure mathematics, and by definition it was not permissible to talk about possible applications of pure mathematics. This view was not invented by G.H. Hardy (1877-1947), but he was certainly one of its most eloquent and influential exponents. In A Mathematician's Apology(Cambridge U. Press, 1940) he wrote (p. 29), "Very little of mathematics is useful practically, a n d . . , that little is comparatively dull"; and (p. 59), "The 'real' mathematics of the 'real' mathematicians, the mathematics of Fermat and Euler and Gauss and Riemann, is almost wholly 'useless' "; and (p. 79), "We have concluded that the trivial mathematics is, on the whole, useful, and that the real mathematics, on the whole, is not." In order to force external reality into his rhetorical model, Hardy decided to include leading theoretical physicists in his c a n o n of "real" mathematicians, but to justify this by saying that their work had no real utility anyway. Thus, he wrote (p. 71), "I count Maxwell and Einstein, Eddington and Dirac among 'real' mathemati-

Good

mathe-

cians. The great m o d e r n achievements of applied mathematics have been in relativity and quantum mechanics, and these subjects are, at present at any rate, almost as 'useless' as the theory of numbers." R e m e m b e r that this was in 1940; and Hardy also wrote (p. 80), "There is one comforting conclusion which is easy for a real mathematician. Real mathematics has no effects on war. No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years." He also asserted (pp. 41-42), "Only stellar astronomy and atomic physics deal with 'large' numbers, and they have very little more practical importance, as yet, than the most abstract pure mathematics." Today, 50 years after Hardy's death, it seems incredible that a b o o k so at odds with reality was so influential for so many years. It is ironic that Hardy's Apologywas in fact not directed to mathematicians at all. After the dreadful carnage of World War I, and the realization that "the War to end Wars"

An earlier version, titled Mathematics Forty Years after Sputnik, appeared in American Scholar, Spring 1998. Permission to reprint portions of that article is gratefully acknowledged.

THE MATHEMATICALINTELLIGENCER9 1999 SPRINGER-VERLAGNEWYORK

hadn't really changed the world, pacifism was very widespread in England, and was effectively the established religion at Oxbridge between the Wars. The extreme attempts by Stanley Baldwin and Neville C h a m b e r l a i n - - w h o between them occupied 10 Downing Street from 1935 to 1940--to avoid antagonizing Hitler can only be understood in this context. It was primarily to the non-scientists at Oxford and Cambridge that Hardy wanted to proclaim the harmlessness of mathematics. Hardy indicates (p. 14) that the Apology, in 1940, was an elaboration of his inaugural lecture at Oxford, in 1920, when the revulsion at the horrors of war would have been particularly vehement; and that he was reasserting his position that "mathematics [is] harmless, in the sense in which, for example, chemistry plainly is not" (p. 15). Chemistry, responsible for the poison gases and disfiguring explosives of the Great War, is Hardy's chief example of a "useful" science, closely followed by Engineering, which does helpful things like building bridges, but destrnctive things as well, like designing warplanes and other munitions. In A Mathematician's Apology, Hardy is anxious to persuade his readership that "real" mathematics (especially the kind done by Hardy himself) is a noble esthetic endeavor, akin to poetry, painting, and music, and has nothing in c o m m o n with merely "useful" subjects like chemistry and engineering, which are also destructive in the service of warfare. A mere two years later, after the "blitz" bombing of London, Hardy's pacifist audience in England would have almost completely disappeared; but as a Mathematician's Manifesto, his Apology remained influential in mathematical circles for decades. David Hilbert (1862-1943), regarded by many as the leading mathematician of the first four decades of the twentieth century, and who largely defined the agenda for twentieth-century mathematics with his famous list of twenty-three outstanding unsolved problems, presented at the International Congress of Mathematicians in Paris in 1900, largely shared and advocated the view advanced in Hardy's Apology. However, Hilbert's list had several problems motivated by numerical analysis, and one asking for a proper, rigorous mathematical formulation of the laws of physics. Coming just ahead of the discovery of relativity and quantum mechanics, this problem led to interesting mathematical work in directions Hilbert could not have anticipated, but in which he actively participated. Another famous professor at G6ttingen during the Hilbert epoch was Felix Klein, who had a m u c h broader appreciation of applications. According to a famous story, a reporter once asked Klein if it was true that there was a conflict between "pure" and "applied" mathematics. Klein replied that it was wrong to think of it as a conflict, that it was really a complementarity. Each contributed to the other. The reporter then went to Hilbert, and told him,

"Klein says there's no conflict between pure and applied mathematics." "Yes," said Hilbert, "of course he's right. How could there possibly be a conflict? The two have absolutely nothing in common." Since Hilbert, unlike Hardy, did work in areas of mathematics with obvious applications, and if the quote is authentic rather than apocryp~hal, the fundamental distinction he may have seen between pure and applied mathematics would likely have involved m o t i v a t i o n - - d o we study it because it is beautiful or because it is useful? Hilbert's illustrious contemporary and leading rival for the title of "greatest mathematician of the age" was Jules Henri Poincar~ (1854-1912), a cousin of R a y m o n d Poincar~ (thrice Premier of France between 1912 and 1929, and President of France for seven years that included World War I). There is no question that Henri Poincar~ worked in some of the most obviously applicable areas of mathematics. Yet even Poincar~ asserted: "The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living." Many leading scientists w h o have made major practical discoveries would share this view, but it is significantly different from Hardy's message. Nowhere does Poincar~ suggest that applicable science, or useful mathematics, is in any way inferior, but rather that the systematic study of nature turns out to be inherently beautiful. Through the ages, the very greatest mathematicians have always been interested in applications. That was certainly true of E.T. Bell's "three greatest mathematicians of all time": Archimedes, Newton, and Gauss. It was equally true of Euler, Lagrange, Laplace, and Fourier. Even in the first half of the twentieth century, it was true of Hermann Weyl, Norbert Wiener, and John von Neumann. As we come to the end of the twentieth century, the earlier insistence on the desired inapplicability of pure mathematics seems almost quaint, though one lingering legacy is that the label "applied mathematics" retains a pejorative taint and an aura of non-respectability in certain circles. I want to examine the questions of when and h o w the concept of inviolable purity became entrenched in many departments of mathematics by the end of the nineteenth century, and what has happened in the past 40 years to weaken this presumption. In the United States, the beginning of the m o d e r n research university dates back only to 1876, with the founding of Johns Hopkins, which was based on a German model that was only a few decades older. Prior to this period, the m o d e r n division of knowledge into departments and disciplines was m u c h less rigid. In Newton's day, the term "natural philosophy" covered all of the natural sciences. The chair which Gauss (1777-1855) held at GSttingen was in

"If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living."

VOLUME 21, NUMBER 4, 1999

39

Astronomy. Only when there were separate, clearly defined departments of mathematics was it necessary to invent a rationale to support their independence from either established or newly emerging fields which sought to apply mathematics. On the other hand, it was not necessary to justify the notion that every university needed a Department of Mathematics. From the time of Plato's Academy, all through the Middle Ages, and into the rise of post-medieval universities, mathematics had always been central. The traditional "scholastic curriculum" consisted of two parts: the more elementary trivium, with its three language-related subjects-logic, grammar, and rhetoric; and the more advanced quadrivium, with its four mathematics-related subjects---geometry, astronomy, arithmetic (i.e., number theory), and music (i.e., harmonic relationships). At a time when Latin and Greek were indispensable parts of a university education, no one would have remotely considered eliminating mathematics as "impractical." Those students seeking a liberal university education, whether at Oxbridge in the U.K. or in the Ivy League in the U.S., were not thought to be concerned with learning a trade and earning a living. That came much later. And high-budget research, with the concomitant requirement to set funding priorities, was not yet a part of the university scene. So, in the late nineteenth century, university mathematics departments had a firm franchise to exist, and considerable latitude to define themselves. Much was happening in mathematics at that time (as well as ever since). The abstract approach was being applied, especially to algebra. The algebraic approach was being applied, especially to geometry and topology; analytic function theory was in full bloom; and a new standard of rigor had emerged. In many areas, mathematics was running so far ahead of applications that it was widely assumed that most of these fields would never have any. This was also true of certain classical areas, like number theory, which was developing rapidly as the beneficiary of new techniques from function theory and modern algebra, and had no foreseeable applications. Rather than be apologetic about the lack of applications for many areas, leading mathematicians and mathematics departments decided to turn this possible defect into a virtue. (In this, they anticipated a basic tenet of Madison Avenue: "If you can't fkX it, feature it.") In fact, the best mathematics consistently found very important applications, but often not until many decades later. Riemann's "clearly inapplicable" non-Euclidean differential geometry (since everyone was certain that we live in a Euclidean universe), from the 1850's, became the mathematical basis for Einstein's General Theory of Relativity some 60 years later. Purely abstract concepts in group theory from around 1900 became central to the quantum m e -

chanics of the 1930's and 1940's, and to the particle physics of the 1950's and onward. Finite fields, invented by Evariste Galois, who died in 1832, were considered the purest of pure mathematics, but since 1950 they have become the basis for the design of error-correcting codes, which are now used indispensably in everything from computer data storage systems to deep space communications to preserving the fidelity of music recorded on compact disks. George Boole's nineteenth-century invention of formal mathematical logic became the basis for electronic switching theory, from 1940 onward, and in turn for digital computer design. Hardy's most precious area of inapplicable pure mathematics was prime number theory. Edmund Landau, in his Vorlesungen i~ber Zahlentheorie ("Lectures on Number Theory", Leipzig, 1927), quotes one of his teachers, Gordan, as frequently remarking, "Die Zahlentheorie ist nutzlich, weil man nfimlich mit ihr promovieren kann." ("Number Theory is useful because you can get a Ph.D. with it.") Today, the most widely used technique for "public key cryptography" is the socalled RSA (Rivest, Shamir, and Adleman) algorithm, which depends on several theorems in prime number theory, and the observation that factoring a very large number into primes (especially if it is a product of only two big ones) is much harder than testing an individual large number for primality. I think it is fair to say that in a very special sense, Number Theory has become a type of applied mathematics, and I'm not referring to number theory's applications in communication signal design and cryptography. Rather, I refer to the fact that Number Theory, which has rather limited methods of its own, has been the beneficiary of powerful applications to it from analytic function theory, from modern algebra, and most recently from algebraic geometry, as with Andrew Wiles's proof of "Fermat's Last Theorem." In 1940, topology would have been high on most people's lists of inapplicable mathematics. Within topology, knot theory would have seemed particularly useless. Yet today, knot theory has extremely important applications in physics (to both quantum mechanics and superstring theory) and in molecular biology (to the knotted structures of both nucleic acids and proteins). The topology of surfaces is also much involved in superstring theory, including the structures which superstrings may take in multidimensional spaces. Even graph theory, the "trivial" onedimensional case of topology, has blossomed into a major discipline where the boundary between "pure" and "applied" is virtually invisible. Until recently, tiling problems were largely relegated to the domain of "recreational mathematics" (an obvious oxymoron to most non-mathematicians, but a pleonasm to true believers). Then, a decade or

Rather than be apologetic about the lack of applications for many years, leading mathematicians and mathematics departments decided to turn this possible defect into a virtue.

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THE MATHEMATICALINTELLIGENCER

so ago, Roger P e n r o s e ' s w o r k on small sets of tiling s h a p e s w h i c h can b e u s e d to tile the entire plane, but only nonperiodically, w a s found to underlie the entire v a s t field of "quasicrystals." I could give many, m a n y m o r e e x a m p l e s of h o w topics and results from the "purest" a r e a s o f m a t h e m a t i c s have f o u n d v e r y i m p o r t a n t applications, b u t I believe I have m a d e m y point. It m a y still be n e c e s s a r y for s o m e M a t h e m a t i c s D e p a r t m e n t s to defend t h e m s e l v e s from being t u r n e d into short-term p r o v i d e r s o f assistance to o t h e r disciplines which are c o n s u m e r s r a t h e r than p r o d u c e r s of mathematics; but the b a s i c principle t h a t g o o d "pure" mathe m a t i c s is a l m o s t certain to have very i m p o r t a n t applications eventually is n o w w i d e l y recognized. F o r m o s t mathe m a t i c i a n s today, the distinction that m a t t e r s is b e t w e e n good m a t h e m a t i c s and bad m a t h e m a t i c s , not b e t w e e n pure m a t h e m a t i c s and applied mathematics. To be fair to Hardy, this w a s the distinction he w a s trying to m a k e in A Mathematician's Apology b e t w e e n "real" m a t h e m a t i c s and "trivial" mathematics. Where he w e n t off the deep end w a s in trying to insist that real m a t h e m a t i c s is useless, and that useful m a t h e m a t i c s is trivial. Like m a n y of my generation, I was attracted to mathematics not b y Hardy's Apology, b u t b y E.T. Bell's Men of Mathematics, and m y early interest in n u m b e r t h e o r y was partly motivated b y the accessibility of the subject. The fwst m a t h e m a t i c s b o o k I ever bought, with m y own almost non-existent disposable income while I was still in high school, was Carmichael's thin volume Theory of Numbers, in hard cover. Two years later, I was systematically reading Landau's Vorlesungen fiber Zahlentheorie, still on my own. When I c a m e to Harvard as a graduate student, ! already a s s u m e d that I w o u l d do a thesis in prime n u m b e r theory. This was a r e s p e c t a b l e branch of m a t h e m a t i c s at Harvard, although none of the faculty there specialized in it. David Widder, w h o had recruited me to be his student m y very first day of classes at Harvard, and w h o included an analytic p r o o f of the Prirae Number Theorem in his b o o k The Laplace Transform, w a s happy to s p o n s o r m y efforts. He had spent time as a post-doc of Hardy and Littlewood in Cambridge, w h e r e the highlight of his sojourn was attending a cricket m a t c h s e a t e d b e t w e e n these two famous gentlemen. F o r help and inspiration, I drove to the Institute for A d v a n c e d Study in P r i n c e t o n one morning in October, 1953, quite u n a n n o u n c e d , a n d w e n t to see Atle Selberg. I h a d d i s c o v e r e d an identity involving von Mangoldt's l a m b d a function, which I thought c o u l d be useful in analytic n u m b e r theory. My identity l o o k e d slightly like Selberg's L e m m a in his f a m o u s E l e m e n t a r y P r o o f of the P r i m e N u m b e r Theorem. His first r e a c t i o n was to think that m y identity w a s false. Trying to disprove it, he c o n v i n c e d himself in the n e x t ten m i n u t e s that it was true. He t h e n s p e n t the rest of the day with me, exploring w a y s I prop o s e d to use this identity, and m a k i n g m a n y helpful suggestions. I l e a r n e d only later that Selberg had a r e p u t a t i o n for being totally r e s e r v e d a n d u n a p p r o a c h a b l e .

The o t h e r n u m b e r - t h e o r i s t w h o w a s very helpful was Paul Erdds, w h o w a s always totally a p p r o a c h a b l e if you w a n t e d to talk a b o u t mathematics. I l e a r n e d only years later of the s u p p o s e d feud b e t w e e n Selberg a n d Erdds. Years later still, in November, 1963, I s a w t h e m b o t h in the s a m e r o o m at a N u m b e r Theory Conference at Caltech. It w a s the d a y J o h n K e n n e d y was assassinated. I r e m e m b e r the session c h a i r m a n announcing the n e w s flash that J F K was dead. After no m o r e than half a minute, the meeting r e s u m e d as before. I s u s p e c t it w a s the only activity in the whole country that F r i d a y afternoon that w a s n ' t shut down. I m e n t i o n e d this long a f t e r w a r d to an insightful mathematician friend, and I c o m m e n t e d that t h e s e mathematicians h a d n ' t r e a c t e d to the news o f J F K ' s death. "No, you don't understand," he told me. "They're mathematicians. That was their reaction." I could have finished up at Harvard in t h e spring of 1955, in time for m y twenty-third birthday, b u t having b e e n a w a r d e d a Fulbright fellowship for s t u d y in Norway, I decided to finish m y thesis writeup there. I w a s n ' t even sure w h o was still active in N o r w a y w h e n I a p p l i e d for the fellowship. There w e r e m a n y famous N o r w e g i a n mathematicians, b u t I k n e w that Niels Henrik Abel, Sophus Lie, and Axel Thue w e r e long dead, and that several others, including Osvald Veblen, Einar Hille, Oystein Ore, and o f c o u r s e Atle Selberg h a d r e s e t t l e d in t h e United States. b o o k I h a d learned minus one. Fa br oo mu t Landau's Viggo Brun's sieve m e t h o d in p r i m e n u m b e r theory, b u t for all I k n e w Brun was also long dead. F o r t u n a t e l y this w a s not the case. Brun t u r n e d 70 the m o n t h I arrived, in June, 1955, but he did not retire until a y e a r later, and he lived well b e y o n d age 90. F r o m L a n d a u ' s austere Satz, Beweis approach, I could p r o v e Brun's Theorem, that the series consisting of the r e c i p r o c a l s o f the twin p r i m e s is either finite or convergent, b u t I h a d no u n d e r s t a n d i n g of w h a t m o t i v a t e d it, o r w h y it w o r k e d . It w a s only w h e n Brun e x p l a i n e d his m e t h o d to m e that it m a d e sense. I i n c l u d e d a sieve-derived result in m y thesis, w h i c h I also p u b l i s h e d in Mathematica Scandinavica. A few y e a r s later, E r d 6 s got an improvem e n t on m y result, w h i c h he p u b l i s h e d in the Australian Journal of Mathematics, in a p a p e r titled "On a P r o b l e m of S. Golomb." Dozens of p e o p l e have p u b l i s h e d p a p e r s tiffed "On a P r o b l e m of Erd6s," b u t since E r d d s p u b l i s h e d this one, "On a P r o b l e m of S. Golomb," I claim that m y "Erdds number" is minus one. F o r four s u m m e r s while a g r a d u a t e student, I w o r k e d at the Martin Company, n o w p a r t o f Lockheed-Martin, and b e c a m e quite i n t e r e s t e d in m a t h e m a t i c a l c o m m u n i c a t i o n theory, including S h a n n o n ' s I n f o r m a t i o n Theory, and especially "shift register sequences," w h i c h w e r e of interest for a variety of c o m m u n i c a t i o n s applications, b u t which I d i s c o v e r e d w e r e m o d e l e d b y p o l y n o m i a l s over fmite fields. Shannon's epic paper, "A Mathematical Theory o f Communication," w a s p u b l i s h e d in 1948, the y e a r after Hardy died, b u t h a d Hardy r e a d a n d u n d e r s t o o d it, he w o u l d have called it "real" mathematics, s e c u r e in the be-

I claim that my "ErdSs number" is

VOLUME 21, NUMBER 4, 1999

4.1

lief that it was not really "useful." After all, S h a n n o n gave e x i s t e n c e proofs that c o d e s c o u l d be c o n s t r u c t e d arbitrarily c l o s e to certain bounds, with no hint o f h o w to find s u c h codes. But w h a t has h a p p e n e d in the p a s t fifty y e a r s is that m a t h e m a t i c i a n s have w o r k e d closely with c o m m u n i c a t i o n s e n g i n e e r s to develop I n f o r m a t i o n Theory and Coding in a w a y that is s i m u l t a n e o u s l y first-rate "real" m a t h e m a t i c s a n d eminently p r a c t i c a l and useful engineering. In several p r o m i n e n t cases, the s a m e individual has s p a n n e d the entire range from developing the theoretical m a t h e m a t i c s to designing the practical h a r d w a r e . I eventually d i s c o v e r e d t h a t an i m p o r t a n t early p a p e r on linear r e c u r r e n c e s over finite fields was p u b l i s h e d in 1934 b y Dystein Ore, but if n o t for the wide range o f applications to several a r e a s of technology, including c o m m u n i cations, I don't think w e w o u l d have a s u b j e c t classification t o d a y in Mathematical Reviews called "Shift Register Sequences," c o r r e s p o n d i n g to literally h u n d r e d s of published r e s e a r c h papers. P e r h a p s Hardy w o u l d have b e e n d i s t u r b e d to learn h o w p r a c t i c a l the p r o p e r t i e s of finite fields have b e c o m e - - b u t then, n o n e of his s a c r e d c o w s has r e m a i n e d untainted. When I r e t u r n e d from N o r w a y in the s u m m e r of 1956, I c a m e to Southern California to w o r k in the C o m m u n i c a t i o n R e s e a r c h Group at the Jet Propulsion Laboratory, in P a s a d e n a . This job, w h i c h g r e w out of the i n t e r e s t I h a d d e v e l o p e d in m a t h e m a t i c a l c o m m u n i c a t i o n s during m y s u m m e r j o b s at the Martin Company, enabled m e to continue m y s e a r c h for a p p l i c a t i o n s of "useless" m a t h e m a t i c s to p r a c t i c a l c o m m u n i c a t i o n s problems. Over the n e x t seven y e a r s I f o r m e d a n d h e a d e d a group of o u t s t a n d i n g y o u n g r e s e a r c h e r s w h o d e v e l o p e d the s y s t e m s that m a d e it p o s s i b l e to c o m m u n i c a t e with s p a c e vehicles as far a w a y as N e p t u n e (three billion miles from Earth) a n d beyond. In 1956, NASA did n o t y e t exist, and JPL w a s f u n d e d b y the O r d n a n c e C o m m a n d o f the U.S. Army. They also supp o r t e d Wernher von Braun's group at Redstone Arsenal in Huntsville, Alabama. The U.S. Army was p r e p a r e d to l a u n c h a small artificial satellite in September, 1956, thirt e e n m o n t h s before Sputnik, using a Redstone missile as the launch vehicle, a n d a small JPL-built p a y l o a d with a radio transmitter, but General J o h n Bruce Medaris, h e a d of the A r m y Ballistic Missile Agency, was unable to get the p e r m i s s i o n of the E i s e n h o w e r administration to p r o c e e d . U n a w a r e that we w e r e in any kind of r a c e with the Soviet Union, the E i s e n h o w e r a d m i n i s t r a t i o n had d e c i d e d that the U.S. s p a c e p r o g r a m should be peaceful, and t h e r e f o r e s h o u l d n o t use an A r m y missile as the launch vehicle. Instead, w e had s o m e t h i n g called Project Vanguard u n d e r development, for w h i c h the launch vehicle w o u l d b e a Navy missile! The Soviet Union's launch of Sputnik 1, on O c t o b e r 4, 1957, t o o k the world b y surprise. It was visible to the n a k e d eye in the night sky, and a fairly simple radio r e c e i v e r c o u l d p i c k up its "beep-beep" signal. A r o u n d N o v e m b e r 12, 1957, General Medaris not only h a d p e r m i s s i o n to launch a satellite using A r m y vehicles, b u t he h a d o r d e r s to p r o c e e d as quickly as possible. Meanwhile, on D e c e m b e r 6, 1957, the

4~

THE MATHEMATICALINTELLIGENCER

first launch of a Vanguard satellite w a s attempted. With the entire w o r l d p r e s s corps watching, t h e r e w a s a spectacular e x p l o s i o n on the launch pad. The vehicle was c o n s u m e d in flames from the b o t t o m upward. Eighty d a y s after the authorization to p r o c e e d , the A r m y satellite w a s r e a d y for launch. The first stage was a liquidfuelled e l o n g a t e d Redstone rocket, from Huntsville. The s e c o n d stage w a s a cluster of eleven solid-fuel Sergeant missiles from JPL. The third stage u s e d t h r e e Sergeant missiles from JPL. A n d the fourth stage, built at JPL, was a cylinder a b o u t 5 feet long and 8 inches in diameter, p a c k e d with electronic equipment, and sitting a t o p a final Sergeant missile. This configuration had n e v e r b e e n tested, but the launch of E x p l o r e r I, on J a n u a r y 31, 1958, w a s a s u c c e s s on the v e r y first try. I had a lab at JPL at that time, w h e r e I studied the p r o p e r t i e s of shift register sequences experimentally. During the w e e k s leading up to E x p l o r e r I, a graduate s t u d e n t o f J a m e s Van Allen w a s a s s e m b l i n g a radiation d e t e c t o r in m y lab, with the a s s i s t a n c e of m y technicians. It w a s this detector, flying on E x p l o r e r I, that "discovered" w h a t c a m e to be k n o w n as the Van Allen Radiation Belts a r o u n d the earth. My o w n special assignm e n t w a s to p a r t i c i p a t e in the "early orbit determination" of the satellite. After E x p l o r e r I w a s l a u n c h e d from Cape Canaveral, no signal w a s p i c k e d up by the d o w n - r a n g e station on the Caribbean island of Antigua. We did n o t k n o w that the signal had b e e n successfully d e t e c t e d a n d r e c o r d e d at our stations in Nigeria a n d Singapore until a few days later, w h e n we w e r e notified b y air mall! A m a t e u r r a d i o groups in Australia a n d Hawaii r e p o r t e d nothing. We were unders t a n d a b l y w o r r i e d that w e had lost o u r satellite. We h a d three t r a c k i n g stations widely s p a c e d in Southern California, c o n n e c t e d to JPL only b y o r d i n a r y t e l e p h o n e lines. The n o m i n a l time for the satellite to c o m e into radio c o n t a c t over California c a m e and went, with no d e t e c t i o n by any of o u r stations. Three m i n u t e s p a s s e d , then a n o t h e r three minutes, a n d still no detection. There w e r e m a n y long faces in o u r orbit d e t e r m i n a t i o n r o o m at JPL. Then, a b o u t eight m i n u t e s late, all three of o u r t r a c k i n g stations called in almost simultaneously. E x p l o r e r I w a s alive and well. One of the u p p e r stages of the launch r o c k e t r y of E x p l o r e r I had over-performed, slightly enlarging the orbit and lengthening the period, and incidentally increasing its lifetime in orbit. That w a s an exciting time to be at JPL. While I w a s at JPL, I learned the real distinction b e t w e e n "pure research" and "applied research." In 1959, the L a b o r a t o r y director, Dr. William H. Pickering, d e c i d e d to form an ad hoc committee, with r e p r e s e n t a t i v e s from all over JPL, to r e p o r t on the "research environment" and w h a t could be d o n e to i m p r o v e it; and he a p p o i n t e d m e to chair it. I d i s c o v e r e d that every m e m b e r h a d strong opinions about w h a t w a s pure r e s e a r c h and w h a t w a s applied research, and surprisingly, it h a d a b s o l u t e l y nothing to do with the s u b j e c t matter. It all b o i l e d d o w n to this. What you w a n t to w o r k on is pure research. What your boss w a n t s you to w o r k on is applied research. Sputnik s h o c k e d the A m e r i c a n public. The notion that

the Soviets w e r e ahead of us in r o c k e t r y c o n t r a d i c t e d a basic t e n e t in Vannevar Bush's influential b o o k Modern A r m s and Free Men, which argued that a c l o s e d n o n - d e m o c r a t i c society like the Soviet Union couldn't p o s s i b l y develop arm a m e n t s a n d a d v a n c e d w e a p o n s as well as we could in the U.S. Bush h a d d e v e l o p e d a very early analog computer, called the "Bush differential analyzer," p r i o r to World War II. During the war, he was Roosevelt's c h i e f advisor on scientific and technological matters. He delivered the letter to FDR, d r a f t e d by Szilard and Wigner, and signed b y Einstein, w h i c h led to the M a n h a t t a n Project a n d the A t o m i c Bomb. Beyond that, m o s t o f the things Vannevar Bush r e c o m m e n d e d - - a t least, the ones I a m a w a r e o f - w e r e ill-advised. F o r example, b e c a u s e of his bias in favor of analog computing, he d e l a y e d r e s e a r c h on digital comp u t e r s until after the war. In Modern A r m s and Free Men, he not only a s s e r t e d that the Soviets c o u l d n ' t p o s s i b l y c o m e up with first-rate weapons, b u t also that neither w e n o r t h e y could e v e r develop intercontinental ballistic missiles (ICBMs). In 1957, I got a letter from Bush on his l e t t e r h e a d s t a t i o n e r y as Chairman of the B o a r d of MIT. Martin G a r d n e r h a d run an article in Scientific American a b o u t m y "polyominoes," and included m y p r o o f that a p a r t i c u l a r covering of the c h e c k e r - b o a r d with p i e c e s of a certain shape w a s impossible, w h i c h was s h o w n b y coloring the b o a r d in a particular way. Bush w a n t e d to k n o w w h y the result w o u l d still hold if you didn't color the b o a r d in that p a r t i c u l a r way! Naturally I w r o t e a very polite and p a t i e n t reply. (By the way, "Polyominoes" is also a s u b j e c t classification in MR.) But p e r h a p s I a m too harsh on Dr. Bush. A newly p u b l i s h e d Bush b i o g r a p h y credits him with creating the post-WorldWar-II s t r u c t u r e of g o v e r n m e n t funding for university research, w h i c h p u t s m a n y of us in his debt. The e x t e n t to which w e w e r e b e h i n d the Soviets in the d e v e l o p m e n t o f large missiles w a s m o s t l y a political matter, and s e c o n d a r i l y an engineering issue. Basic s c i e n c e w a s n o t really involved at all. Nonetheless, b o t h the public a n d the politicians w e r e c o n v i n c e d that a m u c h g r e a t e r c o m m i t m e n t to the s u p p o r t and funding o f research, especially university research, in all the basic sciences, including mathematics, was an urgent national priority. I believe it is very fortunate that this h a p p e n e d , and that it c o n t r i b u t e d significantly to the U.S. winning the Cold War s o m e 30 y e a r s later, but it h a d no relationship to the issue of w h e t h e r the U.S. was b e h i n d in rocketry. As you will r e m e m b e r , in A Mathematician's Apology, H a r d y h a d c o n t e n d e d that "real" m a t h e m a t i c s is m u c h m o r e similar to p o e t r y a n d painting that it is to c h e m i s t r y or engineering. That is s o m e t h i n g that m a n y of us, as mathematicians, might still like to believe; b u t the n e w governm e n t funding didn't e x t e n d to p o e t r y and painting. The rationale for including m a t h e m a t i c s in the n e w governmental largess required a c o m m i t m e n t to the principle that basic

r e s e a r c h in m a t h e m a t i c s , like basic r e s e a r c h in chemistry o r engineering, will ultimately have practical, beneficial consequences. Suddenly, there was a r e a s o n for trying to s h o w that one's m a t h e m a t i c s had p r a c t i c a l uses a n d implications. This was n o t the only r e a s o n for the change in attitude a b o u t w h e t h e r g o o d m a t h e m a t i c s could b e useful, b u t it certainly p l a y e d a part. A n o t h e r m a j o r influence has b e e n the d e v e l o p m e n t of digital technology, w h i c h has p l a c e d n e w e m p h a s i s on areas of discrete m a t h e m a t i c s that w e r e p r e v i o u s l y conside r e d i n a p p l i c a b l e - - l i k e finite fields, w h i c h I've a l r e a d y mentioned. Then t h e r e is C o m p u t e r Science itself, which a s k s questions in p u r e m a t h e m a t i c s like fmding the computational c o m p l e x i t y of various p r o c e d u r e s , w h i c h turns out to be e x t r e m e l y practical. A n o t h e r d e v e l o p m e n t has b e e n Shannon's m a t h e m a t i c a l t h e o r y o f communication, which asks questions m o t i v a t e d by applications, b u t which are m o r e a b s t r a c t m a t h e m a t i c a l l y t h a n anything in physics. An atom, an electron, a photon, or a q u a r k - - t h e s e are all entities in the physical w o r l d w h o s e b e h a v i o r the physicist a t t e m p t s to model. But Shannon's "bit of information" is a purely m a t h e m a t i c a l concept. It has no mass, no spin, no charge, no m o m e n t u m - - a n d y e t the i s s u e s involved in measuring information in bits, in storing information, in moving information from one place to a n o t h e r are so i m p o r t a n t that w e are told t h a t w e live in the "Age of Information." It is also true that s c i e n c e and engineering have c h a n g e d d r a m a t i c a l l y in the fifty years since H a r d y ' s death. S e m i c o n d u c t o r s a n d lasers m a k e nontrivial u s e of quantum mechanics, and the p e o p l e w h o s t u d y t h e m are n o t restricted to using w h a t H a r d y derisively r e f e r r e d to as "school mathematics." Biology at the University level has p r o g r e s s e d from butterfly collecting to g e n o m e sequencing. It is not at all clear-cut w h e t h e r "control theory" is a topic in m a t h e m a t i c s o r a b r a n c h of engineering. I've never called m y s e l f an "applied mathematician." When I'm doing m a t h e m a t i c s as m a t h e m a t i c s I a m a mathematician. When I'm focusing on a p p l i c a t i o n s to c o m m u nications, I'm a c o m m u n i c a t i o n s engineer. F o r the first several years that I w o r k e d on m a t h e m a t i c a l c o m m u n i c a t i o n s p r o b l e m s , I didn't even realize that t h e r e w e r e g o o d journals in which n e w results in these a r e a s c o u l d a n d should b e published. That w a s a lingering aftereffect of m y Hardystyle brainwashing. I h o p e I've finally o u t g r o w n it. In fact, I h o p e we've all o u t g r o w n it. M a t h e m a t i c s isn't "good" j u s t b e c a u s e it's inapplicable, and it isn't "bad" j u s t b e c a u s e it is. In fairness to Hardy, t h e r e are m a n y things in A Mathematician's Apology with which m o s t m a t h e m a t i c i a n s will agree or identify. Hardy a s s e r t s that m a t h e m a t i c i a n s are att r a c t e d to the s u b j e c t b y its inner beauty, r a t h e r t h a n by any overwhelming desire to benefit humanity. Most mathematicians I k n o w w o u l d agree with that. Even m o r e im-

When I'm doing mathematics as mathematics I am a mathematician. When I'm focusing on applications to communications, I'm a communications engineer.

VOLUME 21, NUMBER 4, 1999

43

portant, Hardy identifies h i m s e l f (p. 63) as aRealist (as the t e r m is used in Philosophy) a b o u t mathematics. "I will state m y o w n position d o g m a t i c a l l y . . . . I believe t h a t m a t h e m a t i c a l reality lies o u t s i d e us, that our function is to disc o v e r or observe it, a n d that the t h e o r e m s w h i c h w e prove, a n d w h i c h w e d e s c r i b e grandiloquently as o u r 'creations', are simply our notes o f o u r observations. This view has b e e n held, in one form or another, by m a n y p h i l o s o p h e r s o f high reputation from Plato o n w a r d s . . . . " The great maj o r i t y o f m a t h e m a t i c i a n s s h a r e this view about mathematics. Plato w e n t overboard, trying to e x t e n d m a t h e m a t i c a l reality to physical reality. I m m a n u e l Kant explicitly distinguished b e t w e e n the "transcendental reality" of m a t h e m a t i c s and the (ordinary) reality of the p h y s i c a l universe. My o w n version o f this distinction is that if the Big Bang h a d gone slightly differently, o r if we w e r e able to spy on an entirely different universe, the laws of p h y s i c s could b e different from the ones w e know, but 17 w o u l d still b e a p r i m e number. I r e c e n t l y f o u n d a very similar view attribu t e d to the late great Julia Robinson (1919-1985) in the bio g r a p h y Julia, a Life in Mathematics, by h e r sister, C o n s t a n c e Reid. "I think that I have always h a d a b a s i c liking for the natural numbers. To me they are the one real thing. We can conceive of a c h e m i s t r y that is different from ours, o r a biology, b u t w e c a n n o t conceive of a different m a t h e m a t i c s of numbers. What is p r o v e d a b o u t n u m b e r s will b e a fact in any universe." This is also r e m i n i s c e n t of the f a m o u s dictum of L e o p o l d K r o n e c k e r (1823-1891): "Die ganzen Zahlen hat Gott gemacht; alles a n d e r e s ist Menschenwerk." ("God m a d e the whole numbers; everything else is the w o r k of man.") P l a t o n i s m (i.e., "Realism") a b o u t m a t h e m a t i c s h a s dissenters. Some who, in m y view, are overly influenced by q u a n t u m mechanics, w o u l d argue that 2p - 1, w h e r e P is s o m e very large p r i m e number, is neither p r i m e n o r composite, but in s o m e i n t e r m e d i a t e "quantum state," until it is actually tested. Of course, the Realist view is that it is a l r e a d y one or the o t h e r (either p r i m e o r composite), a n d w e find out which w h e n w e t e s t it. Even less p a l a t a b l e to m o s t m a t h e m a t i c i a n s is the "post-moderu" criticism o f all of "science," that it is j u s t a n o t h e r cultural activity of humans, and that its results are no m o r e absolute o r inevitable t h a n w o r k s of poetry, music, or literature. The e x t r e m e f o r m o f this v i e w p o i n t w o u l d a s s e r t that "4 + 7 = 11" is m e r e l y a cultural prejudice. I will readily c o n c e d e the obvious: it requires a r e a s o n i n g device like the h u m a n brain (or a digital c o m p u t e r ) to p e r f o r m the s e q u e n c e s of s t e p s that w e call "mathematics". Also, culture can p l a y an imp o r t a n t role in determining w h i c h m a t h e m a t i c a l questions a r e asked, and which m a t h e m a t i c a l topics are studied. (Our w i d e s p r e a d use of the d e c i m a l system is u n d o u b t e d l y related to h u m a n s having t e n fingers.) What I will not conc e d e is that, if the s a m e m a t h e m a t i c a l questions are asked, the a n s w e r s would c o m e o u t inconsistently in a n o t h e r culture, on a n o t h e r planet, in a n o t h e r galaxy, or even in a different universe. F o r e x a m p l e , the G r e e k s w e r e i n t e r e s t e d in "perfect numbers," n u m b e r s like 6 ( = 1 + 2 + 3) a n d 28 ( = 1 + 2 + 4 + 7 + 14) w h i c h equal the s u m o f t h e i r e x a c t

THE MATHEMATICAL INTELLIGENCER

divisors (less t h a n the n u m b e r itself). I can readily imagine a "civilization" with a d v a n c e d m a t h e m a t i c s in which the notion of "perfect numbers" w a s n e v e r formulated. What I c a n n o t imagine is a civilization in which perfect n u m b e r s w e r e defined the s a m e w a y as w e do, but w h e r e 28 w a s no longer perfect. It w a s Isaac N e w t o n who wrote, "I do n o t k n o w w h a t I m a y a p p e a r to the world; but to m y s e l f I s e e m to have b e e n only like a b o y playing on the seashore, a n d diverting myself in n o w a n d then finding a s m o o t h e r p e b b l e or a prettier shell t h a n ordinary, whilst the g r e a t Ocean of Truth lay all u n d i s c o v e r e d b e f o r e me." It is h a r d to give a b e t t e r formulation of the Realist view of m a t h e m a t i c a l truth a d h e r e d to by m o s t mathematicians. N e w t o n is firmly e n t r e n c h e d in the p a n t h e o n s of b o t h m a t h e m a t i c s a n d physics. He certainly e r e c t e d no artificial b o u n d a r i e s b e t w e e n theory a n d applications. In the t h r e e centuries since he published the Philosophiae naturalis principia mathematica (the Principia, for short), w e have, with e l e c t r o n m a g n e t i s m and relativity and quantum mechanics, w a d e d d e e p e r into that O c e a n of Truth; b u t Newton's laws o f m o t i o n and gravitation w e r e actually sufficient for the launching of Sputnik and Explorer. Today, 40 y e a r s after t h o s e satellites first circled the earth, m o s t m a t h e m a t i c i a n s - - m y s e l f i n c l u d e d - - h a v e m o v e d farther from H a r d y ' s o u t l o o k and closer to Newton's.

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The Road to the International Math Olympiad: The Philippine Experience Queena N. Lee-Chua

This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of "mathematical community" is the broadest. We include "schools" of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.

Please send all submissionsto the Mathematical Communities Editor, Marjorie Senechal, Department of Mathematics, Smith College, Northampton, MA 01063, USA;

Marjorie

Senechal,

Editor

n a sultry Saturday afternoon in July, 1991, four teenaged boys sat patiently waiting for me in a groundfloor classroom at Faura Hall on the Ateneo de Manila University campus, Loyola Heights, Quezon City, Philippines. I was right on time, but unlike the average Filipino for whom lateness has become quite fashionable, these lads believed in punctuality. Courtesy, too, for that matter--for they rose to their feet en masse as I entered, greeting me with shy smiles. On the board were equations, 3dimensional graphs, lines of p r o o f - evidence of their just-concluded training session. For more than 10 months, Wilbin Chan, 17, and his brother Wyant, 15, of the privately-funded, Chineserun Uno High School, and Jose Ernie Lope, 17 and Ronald Carpio, 15, of the premier public secondary school Philippine Science, had been gearing intensively for the competition of their lives. Every Saturday during the past schoolyear, and five days a week during vacation time, under the aegis of the Program of Excellence in Mathematics (PEM), these chosen four had pored over international math texts, tried novel ways of problem-solving, and learned enough material to put any college math major to shame--number theory, combinatorics, solid geometry, advanced algebra, etc. Their favorite pursuits--computer games, chess, volleyball, basketball, rap music--had been put on hold, all in a bid to bring glory to family, school, and country. For in the following week, accompanied by PEM head coach Dr. Jose Marasigan of the Ateneo and assistant coach Mr. Misael Fisico of the Pamantasan ng Maynila (University of Manila), they would be journeying for the first time in their lives to a country on the other side of the globe, one whose weather is the antithesis to the heat of home: Sweden, specifically the town of Sigtuna, the site of the 32nd International Math Olympiad (IMO).

O

I

On this--the eve of their departure--I was asked to interview them for a major dally (Lee, 1991a and 1995b). I knew that all had been consistent honor students since the primary grades, topping various local contests, most notably the Philippine Math Olympiad (PMO). In a country where math and science are feared and shunned, they had become role models for aspiring students all over. I was half-expecting some aloofness, but they seemed not to have a shred of arrogance, and indeed were awed (with the touching eagerness of the young) at the prospect of representing their country in the international arena. Except for Wyant, who had garnered a bronze medal at the 31st IMO in Beijing, China a year before, they were first-timers. When queried about the factors that led to their success thus far, they had chorused gratitude to the PEM, which had trained them free. They also cited the support of family and friends. They also faced unrelenting pressure. "Sometimes teachers and classmates expect a lot from us," complained Ernie. "They think we should be on top all the time." "Our school administrators do not grant incentives, even exemption from ordinary exams, for our participation. I wish they would. Competing in the IMO is an added responsibility," added Wilbin. Despite this, all four admitted that they were enjoying themselves. Two weeks after our meeting, the Philippines rejoiced. Wyant and Wilbin Chan each brought home a bronze medal, Jose Ernie Lope received honorable mention, and Ronald did commendably enough to merit praise from foreign observers. T h e International M a t h Olympiad Mathematics competitions have existed since the advent of civilization; the modern contest (for secondary school students) began in Hungary in

9 1999 SPRINGER-VERLAG NEW YORK, VOLUME 21, NUMBER 4, 1999

1894. It was the Russians, though, who coined the term "Mathematics Olympiad" in 1934 for the middle school students' math contest in Moscow and Leningrad. This concept spread to other Eastern European countries, where a pyramid-like hierarchical system was formed. Here students competed first in schools, then in regions, then on the national level. It was inevitable that an international competition would develop. In 1959, the first IMO was held in Belasov, Romania, with seven participating countries, including the then Soviet Union. The Olympiad was so successful that it became an annual event. Several countries joined in succession: Yugoslavia in 1963, Mongolia in 1964, Finland in 1965, France, England, Italy, Sweden in 1967. The contest spread out from Europe into other continents. In 1974, the United States joined, and China participated in 1985. The growth of the IMO can be seen from the increase in participating countries. In 1977, there were 21; in 1983, 32; in 1987, 40; in 1988, 54; and presently, more than 70. The number of contestants has also increased: from 40 in the early days to more than 600 now. In 1984, when some local Olympiad winners performed well in the Australian Math Competition, the head at that time--the late Peter O'Halloran-was impressed by a meeting with Dr. Jose Marasigan, the coach of the Philippine delegation. O'Hailoran extended an invitation for Filipinos to compete Down Under, and our youth did well enough that the Philippines was encouraged to send a team to the IMO; because of financial constraints, the Filipinos could only join a year later. Four secondary school students traveled to Canberra, Australia in 1988 as observers, and Victor Luchangco, a senior at the Jesuit-run Ateneo de Manila High School, bagged an honorable mention, missing the bronze by just one point. The Philippines has never missed sending a delegation to the IMO since then.

The Philippines Historically, culturally, and economically, the Philippines does not seem to be a likely candidate for IMO partici-

4'6

THE MATHEMATGAL INTELUGENCER

pation, much less a member of the winners' circle. This archipelago of more than 7,100 islands lies near the equator in the South China Sea, and since ancient times, has been home to Malays and various others, including settlers from neighboring states (e.g., the Chinese). No records of formal prehistoric mathematical systems have been found, though evidence has surfaced concerning the scientific ingenuity of the early Filipinos (Landa Jocano, 1967 and 1984). Basic notions of arithmetic and geometry had been applied in engineering (such as the Banawe rice terraces), boatbuilding (the balangay), timekeeping, metallurgy, and toolmaking. Spain colonized the country in 1521, and for more than three centuries, the Philippines was under the rule of the sword and the cross. Though some benefits, such as infrastructure development and increased trade, might arguably have been derived, native resources were plundered, uprisings savagely suppressed, and religions other than Catholicism outlawed. Even though elementary mathematics was taught in the local colleges and a few textbooks appeared in Manila during the 18th and 19th centuries (e.g., Ignacio Villamor's 1897 works--Geometria Elemental and Aritmdtica Elemental), further study and research in this field was unheard of. As the noted mathematics pioneer Federico Sioson succinctly put it, "Under any criterion accepted by the mathematical establishment, there appears to be no record whatsoever of any Filipino who attained the status of a mathematician through the 350 years of Spanish colonization . . . . And even among the handful of European-educated Filipino intellectuals who flourished during the waning years of Spanish rule and the early years of American occupation, mathematical research was a totally neglected and unknown vocation" (Sioson, 1967). In 1898, the Americans helped the Filipinos fight the Spaniards, only to annex the country to their union. Though American rule was decidedly more benevolent, the Filipinos longed for freedom, and devoted much of their time and effort towards this end. The first glimmerings of higher mathenmt-

ics study appeared in the early 1920s, when the University of the Philippines (UP) in Diliman, Quezon City introduced a master's degree program. The first graduate here was Francisco D. Perez, who obtained his M.A. in 1924 with a thesis on the study of the derivative of an algebraic function without the use of a limit, under the guidance of an American professor, H. L. Smith. With the help of American teachers, a few promising students were sent to universities abroad. Among the pioneers were Emetrio Roa, who graduated from the University of Michigan in 1923 with a doctoral dissertation on statistics; Vidal Tan, from the University of Chicago in 1925 with a study of projective properties of quadrics; Enrique Virata from Johns Hopkins University in 1926 with his work on metric differential geometry, Francisco Perez and Trinidad Jaramillo from the University of Chicago in 1929, the former with a dissertation on integral equations of the Hilbert-Schmidt type, the latter working on elasticity theory. With a growing pool of local talent and international mentors, the Philippines could have started to build a mathematical tradition in the 1930s. However, two factors intervened. Administrative matters took primacy over research studies. "Instead of further cultivating and developing their talents through zealous and sustained research, many of these men chose to divert their energies and efforts to various economic, political, and administrative pursuits" (Sioson, 1967). And the Second World War made education a decidedly low priority. During the Second World War, the Philippines fought on the side of the Allies, and when local military outposts Corregidot and Bataan (the last bastion of the Allies in Asia) fell in 1941, the country suffered under Japanese oppression. For several years, schools remained closed. Liberation came in 1945, and America granted the Philippines political independence the following year. But with freedom came a host of other problems. Partisan politics, cronyism, and corruption exacerbated the already-existing disparity between rich

and poor, the powerful and the disenfranchised, all rooted in the feudalistic structure of Spanish rule. Furthermore, economic independence from the US had not been achieved, the burden of debt was mounting, and to make matters worse, a dictator--Ferdinand Marcos--ruled the country under martial law for more than a decade. The Marcos clan and their cronies systematically looted the country's coffers and exiled many of the best and brightest minds to other lands. On paper, democracy had been achieved, but inequity and inequality were--and still are--the reality. For instance, in 1984, when Dr. Marasigan had his historic meeting with Dr. O'Halloran, data on Philippine income distribution revealed that A familiar sight: a child helping her parents at about 5% of the population were very well off, with 15% in the middle class, and silent acquiescence to social inand the great majority (80%) in the low justice . . . . In a country where the income bracket (Ibon Facts and Fig- principal problem is the concentraures, 1984). The richest 20% of the tion of wealth and power and access population received incomes 15 times to the benefits of S & T [science and higher than the poorest 20%. Further- technology] is limited to one small more, the richest 2% of families got segment of the population, there is ur16.5% of total income (P8,570 million, gent need f o r radical moral and soor $230 million), compared to the poor- cial refo,~n as well as for education est 2.9% who only had a measly share towards social justice (Gorospe and of 0.2% of total income (P104 million, McNamara, 1984). or $5 million). It was wryly noted that even if P8 billion were taken away The Mathematical Society of the from the richest, they would still be 5 Philippines (MSP) was established in times richer than the poorest. the mid-1970s (Nebres, 1973 and 1979), This state of affairs became so lam- but many mathematicians and other entable that in that same year, a pair scientists went underground during of university professors--one a physi- the martial law years, forsaking recist, the other a theologian--jointly search for activism. In the 1986 EDSA aired this impassioned plea: People Power Revolution, academics and students joined their fellow counThe root cause of poverty and in- trymen in the streets, stopping soldiers equality in the Philippines is insti- in tanks with rosaries, flowers, and tutionalized or structural injustice, prayers, and under the glare of interboth on the national and interna- national media, finally managed to tional l e v e l . . , both external neocolo- overthrow the dictator. nialism and internal colonialism-The Marcos tyranny was overthrough domestic capitalism and thrown, but the Corazon Aquino years "feudalism"---have been supported by were far from easy. The national treaa government which uses measures sury was virtually empty, nationwide that keep the majority of the people in power blackouts and lack of basic utila state of poverty and oppression . . . . ities was the norm, and unfinished Structures [exist which] systemati- projects lay strewn in the countryside. cally oppress h u m a n dignity, violate International debt (now $45 billion, basic h u m a n rights and impose gross Ibon Facts and Figures, 1998) was the i n e q u a l i t y . . , promote and facilitate biggest burden, and meeting the deindividual selfishness, complicity mands of the World Bank and the

their sidewalk stall.

International Monetary Fund was top priority. Yet Filippinos were determined to improve their lives. In 1987, the old constitution was replaced by one which was felt to be more responsive to the needs of the citizenry; it included a mandate to give priority to research and development, invention, science and technology education, training, and services. To this end, the national budget was realigned to allocate more funds to these purposes, the ministries of education and science were revamped, and structures put in place to handle specific fields of training and research. Thus, one of the tasks of the Science Education Institute (SEI) of the Department of Science and Technology (DOST) is to fund the training and foreign trips of the Philippine team to the IMO. Another agency--the Department of Education, Culture and Sports (DECS)--is involved in the selection process, primarily in the holding of the local Olympiad, from which potential delegates are chosen. But progress is slow, especially in education. The Philippines has long been proud of its allegedly high literacy rate, but statistics belie the case. True, most Filipinos (7-12 years old) in the National Capital Region (NCR) and other regions do finish elementary school, but the number of enrollees drops substantially in high school

VOLUME 21, NUMBER 4, 1999 4 7

(13-16 years old), and only a fifth of the student population even go to university (17-24 years old). As for graduate school, the number of graduates is a negligible fraction of the entire student population. One reason for this sad state of affairs, aside from their lack of funds for tuition, is the intense pressure on lower- and middle-class Filipino children to learn just enough--basic reading, writing and arithmetic--to help their parents earn a living. Most "whitecollar" jobs are closed to them, but they do manage to eke out a livelih o o d - s e l l i n g flowers in the streets, vending fruit in vegetable stalls and markets, learning the family trade (e.g., shoemaking, embroidery, handicrafts). Though the government and the private sector have tried to help, primarily through scholarships and grants, there has been only a little improvement. This situation looks even worse when we compare the enrollment in the two major types of schools: public and private. Public schools are controlled by the government, which appoints administrators and sometimes key faculty and pays the salaries of everyone in the schools. Tuition fees in elementary and high schools are waived, but sometimes students have to provide for school supplies and uniforms. In government colleges and universities, tuition is subsidized-students pay according to their family's income level, with wealthier ones paying a disproportionate amount to help their financially strapped classmates. In the absence of tuition fees--one of the main sources of income for private schools--standards in public schools are difficult to maintain, let alone improve. Competent teachers are attracted to private schools, which pay better. Add to this the bureaucratic structure inherent in government systems anywhere, the political ploys and machinations of government appointees, the sheer number of students crammed into each class (sometimes more than 50 per class)--it is no wonder that students' needs cannot be adequately met. Math and science are relegated to the bottom of the curriculum,

48

THE MATHEMATICALINTELLIGENCER

since many teachers admit that they are not prepared to handle these subjects, and have not specialized in them in college (Abueva, et al. 1998). In my tours around the country, many public school teachers have complained to me that they are forced to teach math and science because no one else can be found to do so. It is not rare to find an education major specializing in civics or home economics struggling to teach an algebra class. Thus, it comes as no surprise that students from private elementary and high schools normally fare better in nationwide entrance and subject exams. Most private schools are sectarian, and are roughly split 60%-40% between Catholic and non-Catholic (e.g., Protestant, Buddhist, ecumenical) institutions. Education here is far from inexpensive though. In Manila, the average private elementary and high school charges P10,000 to P20,000 per school year, with the top institutions charging more than P30,000! The top two private universities in the country, Ateneo and De La Salle University in Taft Avenue, Manila, charge an average of P25,000 per semester, compared with the top public university, the University of the Philippines, Diliman's high of P10,000. Though public elementary schools cannot compare with their private counterparts, in the secondary and tertiary levels, certain public high schools, colleges and universities do stand out. At the secondary level, these are the vaunted "science high schools," namely, Manila Science High School, Quezon City Science High School, and Philippine Science High School--with four branches---the main campus in Quezon City, and three others in Iloilo, Leyte, Mindanao. Unlike applicants to other government high schools, who are accepted on the basis of an elementary diploma, prospective applicants to these science high schools have to undergo a difficult entrance exam. Normally, less than half of the applicants are admitted. But the rewards upon admission to these selected schools are n u m e r o u s - facilities, teachers, and resources are at par with those of private schools. In the past two years, aside from the ex-

pected number of PMO finalists (thus, PEM participants) from private schools, there are increasing representatives from these public science schools. At the university level, the few public institutions are on a par with, and at times (depending on the particular field) even better, than the private institutions. On the top, of course, is the state university--the University of the Philippines (UP) in Diliman, Quezon City, with its adjunct campus in Baguio City, UP Los Banos, Laguna, UP Visayas, and UP Mindanao. In math and science, another top-ranked school is the Mindanao State University-Iligan Institute of Technology (MSU-ITT) in Lanao del Norte, Mindanao, with its sister school in its sister province, MSU in Marawi City, Lanao del Sur, Mindanao. In its effort to promote science and technology, the government recently bestowed on four universities the title of "Centers of Excellence" (for math and science). Two are public: UP Diliman and MSU-ITT; two are private: Ateneo de Manila and De La Salle universities. Four other schools were deemed "Centers of Development"-two public: UP Los Banos and Mindanao Polytechnic State University; two private: Ateneo de Cagayan and University of San Carlos, Cebu City. The mathematics departments of these universities boast excellent faculty, many of whom have doctorates (mostly from abroad): UP-Diliman has approximately 25 Ph.D.s, MSU-ITT, 20, Ateneo de Manila, 15, De La Salle, 10. (Of course, there are always other doctoral candidates waiting in the wings.) However, research is mostly on applied math, with combinatorics and graph theory the overwhelming favorite, followed by computer science (algorithmic design and analysis) and statistics. Analysis is a favorite among the "pure math" researchers, with algebra having a few devotees. It is instructive to look at the roster of the country's most outstanding scientists to date (deemed "Academicians" or "Outstanding Young Scientists" by the DOST). The mathematics awardees, their universities, and their fields are given in Table 1. Other noted mathematicians in-

UP Diliman --Sergio S. Cao, Rene P. Felix, Polly W. Sy (analysis) --Jose Maria A. Balmaceda (algebraic combinatorics) --Luz R. Nochefranca (graph theory and combinatorics, algebra) --Mark E. Encarnacion (design and analysis of algorithms) UP Los Banos --Eliezer A. Albacea, Rolando E. Ramos (design and analysis of algorithms) --Aria Inez N. Gironella, Filipino P. Lansigan (statistics) Ateneo de Manila University --Bienvenido F. Nebres, S.J. (analysis, math education) --Jose A. Marasigan (geometry, math education) --Felix P. Muga II (design and analysis of algorithms)

clude Harry Carpio (MSU-ITT, analysis), Jimmy Caro (UP Diliman, design and analysis of algorithms), Severino Diesto (De La Salle, topology), Reginaldo Marcelo (Ateneo de Davao, analysis), Milagros Navarro (UP Diliman, analysis), Marijo Ruiz (Ateneo de Manila, graph theory and combinatorics). Filipino mathematicians are striving to be heard--in international conferences, symposia, and journals. But the Philippines still trails its Asian neighbors in this respect. For instance, the nearly 1,000 science and math profes-

Dr. Jose Marasigan with the author.

sors in UP authored only 60 ISI (Institute for Science Information)indexed journal articles in 1995. The whole country turned out only 224 such publications that year, compared with Thailand's 574 and Singapore's 1270 (Abueva, et al. 1998). Manpower is also limited--as of 1996, there were only about 1,000 involved in research, about 55 per million people, way below the 300 per million prescribed by the UNESCO. Very few have advanced degrees, and very few students are enrolled in science programs, including math. The government cannot offer high salaries to attract qualified personnel, and academic institutions do not do much better. The quality of teaching leaves much to be desired. Only 7 to 8% of those teaching science and math are majors in those fields (Abueva, et al. 1998). In the recentlyconcluded Third International Math and Sciences Study, the Philippines fared dismally--39th out of 41 countries among students 13 years old or older (Nebres, 1997). Behind PEM: Dr. Jose Marasigan Against this backdrop, the achievements of PEM become even more remarkable. Let us now meet the guiding spirit behind PEM: Dr. Jose Marasigan. I first met Dr. Marasigan in March 1983--briefly--when he awarded our high school team silver medals during the national f'mais of the Metro Manila Math Competition (MMMC). Later that

year, I would be a freshman at the Ateneo, and our paths would cross again. Dr. Marasigan was our professor for the advanced placement math course. We were a class of 60, mostly honor students from different sch~ools, and we thought we already knew a lot of math. Dr. Marasigan had other ideas. On the fLrst day of class, he launched into a lecture on the Buffon needle probl e m - a n d we were hooked. With his guidance, the class went beyond the calculus text. We delved into braintwisters, proofs, IMO problems. Dr. Marasigan also had a sense of humor. Once he announced that he would exempt from the final exams any student who could come up with a good proof of Fermat's Last Theorem. Never in the history of our university library had so many students crammed the aisles of the math section! We failed in this particular endeavor, of course, but in the process, our class became perhaps the most knowledgeable on Fermat in the Philippines. After graduation, I entered the Math Department, and am now proud to call Dr. Marasigan a colleague. Born on May 26, 1943, Jose Abril Marasigan never thought he would grow up to be a mathematician. Though his grandfather was a mathematics teacher, the young Jose preferred tinkering with gadgets, and thus decided to take up chemistry at the Ateneo. There he met the American Prof. Wallace Campbell and the Filipino Dr. Federico Sioson, both great theoreticians, and great teachers, besides. "They became my role models, and thus I decided to shift to mathematics." It was said that the chemistry department chair tried to entice the young lad back into that field, to no avail. Philippine chemistry's loss was mathematics' gain. In 1962, Jose Marasigan graduated, cure laude, with a bachelor of science degree in mathematics. In 1968, he graduated from the Technische Hochschule in Darmstadt, Germany, with a doctorate degree. Under the supervision of Prof. Helmut Maurer and the late Prof. Roff Lingenberg, Marasigan entitled his dissertation Charackterisierung metrischer Ebenen durch Beweglichkeitsaxiome (Characterizing metric planes through axioms of motion).

VOLUME21, NUMBER4, 1999 49

Marasigan is now a full professor in the mathematics department at the Ateneo. He has won several national awards, including the Outstanding Young Scientist of the Year and the Juan Salcedo Jr. Science Education Award.

The Development of PEM After the Filipinos joined the Canberra IMO in 1988 as observers, Dr. Marasigan established the formal training program for possible delegates. Finalized in 1989, the PEM started with one center--the Ateneo campus in Quezon City, Metro Manila. Now three other centers are in the works: in Baguio, in Cebu, in Iligan. A partnership among academe, government a n d - - t o some degree--private business, the PEM has several goals: 1. to provide intensive and comprehensive training for mathematically gifted secondary students; 2. to encourage and nurture the study of math in the Philippines; 3. to promote cooperation among mathematicians, mathematics education teachers, and different agencies in improving the quality of math education here; 4. to promote excellence as a way of life; 5. to raise the Philippines' standing in the international educational and scientific community. As director, Dr. Marasigan invites qualified and interested instructors

from secondary and tertiary level institutions to become members of the Technical Committee or co-trainers, and screens potential IMO participants from all over the country. The screening process is patterned after that of Germany, and involves a two-pronged process: one to attract students from the National Capital Region (NCR), and the other to entice those in the other regions. At the start of each schoolyear in June, six challenging questions are formulated--three for freshmen and sophomores, the other three for juniors and seniors. Through the network of DECS local offices, these questions are distributed to high schools in the region. Students taking the test have to be bona fide Filipino citizens, and have to sign a declaration (attested to by their teacher or adviser) affirming that they have worked independently in solving the questions. Textbooks and references used must be indicated in their answer sheets. All solutions are then submitted to the SEI before the September deadline, and the Technical Committee of the PEM selects the top 30 scorers for each level. These 60 students are then invited to undergo the training program of the PEM, which starts every October and lasts till July of the following year. At the end of the training, the members of the Philippine team to the next IMO are selected. The questions range from number theory and combinatorics to functions, solid geometry, advanced al-

gebra. Patterned after questions from previous IMOs and other math competitions, such as the Putnam Examination, these problems are beyond the scope of the average Filipino secondary math curriculum, which centers on elementary algebra, geometry, trionometry, statistics. Sample questions are given in Table 2.

The Philippine Mathematical Olympiad (PMO) For students in the 15 other regions of the country outside the NCR, the route to the IMO is just as challenging. They have to be winners in the premier local math fest: the PMO. Under the aegis of Professor Josefina Fonacier of the Institute for Science and Math Education at UP Diliman, the first PMO was conducted in 1984-85 on a feasibility basis in two regions, the NCR and Region IV (Southern Tagalog). Following this successful tryout, the PMO was expanded to a nationwide scale, and since 1986 has been conducted every two years. The Department of Science and Technology (DOST) is now the major sponsor, and for the past two competitions has shelled out an average of half a million pesos for each event. Business also stepped into the picture, with a major bank sponsoring trophies and cash awards. "Today, the mechanics are all in place," affirms Prof. Fonacier. Aside from identifying and motivating mathematically gifted students, the PMO

TABLE 2. Sample questions for IMO candidate Level I (First and S e c o n d Y e a r s ) 1. Write 30 as the sum of four positive integers a, b,

c, d

with a >- b -> c > d so that the product

abcd

is a maximum.

2. Prove that the product of four consecutive integers cannot be a perfect square. 3. Given five points P1, P2, P3, P4, P5 in the plane having integer coordinates, prove that there is at least one pair (Pi,P]) with i C j such that the line segment PiPj contains a point Q having integer coordinates and lying strictly between P~ and Pj. 4. Prove that (21n - 3) / 4 and (15n + 2) / 4 cannot both be integers for the same positive integer n. Level II (Third and F o u r t h Y e a r s ) 1. Find all polynomials

g(x) = x n + s i x n 1 4-

"

'

"

+ an

with the following properties: a. all the coefficients al, a2 . . . .

an belong to the set { - 1 , 1}; b. all the roots of the

equation g(x) = 0 are real. 2. Let ABC be any triangle and D, E, F be points on BC, CA, AB, respectively. If AF -< FB, BD -< DC, CE -< EA, show that the area of triangle DEF -> 1/4 area of triangle ABC. 3. Ten distinct numbers from the set {0, 1, 2 . . . . .

13, 14} are to be chosen to fill in the ten circles in the diagram. The absolute values of the differences of the

two numbers joined by each segment must be different from all other segments. Is it possible to do this? Justify your answer.

4. Prove that for any set of n integers, there is a subset whose sum is divisible by n.

50

THE MATHEMATICALINTELLIGENCER

Professor Josefina Fonacier.

aims to provide a vehicle for the professional growth of teachers, and to stimulate the improvement of math education (Fonacier, 1996). The indefatigable Prof. Fonacier graduated with a bachelor of science degree in educa-

tion, major in math, c u m laude, from UP Diliman in 1949. She never had the time to pursue a doctorate. "I am waiting for some institution to give me an honorary degree," she laughs. "But my biggest regret is not having spent enough time teaching in the classroom." Now retired, she recently passed on her post as PMO director to younger faculty. However, she is still actively working behind the scenes, networking and encouraging more schools to join the competition. One major problem in the beginning was the wide disparity of mathematical ability of students and coaches in the different regions. Most of the winners normally hall from the NCR, with big centers in the other island groups a distant second or third. To address this need, expert mathematicians started conducting free professional development seminars for coaches in 1994. The sessions, which became very popular, dealt with specific problem-solving skills and content.

Ill

f . ~ : | II : I t

1988

m l "J ,'Tli i ,1~.] h [ : a ,, [:~. h " l l ~

~.lz,n

hVj [ l . ~ I

Canberra, Australia Victor Luchangco-- Honorable Mention

1989

Braunschweig, Germany Jerome Kholhayting--Silver Medal

1990

Beijing, China Wyant Chan--Bronze Medal

1991

Sigtuna, Sweden Wyant Chan--Bronze Medal Wilbin ChaD-Bronze Medal Jose Ernie Lope--Honorable Mention

1992

Moscow, Russia Wyant ChaD--Bronze Medal Michael Sin Sy--Honorable Mention

1993

Istanbul, Turkey Wyant Chan--Bronze Medal

1994

Hong Kong Glen Ong--Honorable Mention

1995

Toronto, Canada

1996

Bombay, India

Bernard Chan--Bronze Medal none 1997

Plata Del Mar, Buenos Aires, Argentina

1998

Taipei, Taiwan

Mark John Hermano--Honorable Mention noDe

The Philippine team at the 30th IMO in Braunschweig, Germany. Front row: Michael Garcia, Eric Angelo France, Jerome Khohayting (silver medalist), Dr. Jose Marasigan (head coach), Justine Leon Uro (assistant coach), Marisciel Litong. Back row: Rommel Regis, Ceferino Sanchez, Dr. Jose Khohayting (father of Jerome), Angeles Franco (mother of Eric).

VOLUME21, NUMBER4, 1999 51

Prof. Fonacier is heartened by the increasing success of participants from other areas, especially those far from the town centers. For instance, in war-torn Cotabato, where conflicts between Muslims and Christians still sporadically erupt, a teenager from the Albert Einstein School made it all the way to the 1998 national finals, finally clinching third place. More boys than girls reach the national finals. "Perhaps it is the tension of the competition, perhaps boys tend to perform better under pressure." Or perhaps cultural factors still pervade. Many people in Filipino society still believe that girls should excel in art, not science. (With the advent of a new millenium and the increasing exposure of the Philippines to the West, it is hoped that this myth will steadily erode.) Added to the gender issue---so far only one teenaged girl has become a Filipino delegate to the IMO--is the topic of race. Of the 9 winners to date, a disproportionate number are of Chinese origin. Dr. Marasigan attributes this to the value of education for the Filipino-Chinese, to their exceptional patience and determination, and to the support of the contestants' parents. T h e S u c c e s s of t h e P E M

"From the start, the PEM was well-received," asserts Dr. Marasigan. From one section of 20 students in 1989, the PEM now trains three sections of more than 40 students each. Funding was--and remains--a perennial problem. During the first few years, the charity arm of a well-known utility company helped shoulder some expenses. For the Philippine team's trip to the 1994 IMO in Hong Kong (where one of the contestants received honorable mention), the national airline carrier issued free tickets. However, for the trip to Argentina (far from the Philippines) in 1997, funding could only be found for two contestants! The quality of the teachers is another problem. "We still have difficulty finding qualified trainers. Most of the teachers, even college professors, are intimidated by the brilliance of these teenagers, and so are reluctant to teach them," reflects Dr. Marasigan. This problem stems from a larger one--the

52

THE MATHEMATICAL INTELLIGENCER

overall poor quality of math education in the country. The greatest sacrifice is demanded from the students themselves. "Every Saturday, they have to come to the Ateneo campus, many of them often stuck in traffic for several hours. We do not have enough money to give them snacks, so they bring their own food or use their allowance to purchase cheap snacks from the canteen." (Recently however, some schools have showed support by providing their own van to bring their students to the training center.) Fueled by determination and hard work, the PEM has boosted national pride by delivering exemplary performances year after year (Table 3) However, recent performances of the Philippine team suggest that in some cases motivation has been a problem. Since PEM training is not part of their formal school curriculum, some participants have not been dedicated in solving exercises. Dr. Marasigan plans to institutionalize the program and, supported by the SEI, to make the training part of their academic curriculum. "In short, the trainees will be given grades. We want them to take it seriously." By far the bigger problem is "brain drain." Most of the Filipino winners have been offered full scholarships by universities abroad, and they jump at the chance to fulfill childhood dreams of becoming an engineer, a mathematician, a scientist. Victor Luchangco finished his doctorate degree in computer science at the Massachusetts Institute of Technology (MIT). Jerome Khohayting and Michael Sin Sy both finished their undergraduate degrees in electrical engineering at the same university, and Jerome is now a trader at Merrill Lynch on Wall Street. Still another MIT graduate is Wyant Chan, who finished s u m m a c u m l a u d e in physics, and is now in graduate school. Jose Ernie Lope completed his doctorate in mathematics at the Sophia University in Japan. Glen Ong is currently taldng up electrical engineering at Carnegie Mellon University. This last fact is particularly poignant for me, for during an interview, he sadly confided, "I think Filipino scientists go away because foreign countries have more resources. Money is one thing but necessary equip-

merit and technology is another. I would personally stay here if there was any opportunity. Scientists staying in the Philippines need a two-way interaction: scientists work for the Philippines and in return, the Philippines provides for them" (Lee, 1992). Despite these setbacks, the PEM has no plans to slow down. The Philippines has been hard hit by the recent Asian economic crisis, and education is the key for many citizens to raise their living standards. Also, competence in math and science is sorely needed for globalization. Dr. Marasigan is still not satisfied. "Compared to our other Asian neighbors, like Singapore and Japan, our performances are below par. We will continue to strive until we reach the degree of competence needed in the next millenium."

R

E

F

E

R

E

N

C

E

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Landa Jocano, F. (1967). The beginnings of Filipino society and culture. Philippine Studies 15(1), 9-40. (1984). Scientific thinking in traditional practices. Solidarity 101, 3-7. Lee, Queena N. (1991a). The numbers olympiad. Philippine Starweek, 7 July. -(1991 b). Why Be Afraid of Math and Other Intellectual Adventures. Manila: Bookmark, Inc. -(1992). A math champ's secret. Sunday Inquirer Magazine, 1 March. (1995a). Eureka/ Manila: Anvil Publishing, Inc. (1995b). Popularizing mathematics in the Philippines: A personal account. In S. Tangmanee and E. Schulz (Eds.), Proceedings of the Second Asian Mathematical Conference 1995, 589-93. Singapore: World Scientific. (1996). A multiple regression analysis of college mathematics scores in the Philippines. In Nguyen Dinh Tri, Pham The Long, -

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-

et al (Eds), Proceedings of the Seventh Southeast Asian Conference on Mathematics Education, 167-71. Hanoi, Vietnam: Vietnamese Mathematical Society. Marasigan, Jose A. (1993). Introduction to problem solving. Ateneo de Manila UniVersity: Unpublished paper. -and Queena N. Lee. (1996). Concept paper on the International Math Olympiad. Quezon City: Ateneo de Manila University. Unpublished paper. Nebres, Bienvenido F. (1973). Mathematics and mathematicians in the Philippines. Philippine Studies 21, 409-23. --. (1979). Report of math activities in the Philippines and Southeast Asia. Matimyas Matematika 3(3), 27-30. --. (1997). Prerequisite for global competitiveness. Philippine Daily Inquirer, 18 April. Sioson, Federico M. (1967). Mathematical research in the Philippines. Philippine Studies 15(2), 241-258.

A mathematician named Erd6s, eccentric, perhaps even weird6s, though ever so clever, in all his life never c o ~ d pubfish a paper in Kurd6s. Peter B. Borwein Department of Mathematics & Statistics Simon Fraser University Burnaby, BC V5A 1S6 Canada e-mail: [email protected]

VOLUME 21, NUMBER 4, 1999

53

m,',r:-~.~ .,..-~w:9 [ . i n J e r e m y

Gray,

Editor

]

ven the briefest acquaintance with Leonard Eugene Dickson's career suggests that he was a man of immense energy and determination. In the 15 years from 1891 to 1906, he amassed a research record which included at least 74 talks, 67 published manuscripts, and 2 service contributions. 1 This made him the most active member of the American mathematical research community at the time. He went on to write 18 books and roughly 300 manuscripts, serve as editor of the

LeonardEugene E Dickson

(1874-1954): An American Legacy in Mathematics Della D. Fenster

Column Editor's address: Faculty of Mathematics, The Open University, Milton Keynes, MK7 6AA, England

Transactions of the American Mathematical Society, and Monthly, and guide 67 graduate students, 18 of whom were women, to doctorates in mathematics during his 40-year tenure on the faculty at the University of Chicago. 2 He was President of the American Mathematical Society from 1916 to 1918 and represented American mathematics twice at International Congresses. In this paper, I use previously undiscovered archival materials to provide new insight into Dickson's early life and its subsequent implications for his mathematical career. The members of the original mathematics department at the University of Chicago (E. H. Moore, Oskar Bolza, and Heinrich Maschke) exerted a profound influence on Dickson as he pursued his doctorate there from 1894 to 1896. 3 Collectively, the department's sustained commitment to research, their high standards for publication, and their vision of an American (as opposed to New England) mathematical community came to permeate Dick-

son's mathematical persona in these formative years. Individually, Moore, in particular, shaped Dickson's early mathematical pursuits. Moore initially inspired a young Dickson to write an algebraic thesis (Dickson [1897]), and a few years later he served as the impetus behind Dickson's foundational work in mathematics (Dickson [1903a], [1903b], and [1905]). Dickson furthered the strong algebraic tradition at Chicago as he impressed his own version of these characteristics on the second generation of Chicago-trained Ph.D.'s (Fenster [1997]). He untiringly (and successfully) pursued mathematical research in group theory, invariant theory, f'mite field theory, the theory of algebras, and number theory. Through his research, his teaching, and his scholarship--including his three-volume compendium of the history of the theory of numbers (Dickson [1919, 1920, 1923])--he showed his students (and colleagues, for that matter) what the life of a professional mathematician could and should be, and greatly helped to consolidate and strengthen American mathematics in the opening decades of the twentieth century (Fenster [1997], p. 21).

The Dicksons: Versatile, Determined, Energetic People of Excellence Undoubtedly Dickson acquired many of his characteristics from his parents. When just 22 his father, Campbell "Cam" Dickson, left the Dickson family homestead in New York to heed the

1Fenster and Parshall [1994a], p. 186. "Talks" denotes papers read either personally or in absentia at AMS meetings, AAAS meetings, the Chicago Congress, etc. "Papers" tallies articles published in the Bulletin of the American Mathematical Society (AMS), Transactions of the AMS, Annals of Mathematics, American Journal of Mathematics, or Papers Read at the Chicago Congress. "Service" refers to holding office in or serving on committees of the AMS or AAAS (1 = one year's service), serving on the editorial board of one of the above journals (1 = one year's service), serving as an officer or organizer of a teachers' group, etc. These years 1891-1906 were the first 15 years of publication of the Bulletin. 2See Albert [1955} and Archibald [1938] for the "standard" biographical references to Dickson. Other biographical information was found in Abernathy [1937a] & [1937b] and T. Dickson [no date]. 3Fenster [1997], pp. 11-13; and Parshall & Rowe [1994, pp. 379-381].

THE MATHEMATICALINTELLIGENCER9 1999 SPRINGER-VERLAGNEW YORK

unofficial national call to "go west" in 1858. He became a sheep farmer in Texas, found that range life suited him, and remained there for three years. Only his strong opposition to the break-up of the Union drove him north again. Once he reached New York in September of 1861, Cam volunteered for the Federal Army, and fought bravely in the Civil War until he sustained severe injuries while leading his company to fire the first shot at the battle of Gettysburg in 1863 (Abernathy [1937a], pp. 21, 26). Just as his wounds healed, the news of oil wells in Pennsylvania helped to rekindle his adventurous spirit, and he set off, hoping to earn the necessary funds for a sheep ranch in Missouri. The oil business filled Cain's pockets, but not his soul. He soon sold his interest in the oil field, and tempering his adventure with practicality, he set aside his dream of a Missouri ranch for a tract of level, fertile land offered by the Federal Government to Civil War veterans. He staked his claim near Independence, Iowa and stocked his ranch with fine sheep (Abernathy [1937a], pp. 28-29). In the fall of 1866, Cam came across the picture of his neighbor's sister, the college-educated Lucy Tracy, who taught school in Salina, New York ("a small village near Syracuse"). He implored his neighbor to invite Lucy to Iowa. Although Lucy accepted her sister's invitation and made plans to visit Iowa that summer, Cam grew impatient, took matters into his own hands, and traveled to New York in March of 1867 to meet Lucy himself. Lucy came to Iowa that summer, after all, and in the fall of 1867, Lucy and Cam married (Abernathy [1937a], pp. 29-30). In the winter of 1878, Cam lost his fortune to a poor speculation in wheat. He left Lucy with the four children they had now and immediately set out for Texas to inquire about land deeded to him 17 years earlier. While en route to the purportedly promised land, Cam visited his former neighbors, Lucy's relatives, who now made their home in Cleburne, Texas. There, by a strange

Leonard Dickson. (Frances Dickson Abernathy Papers and Texas Library Collection, CN 10200. The Center for American History, The University of Texas at Austin.)

set of events which involved Cain's decision to lend his money to Lucy's relatives who owned a hardware store on the verge of bankruptcy, Cam awakened one morning to find himself the new sole proprietor of a hardware store in Cleburne (Aberuathy [1937a], p. 34). He seized the opportunity and developed the company into a hardware and furniture store which served the Cleburne community for over half a century. On the back of his business success, Campbell helped bring the first railroad and roads, water, sewage treatment, electricity, and the street car line to Cleburne. He served as a director of the First National Bank and used his position on the city council to secure a building for the first public school in Cleburue (Abernathy [1937a], p. 41). Lucy and the children (the sons Tracy, Frederick, and Leonard, and the daughter Evelyn) joined Cam in Cleburne in the fall of 1878. Thus began the Dicksons' long association with the town of Cleburne, and, most relevant to our story, Leonard Dickson's impressionable boyhood years. It is clear that Cam gave the Dick-

son children an impressive example of how to pursue successfully several interests at once--at home, in the town, and in the country at large. By taking the initiative to tackle whatever project lay before him, he showed his cl~ldren what a determined, forward-looking man of action could accomplish. Lucy exhibited a complementary set of qualities. She promoted a sense of perfection in the family by demanding it of herself in daily routines and spiritual concerns (Abernathy [1937b], pp. 31-32). Like Cam, Lucy originally hailed from New York, where she had lived on a farm with her parents and four sisters. A good student, Lucy enrolled in the State of New York Normal School in Albany, New York in the fall of 1864, just before she turned 17. She graduated in July of 1865 and began teaching that fall (Abernathy [1937b], p. 6). But her health began to decline in the late 1870's. In the early 1880's, Lucy gave birth to her second daughter, Frances, the only "real" Texan in the family. After Frances's birth, Lucy's health took a marked turn for the worse and she began to live more and more in her spiritual world. In the spring of 1896, just before Leonard Dickson received his doctorate from the University of Chicago summa cum laude, Lucy Tracy Dickson died. After an unsuccessful attempt to make money when he was 70 by mining in Mexico, Cam died in June of 1911. L e o n a r d Dickson's C a r e e r Leonard Dickson's strong family ties to Texas motivated him to pursue his undergraduate and master's degrees at the University of Texas. There he came under the influence of G. B. Halsted, who cultivated and furthered his mathematical prowess, and he graduated at the top of his class. 4 With his "rugged individualism and dynamic energy," (Birkhoff [1977], p. 34), inspired, in part at least, by his pioneering parents, Dickson joined the first generation of aspiring American mathematicians who chose to pursue their doctorates at institutions in their home country. 5

4Fenster [1997], pp. 11-13; and Parshall & Rowe [1994, pp. 379-381]. 5parshall & Rowe [1994] have written about the first generation of doctoral students at the University of Chicago on pp. 372-401.

VOLUME21, NUMBER4, 1999 55

Lucy Dickson (I); Campbell Dickson (r). (Frances Dickson Abernathy Papers and Texas Library Collection, CN 10202 and CN 10201. The Center for American History, The University of Texas at Austin.)

Thus w h e n Dickson a r r i v e d at the two-year-old University o f Chicago in 1894, he brought along the intangibles o f a strong family heritage. He saw in Moore, Bolza, and M a s c h k e precisely w h a t he had s e e n in his o w n father. That is, he saw their c o m m i t m e n t to establishing a strong d e p a r t m e n t at Chicago, to training future researchers, a n d to cultivating a truly A m e r i c a n (as o p p o s e d to New England) mathematical community. Cam D i c k s o n ' s comm i t m e n t to local a n d n a t i o n a l c o n c e r n s w a s a w a y of life for L e o n a r d Dickson in Texas. The d e d i c a t i o n of Moore, Bolza, and M a s c h k e to the cause of m a t h e m a t i c s at Chicago a n d the country at large easily b e c a m e a w a y of life for the young Dickson. Following a y e a r of s t u d y in E u r o p e with, among others, the mathematicians Camille Jordan, Sophus Lie, ]~mile Picard, and Charles Hermite (No A u t h o r [1899]), and t w o y e a r s as an ins t r u c t o r at the University o f California, D i c k s o n a c c e p t e d a t h r e e - y e a r associ-

ate p r o f e s s o r s h i p at the University o f T e x a s (Lewis [1989], p. 214). But after only a y e a r at Texas, Dickson a c c e p t e d an a s s i s t a n t p r o f e s s o r s h i p at the University of Chicago in the fall of 1900. This move, which d i s p l e a s e d the B o a r d o f Regents at Texas, recalls the w a y L e o n a r d Dickson's p a r e n t s h a d m a d e the c i r c u m s t a n c e s w o r k for them, s e i z e d the available o p p o r t u n i ties, a n d b l a z e d i n d e p e n d e n t p a t h s for themselves. Moreover, Chicago prov i d e d D i c k s o n with a c c e s s to a b e t t e r library, a b e t t e r d e p a r t m e n t overall, and, m o s t importantly, a c h a n c e to b e close to E. H. Moore, his m o s t crucial s o u r c e of i d e a s for research. A p o r t i o n of Dickson's w o r k in the t h e o r y of algebras serves as one exa m p l e o f h o w Dickson b e g a n to w e a v e his family attributes into his professional pursuits. Although D i c k s o n ' s c e l e b r a t e d w o r k in this a r e a a p p e a r e d in the early 1920s (see F e n s t e r [1998]), his r e s e a r c h actually b e g a n in the first few y e a r s of the twentieth c e n t u r y

when Dickson followed E. H. M o o r e ' s lead and s t u d i e d the foundations of mathematics. A m o n g o t h e r concepts, he p u t forth his first definition of a linear associative a l g e b r a in 1903 (Dickson [1903b]). In this paper, Dickson reflected the influence of Moore and the other American Postulate Theorists [Scanlan], when he not only p r o p o s e d a defmition of an a l g e b r a b y i n d e p e n d e n t postulates but also gave a m o r e a b s t r a c t repr e s e n t a t i o n of the concept. At this s a m e time, he e x t e n d e d the definition of an a l g e b r a f r o m one over the reals or the c o m p l e x e s to one over an arbitrary field F. This r e p r e s e n t e d a significant a d v a n c e at the time, occurring j u s t as the n o t i o n o f an arbitrary field was itself being p l a c e d on firm foundations. 6 Over the c o u r s e o f the next 20 years, Dickson w o u l d p r o p o s e three o t h e r def'mitions of an algebra, m o t i v a t e d b y J o s e p h H. M. W e d d e r b u r n ' s 1904-1905 visit to Chicago a n d Henry Taber's pa-

6At this time, the concept of an arbitrary field probably referred to those of characteristic zero, or, at least, those with (in modern terminology) only finite separable extensions. The ideas of perfect and imperfect fields, for example, did not arrive until the work of Ernst Steinitz in 1910. See Parshall [1985] and Gray [1997]. For the work by Steinitz, see Steinitz [1910] and [1950].

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pers in those same years, the appearance of Dickson's Linear Algebras in 1914, and his own work in the arithmetic of algebras in the early 1920's (Dickson [1903b], [1905], [1914], [1923]; Fenster [a]). As he re-evaluated this concept in light of his mathematical researches (especially in division algebras and, later, in the arithmetic of algebras), Dickson emphasized the linear independence of the basis elements with respect to the field of coefficients rather than the complex numbers. This allowed for a more farreaching definition, one which included the desirable example of the complex numbers. Ultimately, he excluded the properties of division and associativity and, in 1923, put forth a defmition of an algebra as we know it today (Fenster [a]). Dickson's contributions to the development of the concept of an algebra lay principally in their breadth and depth. In a broad sense, he lent a certain consistency to the foundations of the theory, pursuing research on this aspect of the field for over two decades. Even with more celebrated results in division algebras and in the arithmetic of algebras, he seemed to remain committed to finding the "best" definition of an algebra. By reconsidering and adjusting the fundamental concept, Dickson strove to produce the theory most conducive to further growth. He sustained this search for the "best" definition along with his concurrent research in group theory, invariant theory, division algebras, and history of n u m b e r theory, an untiring and many-sided activity that recalls Cam Dickson's boundless energy for pursuing various interests with vigor. Perhaps no area of Dickson's professional life reflects (and required?) the dogged "Dickson determination" more than his extensive historical study of n u m b e r theory. At the time Dickson undertook this w o r k in 1911, he had already established himself as a distinguished American research mathematician. As early as 1903, for example, E. H. Moore had described Dickson as "one of the very strongest research m e n in mathematics in the whole c o u n t r y . . . " (Moore [1903]). Moreover, in the balloting for the sec-

ond edition of American Men of Science (1910), Dickson was almost exclusively ranked as one of the top seven American mathematicians. In February of that year, with his mathematical research interests focused primarily in invariant theory, Dickson initially proposed the idea of an "extensive report" of the theory of numbers to R. S. Woodward, president of the Carnegie Institution (Dickson [1911a]). In October of the same year, Dickson again wrote Woodward, this time in support of D. N. Lehmer's "Table of Primes up to 10 Million." He supplemented this equivalent of a letter of recommendation for Lehmer with an almost two-page report regarding the project of which I wrote you at the beginning of this year--viz, a History of the Theory of Numbers, to give in f o r m convenient f o r reference every result that has been discovered in the theory of numbers, beginning with the Greeks. You closed your reply of Feb. 13th by "urging that the work should be as thoro[ugh], searching and exhaustive as time and effort can make it." I beg to report much progress on this very ambitious project. Some months ago I f i n i s h e d the preliminary task of collecting the references--by a direct, first-hand examination of the literature, independent of extant encyclopedias and reports, using the same only as a check . . . . I a m devoting 40 to 45 hours weekly to this work giving it m y entire attention apart f r o m 8 h[ou]rs of class work and irregular work as Editor of the Transactions. Those who know of the project think highly of i t s usefulness--but apparently do not envy the labor. At first I could not hold in check m y research tendencies and have published f o u r papers completing certain gaps in the theory--but now I a m sticking f i r m l y to the definite task . . . . (Dickson, [1911b]). Thus while Dickson pursued this new area of research, he simultaneously attempted to secure a publisher for this work. In particular, from this letter,

along with m a n y others of the same genre, we learn that Dickson viewed the labor-intensive c o m p o n e n t of his study as a key point in his argument to convince the Carnegie Institution of the merits of this work. This 16tter shows plainly his o w n high rating of its importance. In addition to the number-theoretic forays inspired by Dickson's historical work (see, for example, Dickson [1913a] and [1913b]), he continued to work in invariant theory and the theory of algebras, and managed to publish seven books (including his celebrated Algebras and their Arithmetics) and a colloquium series, along with more than 50 manuscripts, from the time he began this historical project in 1911 until the third and final volume appeared in 1923. Dickson then worked until at least 1928 to secure publication of a fourth volume of this historical project, but it never appeared. (See Fenster [b].) The Dickson family characteristics influenced his professional life in other ways beyond his actual mathematical researches. While at Chicago, Dickson's prolific researches and subsequent publications and training of the next generation of students furthered the strong algebraic tradition begun by E. H. Moore. In particular, Dickson's eighteen w o m e n students (roughly 27% of his total n u m b e r of students) were no small part of the twentieth-century heritage of American w o m e n in mathematics. Dickson advised 40% of the w o m e n doctorates in mathematics at the University of Chicago and 8% of all the w o m e n doctorates in the United States from 1900 to 1940 (Green & LaDuke [1987]). Dickson's w o m e n doctoral students confirm the University of Chicago's favorable position among institutions for training future w o m e n mathematicians (Fenster [1997], pp. 14-17). But why Dickson? Why did he serve as a "cluster-point" [Green & LaDuke [1987], p. 20) for w o m e n in mathematics? Indeed the climate at the University of Chicago proved conducive for w o m e n seeking higher degrees in mathematics. Perhaps, as Alice Schafer has suggested, a "snowball" effect may have influenced Dickson's large num-

VOLUME 21, NUMBER 4, 1999

57

ber of women students. That is, once a few women completed their degrees under his guidance, favorable word spread about his ability as an adviser (Fenster [1997], p. 16). Moreover, Dickson's mother Lucy Tracy Dickson, had herself held a college degree and, for a short time, a teaching position. It is likely that instilled a love of teaching in her children. 7 Since Leonard grew up with a college-educated mother who fondly recalled her one-time profession, he apparently found it quite natural to advise women graduate students in mathematics and train them, in general, for teaching positions at the collegiate level s Moreover, Dickson's family legacy offers insight into two frequently asked questions about Dickson's life: (1) Why did Dickson burn his papers on retirement? (2) Why, except for returning to Chicago to teach a few summer school classes, did Dickson leave mathematics and never look back? As for the former question, his sister Frances Dickson Abernathy's biographical accounts of Cam and Lucy Dickson appeared in 1937, just two years before Leonard Dickson retired. Since Dickson maintained close ties with his siblings and provided biographical information for the history, it seems likely that Dickson knew about his sister's project. In particular, she devoted far more than half of The Life of Lucy Tracy Dickson to Lucy's letters and diary accounts and about a fourth of The Life of Campbell Dickson to his letters. Many of these letters contain personal thoughts and musings. And therein lies the secret. Although Dickson voraciously crusaded for American mathematics at the University of Chicago and in the mathematical world in general, he maintained an intensely private personal life. This fact, coupled with what he most certainly read in the biographies of his parents, suggests, perhaps, that he chose to obliterate his more

personal records rather than risk possible public exposure (and misunderstanding?) of them. As for the second question, Dickson felt he had seen too many world-class mathematicians do mediocre work long after they reached retirement. 9 He wanted to leave mathematics with a solid, unblemished record. He had seen mathematicians wane in their later years and he had seen his 70-yearold father invest his time and financial resources in a mining opportunity in Mexico which ultimately cost him dearly. In the spring of 1939, a 65-yearold Dickson found himself with a long series of papers by him and his students which provided an almost complete verification of the so-called ideal Waring theorem, and a discerning spirit which told him from professional and personal experience to quit while he was ahead. He did. He returned to Texas after his retirement from Chicago where he lived for 15 more years. He died just five days shy of his eightieth birthday in 1954.

Epilogue Indeed, Dickson left mathematics as he intended. He built his distinguished record in a little more than 40 years and he left it intact when he departed. He thus integrated his rich family legacy with his mathematical one. He brought a heritage firmly grounded in American history, encompassing the passion which swept the country to travel west in the mid-1800's, the difficulties of the Civil War, the lure and the challenges of frontier life, the issues surrounding women's education and women's roles in society, and the "modernization" of America. Dickson gave more to the mathematical community than mathematical prowess, leadership, prolific publications, and unwavering devotion to high standards: he brought the passion and adventure of the pioneering spirit. One

might even say he brought a truly American spirit to American mathe-

matics. BIBLIOGRAPHY

Abernathy, F.D., 1937a: The Life of Cambell

Abernathy (Frances Dickson) Papers, The University of Texas Archives, The Center for American History. Abernathy, F.D., 1937b: The Life of Lucy Tracy Dickson, Abemathy (Frances Dickson) Papers, The University of Texas Archives, The Center for American History. Albert, A.A., 1955: "Leonard Eugene Dickson, 1874-1954," Bulletin of the American Mathematical Society 61,331-345. Archibald, R.C., 1938: A Semicentennial HisDickson,

7Both of the Dickson daughters attended college; in fact, Frances, the author of the biographies of Cam and Lucy, completed a Master's degree in 1935 with a thesis on the History of the Builders of Johnson County (TX) (Abernathy [1937a], p. 65). 8This would mean teaching at the women's colleges, primarily. See (Fenster [1997], pp. 14-16) for more on Dickson's women students and their employment opportunities after graduation. For a broader perspective on women in the early American mathematical research community (1891-1906), see (Fenster & Parshall [1994b]). The current study, then, begins to address two questions posed in Fenster [1997], p. 16; namely (1) Were t h e r e . . , reasons for which Dickson became a "clusterpoint" for women graduate students at Chicago? and (2) Did he have personal motivations for wanting to direct women? The influence of Lucy Dickson on Leonard points to a decisive "yes" to these questions. 91 thank Saunders Mac Lane for calling this to my attention in (Mac Lane, [1992]).

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tory of the American Mathematical Society, 1888-1938, American Mathematical Society, New York.

Birkhoff, G.D., 1977: Some Leaders in American Mathematics: 1891-1941, pp. 25-78 in The Bicentennial Tribute to American Mathematics, 1776-1976, Dalton Tarwater, editor. The Mathematical Association of America. Cattell, J.M., (no date): Library of Congress, James McKeen Cattell Papers, Box 61. Cattell, J.M., ed., 1906, 1910: American Men of Science, 1st and 2d eds., Science Press, New York. Dickson, L.E., 1897: "The Analytic Representations of Substitutions on a Power of a Prime Number of Letters with a Discussion of the Linear Group," Annals of Mathematics 11, 65-143. Dickson, L.E., 1903a: "Definitions of a Field by Independent Postulates," Transactions of the American Mathematical Society 4, 13-20. Dickson, L.E., 1903b: "Definitions of a Linear Associative Algebra by Independent Postulates," Transactions of the American Mathematical Society 4, 21-26. Dickson, L.E., 1905: "Definitions of a Group and Field by Independent Postulates," Transactions of the American Mathematical Society 6, 198-204. Dickson, L.E. to R.S. Woodward, 11 February, 1911a, Carnegie Institution Archives, Washington, D.C., Dickson Papers. Dickson, L.E. to R.S. Woodward, 23 October, 1911 b, Carnegie Institution Archives, Washington, D.C., Dickson Papers. Dickson, L.E., 1913a: "Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors," American Journal of Mathematics 35, 413-422. Dickson, L.E., 1913b: "Even Abundant Numbers," American Journal of Mathematics 35, 423-426.

Dickson, L.E., 1914: Linear Algebras, Gray, J.J., 1997 K6nig, Hadamard and Cambridge University Press, Cambridge; KQrshak, and abstract algebra, Mathematical reprint, 1930, Cambridge University Press, Intelligencer 19.2, 61-64. Cambridge. Green, J. and J. LaDuke, 1987: Women in the Dickson, L.E., 1923: Algebras and their American Mathematical Community: The Arithmetics, University of Chicago Press, Pre-1940 Ph.D.'s, Mathematical Intelligencer Chicago. 9, 11-23. Dickson, L.E., 1919, 1920, 1923: History of the Lewis, A., 1989: The Building of the University Theory of Numbers, 3 vols., Carnegie of Texas Mathematics Faculty, pp. 205-239 Institution, Washington, D.C. in vol. Ill of A Century of Mathematics in Dickson, T., (no date): Campbell Dickson, pp. America, Peter Duren editor. American 206-212 in vol. I (B) of The Scrap Book Mathematical Society, Providence. History Series of Cleburne. The University of Mac Lane, S., 1992: Interview by author, 5-6 Texas Archives, The Center for American March, Charlottesville, Virginia, Tape ReHistory. cording. Fenster, D.D., [a]: The Development of the Moore, E. H. to Carnegie Institution, 16 Concept of an Algebra: Dickson's Role, September, 1903, Carnegie Institution Arforthcoming. chives, Washington, D.C., Dickson Papers. Fenster, D.D., [b]: Why Dickson left Quadratic No Author Cited, 1899: The University of Texas Reciprocity out of his History of the Theory Record, vol. 1, Number 3, August, University of Numbers,American Mathematical Monthly, of Texas Memorabilia Collection, The Center to appear. for American History, Austin. Fenster, D.D., 1997: Role Modeling in No Author Cited, 1914: "Who's Who at Mathematics: The Case of Leonard Eugene Texas?" The Alcalde, January, 266-268, Dickson (1874-1954), Historia Ma thematica University of Texas Memorabilia Collection, 24, 7-24. The Center for American History, Austin. Fenster, D.D., 1998: Leonard Eugene Dickson Parshall, K.H., 1985: "Joseph H. M. and his work in the Arithmetics of Algebras, Wedderburn and the Structure Theory of Archive for History of Exact Sciences 52, Algebras," Archive for History of Exact 119-159. Sciences 32, 223-349. Fenster, D.D. and K.H. Parshall, 1994a: "A Parshall, K.H. and D. Rowe, 1994: The Profile of the American Mathematical Emergence of an American Mathematical Research Community: 1891-1906," pp. Research Community: J. J. Sylvester, Felix 179-227 in vol. III of The History of Modern Klein, and E. H. Moore, American and Mathematics, Eberhard Knobloch and David London Mathematical Societies, Providence E. Rowe, editors. Academic Press, San and London. Diego. Scanlan, M., 1991: "Who were the American Fenster, D.D. and K.H. Parshall, 1994b: Postulate Theorists?" Journal of Symbolic "Women in the American Mathematical Logic 56, 981-1002. Research Community: 1891-1906," pp. Steinitz, E., 1910: Algebraische Theorie der 229-261 in vol. Ill of The History of Modern Korper, Journal f(Jr Mathematik 116, 1-132. Mathematics, Eberhard Knobloch and David Steinitz, E., 1950: Algebraische Theorie der E. Rowe, editors. Academic Press, San Korper, ed. Reinhold Baer & Helmut Hasse, Diego. Chelsea Publishing Co., New York.

VOLUME 21, NUMBER 4, 1999

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which Pielou has offered as a discrete analogue of the w e l l - k n o w n delay logistic differential equation, and

Global Behavior of Nonlinear Difference Equations of Higher Order with Applications

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by G. L a d a s and V. L. Kocic MATHEMATICS & ITS APPLICATIONS SERIES, VOL. 256 DORDRECHT: : KLUWER ACADEMIC PUBLISHERS, 1993. xi + 228 PP. US $112.00, ISBN 0 7923-2286-X

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1

which arises in n u m b e r theory. This c h a p t e r also i n t r o d u c e s the basic necessities for the discussion of the oscillation, stability, periodicity, and attractivity of solutions. Chapter 2 d i s c u s s e s the global asymptotic stability a n d global attractivity of the equilibrium of Xn+ 1 =

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he last few years have w i t n e s s e d a strong r e s u r g e n c e in interest in difference equations as a study in their own right, including the initiation in 1994 o f a n e w journal, Journal o f Difference Equations and Applications. In fact, one of the authors, G e r r y Ladas, o f the m o n o g r a p h u n d e r review is a co-editor o f this journal. There have b e e n a n u m b e r of b o o k s published in r e c e n t y e a r s dealing with this topic; s o m e are e l e m e n t a r y in n a t u r e (Kelley a n d P e t e r s o n [3], Lakshmikant h a m a n d Trigiante [4], Mickens [5]), giving a f o u n d a t i o n to the general subj e c t a r e a along with an i n t r o d u c t i o n to m o r e a d v a n c e d topics, w h e r e a s o t h e r s (Agarwal [1], Chapter 7 of GyOri a n d Ladas [2], a n d the m o n o g r a p h u n d e r review) c o n t a i n surveys of results from the literature that are of special interest to the authors. The focus of this w o r k is the s t u d y of n o n l i n e a r difference equations of the general form

T

Xn+ 1 =

F(x~,

Xn-1,

. . . , Xn-k),

(E)

n = O, 1 , . . . , w h e r e k is a fixed positive integer and F is a c o n t i n u o u s non-negative function. The first of the six c h a p t e r s presents s o m e motivating examples, s u c h Column Editor's address: Department of Mathematics, Drexel University, Philadelphia, PA 19104 USA.

60

as

Xn+l -

THE MATHEMATICAL INTELLIGENCER 9 1999 SPRINGER VERLAG NEW YORK

bXn

A + Xn-k '

b > A,

x,f(x~,

Xn-kl

, Xn-k2,

. . . , X.n-k,.)

and s o m e of its s p e c i a l cases. It is in this c h a p t e r that the i m p o r t a n t notion of p e r m a n e n c e is i n t r o d u c e d and studied. Although it m a y n o t b e obvious to the casual r e a d e r (it is m a d e clear in the notes in the last section of the chapter), m u c h o f w h a t is here is a survey of previously k n o w n results. In Chapter 3, the a u t h o r s c o n s i d e r rational equations; that is, t h o s e of the form

a + ~,k=o aix~, i Xn+l

w h e r e the coefficients are non-negative real n u m b e r s a n d k is a positive integer. Once again, m u c h of w h a t app e a r s here can a l r e a d y be found in the literature. Chapter 4, entitled "Applications," is in s o m e sense the m o s t interesting. Various m o d e l s from biological phen o m e n a are described, and the exist e n c e and stability of equilibrium solutions are discussed. The m o d e l s have mostly a p p e a r e d elsewhere, but t h e y m a k e an interesting collection. Chapter 5 d e a l s with p e r i o d i c cycles of various c a s e s o f the equation 1 + xn + x n - 1 ~- "'" ~- X n - k + 2 Xn+ 1

:

Xn -k+l

w h e r e k is a non-negative integer. The final chapter, "Open P r o b l e m s and Conjectures," c o n t a i n s a n u m b e r of

problems, partial results, a n d directions for further research. The b o o k also contains a p p e n d i c e s on the Riccati equation, a c o n t r a c t i o n principle, a n d global b e h a v i o r of systems. The first five c h a p t e r s in the b o o k are i n t e r s p a c e d with a variety o f "Res e a r c h Projects." Some of t h e s e are obvious suggestions for p o s s i b l e generalizations of k n o w n results, w h e r e a s o t h e r s a r e m o r e subtle and b r o a d e r in scope. Although the b o o k is not suitable as a first i n t r o d u c t i o n to difference equa-

tions, the m a t e r i a l is a c c e s s i b l e to anyone who has a firm b a c k g r o u n d in analysis and a p a s s i n g k n o w l e d g e o f s o m e of the f u n d a m e n t a l ideas f o u n d in the i n t r o d u c t o r y t e x t s m e n t i o n e d above. REFERENCES

1. R. P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, New York, 1992. 2. I. Gy6ri and G. Ladas, Oscillation Theory of

Delay Differential Equations with Applications, Oxford University Press, Oxford, 1991. 3. W. G. Kelley and A. C. Peterson, Difference

Equations, Academic Press, New York, 1991. 4. V. Lakshmikantham and D. Trigiante, Theory

of Difference Equations: Numerical Methods and Applications, Mathematics in Science and Engineering Vol. 1811 Academic Press, New York, 1988. 5. R. E. Mickens, Difference Equations, Van Nostrand Reinhold, New York, 1990. Department of Mathematics and Statistics Mississippi State University Mississippi State, MS 39762 USA

VOLUME 21, NUMBER 4, 1999

(}1

II-"le..,,~.m.~

. ] ., , t . a

Robin

Calendars

Greenwich

Gregorian calendar

Julius Caesar

Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics, The Open University, Milton Keynes, MK7 6AA, England

62

Wilson,

Editor

J

efore the time of the Romans there were many different calendars in use. As early as 4000 BC the ancient Egyptians used a 365-day solar-based calendar of twelve 30-day months and five extra days added by the god Thoth. The Greek, Chinese, and Jewish lunarbased years consisted of 354 days with extra days added at intervals, while the early Roman year had just 304 days, although this was extended to 355 days in 700 BC with the addition of the new months Januarius and Februarius. In 45 BCJulius Caesar introduced the "Julian calendar." This had 3651/4 days, with the fraction taken care of by the addition of an extra "leap day" every four years; in order to synchronize with the solar year, the year 46 BC (the "year of confusion") needed an unprecedented 445 days. The beginning of the year was moved to January and the lengths of the months alternated between 30 and 31 days (apart from a 29day February in non-leap years). Regrettably this was later altered by Augustus Caesar who stole a day from February to add to August and changed September to December accordingly. Later writers determined the length of the solar year with increasing accuracy. In particular, the Islamic scholars Omar Khayyam and Ulugh Beg independently measured the solar year as about 365 days, 5 hours and 49 minu t e s - j u s t a few seconds adrift. The Julian year was 11 minutes too long, and by 1582 the calendar had drifted by 10 days with respect to the seasons. In that year Pope Gregory XIII removed the extra days and solved the problem of the over-length year by omitting three leap days every 400 years--thus, 1700, 1880 and 1900 were not leap years, but 2000 is. Other countries eventually followed suit--Germany in 1700, Britain in 1752, Russia in 1917, and eventually China in 1949.

B

THE MATHEMATICAL INTELLIGENCER 9 1999 SPRINGER-VERLAG NEW YORK

Meanwhile, the line from which time is measured had been set at Greenwich in 1884. Then, in 1972, atomic time replaced Earth time as the official standard, and the year is now measured at about 2.9 • 1017 oscillations of atomic cesium.

Omar Khayyam

Pope Gregory Xlll

Ulugh Beg