Wolf Prize in Mathematics Volume 2
Wolf Prize in Mathematics Volume 2
Edited by
S S Chern Nankai Institute of Mathem...
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Wolf Prize in Mathematics Volume 2
Wolf Prize in Mathematics Volume 2
Edited by
S S Chern Nankai Institute of Mathematics, Nankai University, China
F Hirzebruch Universitat Bonn & Max-Planck-lnstitut fur Mathematik, Germany
V f e World Scientific «k
NewJersey Jersey •• London London •• Singapore Sn • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data Wolf prize in mathematics / edited by S.S. Chern, F. Hirzebruch. p. cm. Includes bibliographical references. ISBN 9810239467 (v. 2) 1. Mathematicians-Biography. 2. Mathematics. 3, Wolf Foundation Prizes. I. Chern, Shiing-Shen, 1911- . I I . Hirzebruch, Friedrich, 1927QA28.W65 2000 510.92'2-dc21 [B]
00-031994
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
The editors and publisher would like to thank the following organisations and publishers of the various journals and books for their assistance and permission to reproduce the selected reprints found in this volume: Academic Press American Institute of Physics American Mathematical Society B. G. Teubner Stuttgart Birkhauser Publishers Duke University Press Gautheir-Villars Hermann Publishing IEEE International Mathematical Union The Johns Hopkins University Press
MIT Press National Academy of Sciences (USA) Plenum Pubishing Corporation Portuguese Mathematical Society Publish or Perish, Inc. The Royal Society of London The Royal Swedish Academy of Sciences Springer-Verlag Vandenhoeck & Ruprecht The Wolf Foundation
While every effort has been made to contact the publishers of reprinted papers prior to publication, we have not been successful in some cases. Where we could not contact the publishers, we have acknowledged the source of the material. Proper credit will be accorded to these publishers in future editions of this work after permission is granted.
Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
Printed in Singapore by Mainland Press
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VII
Preface
There is no Nobel prize in mathematics. Perhaps this is a good thing. Nobel prizes create so much public attention that mathematicians would lose their concentration to work. There are several other prizes for mathematicians. There is the Fields medal (only for mathematicians). This medal is awarded to mathematicians who are at most 40 in the year of the International Congress of Mathematicians where the medals are presented. Thus it honours outstanding work and encourages further efforts. The Fields medal is perhaps best known and is often called the Nobel prize in mathematics. World Scientific has published a book of the Fields medallists' lectures. Then there is the Wolf prize. The Wolf foundation describes the prize as follows: "The WOLF FOUNDATION began its activities in 1976, with an initial endowment fund of 10 million U.S. dollars donated in its entirety by the Wolf family. The main founders were Dr. Riccardo Subirana Lobo Wolf and his wife Francisca . . . . Since 1978 five or six annual prizes are awarded to outstanding scientists and artists, irrespective of nationality, race, colour, religion, sex or political view, for achievements in the interest of mankind and friendly relations among people. In Science, the fields are: AGRICULTURE; CHEMISTRY; MATHEMATICS; MEDICINE; PHYSICS, and in ARTS, the prize rotates annually among Music, Painting, Sculpture and Architecture . . . . The official presentation of the prizes takes place at the Knesset building (Israel's parliament) and the winners are handed their awards by the President of the State of Israel at a special ceremony . . . ." The Fields medal goes to young people, and indeed many mathematicians do their best work in the early years of their life. The Wolf prize often honours the achievements of a whole life. But it may also honour the work of young people. The first Wolf prize winners in mathematics were Izrail M. Gelfand and Carl L. Siegel (1978). Siegel was born in 1896 and Gelfand in 1913. Gelfand is still active at Rutgers University. Several prize winners were born before 1910. Thus the achievements of the prize winners cover much of the twentieth century. The documents collected in these two volumes characterize the Wolf prize winners in a form not available up to now: bibliographies and curricula vitae, autobiographical accounts, early papers or especially important papers, lectures and speeches, for example at International Congresses, as well as reports on the work of the prize winners by others. Since the work of the Wolf laureates covers a wide spectrum, a large part of contemporary mathematics comes to life in these books. Mathematical prize winners are usually quite modest. They know that the selection committee had to choose from a large list of excellent candidates and that not only merit is needed to receive a prize, but also much luck. Quite different sets of mathematicians could illustrate with their work just as well the development of mathematics in the period covered by the Wolf prizes.
viii The volumes are also a symbol of thanks to the donors who made the Wolf foundation possible and to all who worked, and work, for the success of the foundation. The editors also thank all mathematicians who prepared the material for deceased Wolf prize winners. Without their generous help the two volumes would be very incomplete. The editors want to express their thanks to World Scientific for a splendid cooperation. They especially profited from the work and valuable advice of Dr. Sen Hu of World Scientific office in the USA and thank him most cordially. S. S. Ckern F. Hirzebruch
Contents
List of Wolf Prize Winners Preface Raoul Bott Autobiographical Sketch Impedance Synthesis without Use of Transformers (with R. J. Duffin) Two New Combinatorial Invariants for Polyhedra On Torsion in Lie Groups The Cohomology Ring of G/T (with H. Samelson) The Stable Homotopy of the Classical Groups On the Parallelizability of the Spheres (with J. Milnor) On a Theorem of Lefschetz A Lefschetz Fixed Point Formula for Elliptic Differential Operators (with M. F. Atiyah) On a Topological Obstruction to Integrability On Characteristic Classes of T-Foliations (with A. Haefliger) An Equivariant Setting of the Morse Theory
v vii
1 8 10 16 19 23 26 29 35 41 46 52
Alberto P. Calderon (prepared by E. M. Stein) Curriculum Vitae List of Publications Singular Integrals: The Roles of Calderon and Zygmund (by E. M. Stein) Uniqueness in the Cauchy Problem for Partial Differential Equations Commutators of Singular Integral Operators Cauchy Integrals on Lipschitz Curves and Related Operators (with A. Zygmund) On the Existence of Certain Singular Integrals (Introduction only)
114
Andrei N . Kolmogorov (prepared by V. M. Tikhomirov) A. N. Kolmogorov (by V. M. Tikhomirov) On Normability of a General Topological Linear Space List of Works by A. N. Kolmogorov Obituary (Mr. Andrei Kolmogorov - Giant of Mathematics)
119 141 144 162
Mark Grigor'evich Krein (prepared by J. Gohberg) Mark Grigorievich Krein: Recollections (by I Gohberg) Mark Grigor'evich Krein (Recollections) (by B. Ja. Levin)
165 177
61 63 70 81 102 110
Abstract Solution on the Inverse Sturm-Liouville Problem On a Fundamental Approximation Problem in the Theory of Extrapolation and Filtration of Stationary Random Processes On the Determination of the Potential of a Particle from Its S-Function Continual Analogues of Propositions on Polynomials Orthogonal on the Unit Circle Analytic Problems and Results in the Theory of Linear Operators in Hilbert Space Notes by I. Gohberg Peter D . Lax Curriculum Vitae Short Mathematical Essays List of Publications A Sample of Lax's Contributions to Classical Analysis, Linear Partial Differential Equations and Scattering Theory (by P. Sarnak) Hans Lewy (prepared by S. Hildebrandt) Curriculum Vitae Mathematicae (by D. Kinderlehrer) The Music in Hans Lewy's Life (by Helen Lewy) Crises in Mathematics Further Reading Bibliography (by D. Kinderlehrer) Publications Dedicated to Hans Lewy Neuer Beweis des analytischen Charakters der Losungen elliptischer Differentialgleichungen On the Existence of a Closed Convex Surface Realizing a Given Riemannian Metric On the Boundary Behavior of Minimal Surfaces An Example of a Smooth Linear Partial Differential Equation without Solution
183 195 199 204 208 212 217
219 221 240 252
264 269 272 276 277 284 285 296 299 307
Laszlo Lovasz Curriculum Vitae List of Publications Kneser's Conjecture, Chromatic Number, and Homotopy On the Shannon Capacity of a Graph Algorithmic Mathematics: An Old Aspect with a New Emphasis One Mathematics: There is No Natural Way to Divide Mathematics
311 313 327 333 340 353
John W . Milnor Curriculum Vitae Publications of John Milnor (Through 1992) On Manifolds Homeomorphic to the 7-Sphere
359 360 367
XI
Some Consequences of a Theorem of Bott Algebraic K-Theory and Quadratic Forms Groups which Act on Sn without Fixed Points Eigenvalues of the Laplace Operator on Certain Manifolds
374 380 407 415
Jiirgen K. Moser (prepared by E. Zehnder) Curriculum Vitae Bibliography Cantor-Medaille fur Jiirgen Moser (by E. Zehnder) Dynamical Systems - Past and Present
417 419 428 438
Ilya Piatetski-Shapiro Curriculum Vitae List of Publications Etude on Life and Automorphic Forms in the Soviet Union Regions of the Type of the Upper Half-Plane in the Theory of Functions of Several Complex Variables Automorphic Functions and Arithmetic Groups Tate Theory for Reductive Groups and Distinguished Representations Two Conjectures on L-Functions
461 462 474 487 496 513 519
Jean-Pierre Serre Short Note Curriculum Vitae Cours au College de France (1955-1994) Publications Lettre a John McCleary, 11 Mars 1997 Lettre a David Goss, 12 Decembre 1991 Lettre a Pierre Deligne, 24 Juillet 1967 Lettre a David Goss, 30 Mars 2000 Extrait du discours de reception du prix Balzan, Berne, 1985 Entretien avec Jean-Pierre Serre Lettre a Jacques Tits, 13 Mars 1993
523 524 525 526 527 530 532 537 540 542 550
Carl L. Siegel (prepared by H. Klingen) Curriculum Vitae (by H. Klingen) Honours and Awards (by H. Klingen) Nachruf auf Carl Ludwig Siegel (by Th. Schneider) Bibliography Approximation algebraischer Zahlen Analytische Theorie der quadratischen Formen Meroniorphe Funktionen auf kompakten analytischen Mannigfaltigkeiten Iteration of Analytic Functions
553 554 555 565 573 578 585 592
Xll
Yakov G. Sinai Curriculum Vitae Bibliography Markov Partitions and C-Diffeomorphisms Structure of the Spectrum of the Schrodinger Operator with Almost-Periodic Potential in the Vicinity of Its Left Edge A Random Walk with Random Potential Probabilistic Approach to the Analysis of Statistics for Convex Polygonal Lines Elias M. Stein Curriculum Vitae Bibliography Some Problems in Harmonic Analysis Suggested by Symmetric Spaces and Semi-Simple Groups Problems in Harmonic Analysis Related to Curvature and Oscillatory Integrals
599 601 609 631 637 641
647 648 660 677
Jacques Tits Short Note Curriculum Vitae Resume des travaux anterieurs a 1972 (ref. [1] a [92]) Publications Groupes et Geometries de Coxeter
703 704 706 730 740
Andre Weil (prepared by J-P. Serre) Curriculum Vitae Weil's Work Publications Andre Weil and Algebraic Topology (by A. Borel) Andre Weil (by J-P. Serre)
755 756 757 763 769
Hassler W h i t n e y (prepared by J. Milnor and J. Eells) Academic Appointments and Awards Bibliography of Hassler Whitney Moscow 1935: Topology Moving Toward America A Function Not Constant on a Connected Set of Critical Points Singularities of Mappings of Euclidean Spaces Elementary Structure of Real Algebraic Varieties Hassler Whitney (23 March 1907-10 May 1989) (by S. S. Chern)
781 782 785 806 810 827 839
Andrew J. Wiles Bibliography Modular Forms, Elliptic Curves, and Fermat's Last Theorem Twenty Years of Number Theory
845 847 850
Xlll
Oscar Zariski (prepared by D. Mumford) Curriculum Vitae Preface (to Volume IV of the Collected Works) Bibliography Oscar Zariski (1899-1986) The Fundamental Ideas of Abstract Algebraic Geometry The Compactness of the Riemann Manifold of an Abstract Field of Algebraic Functions Analytical Irreducibility of Normal Varieties
900 909
Correction (for Lars Hormander's Contribution, Vol. 1)
919
Photos of Wolf Prize Winners
921
865 867 875 883 887
Wolf Prize in Mathematics, Vol. 2 (pp. 1-59) eds. S. S. Chern and F. Hirzebruch © 2001 World Scientific Publishing Co.
Autobiographical Sketch
Raoul Bott
I was born in Budapest on the 24th of September 1923 and lived the first 16 years of my life in the Hungarian part of Slovakia. My schooling was somewhat unorthodox from the very start and really continued to be so until I received my final degree in 1949. Until the age of eight I was educated at home, where I was taught the rudiments of arithmetic, German and English. Hungarian was spoken but I recall no formal education in this tongue. In 1932, when we moved to Bratislava — the capital of Slovakia — I finally went to a regular school, learned Slovak and had progressed to the third year of the "Real Schule" when the pre-war events of 1938 disrupted my schooling altogether. I have written a detailed account of the period "23-39" in my article "The Diozeger Years" and hope to write in greater detail on the period "39-49" at some future time. Here, I have in mind just a brief, mainly "educational" sketch, leaving aside the more personal side insofar as I can manage that. In the early June of '39, my stepparents flew me to England and there enrolled me in "The Beltane School," a "private," as opposed to a "public school," in the paradoxical sense in which the English use these terms. This boarding school was quite unusual in that it was coeducational and was run along very progressive lines by its charismatic headmaster, Mr. Tomlinson. Beltane was of course even more unusual during my period there because of the onset of the war. Somehow, at the eleventh hour Mr. Tomlinson had managed to secure a beautiful old manor-house with fine gardens, fields, stables etc., in the small town of Melksham, Wiltshire, a few miles from Bath, and he then very courageously moved his whole establishment there. To serve as a school, the place needed major rearrangements and as a result, we older boys spent much of our time hammering, carrying and digging. I now certainly recall shingling the stables, which were being altered into classrooms, more than attending classes. Nevertheless, or maybe precisely because of these romantic circumstances, I remember my time at Beltane as one of the truly formative periods of my life. In one stroke it made me into a lifetime Anglophile as well as a great admirer of the opposite sex. Mr. Tomlinson was also a natural role model with his great faith in progressive education and his cheerful good looks as he placed his empire in the red shirt which he so often wore, and which, I expect, reflected his political leanings. I especially remember the Wednesday evenings, when, after supper, the boys and girls gathered in the main hall ostensibly for the purpose of mending their clothes — mainly socks in my case — while he read to us from the short stories of Damon Runyon! I say ostensibly, because the whole occasion abounded in opportunities for meaningful glances and more or less innocent assignations.
Raoul Bott Academically I was in a twilight zone between the 0 and A levels and, because I was not forced to attend classes, I rarely showed up for them. All the boys and girls of my age seemed hopelessly ahead of me in every academic pursuit. The unusual regimen of the school had attracted a highly gifted clientele; many of the students were also refugees from Hitler, and most of the others came from English families with interests in the arts, music and the theatre. Mathematics was not so much in the air, and what there was, again seemed out of reach. I had never been taught Euclidean geometry, while the boys around me were solving complicated problems in that subject. My stepparents only had a transit visa for England and they left for Canada early in 1940. In July of that year, just a week or two before the bombing of England started, my stepsister and I set sail on the liner Scythia from Liverpool to Halifax, to join them. On this trip I had dreams of becoming a cowboy, but instead, and really quite ironically, the harsh fate of going to a British public school, which I had so miraculously avoided in England, now caught up with me in Canada. I was sent to "Ridley College" in St. Catherines, Ontario, a school modeled precisely on the strict regimen of classical English education. Classes were obligatory, rules were strictly enforced, "new boys" were hazed. There were prefects, there were canings and "strappings" and, of course, there were no girls. Due to the indefinite nature of what I might or might not know, the authorities at Ridley decided to put me in the penultimate class, so that I would have had two full years before graduation. At this point mathematics entered my life for the first time, and quite decisively. The very first class I attended at Ridley was a mathematics class and, to put us at ease, the kindly teacher started the class with a "proof that 1 equals 2! I had seen this trick before and immediately put up my hand to expose the hidden division by zero. After class, the teacher came over excitedly and on the spot moved me up a whole year to join the seniors! Thus, largely by avoiding school altogether, I managed to graduate in 1941, a full year ahead of my poor classmates in Bratislava. At Ridley we were taught trigonometry and analytic geometry and these became my best subjects. Our teacher was a clear expositor, but reserved and very even-mannered. The far away war touched our innocent existence tragically when one day he was called out of the classroom. He returned nearly immediately and finished the class in his usual manner. Later, we learned that he had been told that his only son had fallen in action. After that, we respected him even more. It was also rumored among the students that he "knew the calculus." This one year at Ridley, although so different from Beltane, was not a bad one. At worst it taught me diplomacy, for I found myself in the contradictory position of being a "new boy" and a "senior" at the same time. In the spring I passed the entrance exams to McGill University in the sciences, but failed in the humanities. Still, McGill gave me the benefit of the doubt so that in the fall of '41, I started a quite new existence in Montreal, enrolled as an electrical engineering student, heading in the direction of my early hobbies and interests. The sense of "freedom" after the regimented life of Ridley was quite overwhelming. That first year I had a small room in a boarding house and apart
Autobiographical Sketch from the rather formidable engineering curriculum, I felt a great exuberance at being my own master. I see now that it is difficult to say just a "few words" about the four years of one's graduate days. In the language of the physicists, each of them has the "effective" weight of 10 later ones. There is, first of all, the excitement, the gleam in the eye, "the world is my oyster" of the freshman year. For me this was the year I tried hard to hold my own with the "engineering tradition" of hard drinking, loud, boisterous, mischievous and macho behavior. But I actually also took my classes seriously and did rather well in my studies that year. The sophomore year saw a sharp drop in my grades as my interests veered from engineering and science altogether to more "existential" matters. The third and fourth year I managed to improve my grades a little and think I graduated with a quite respectable ranking in my class. At McGill final exams were taken very seriously and the results appeared only one or two months later, when in each subject, our names were posted on a bulletin board at the appropriate place. The names were in linear order according to our performance. That is, provided one had passed. If one had failed one's name would not appear at all! Thus we tended to fall into two categories. Those who started looking for their names from the top and became more and more alarmed the lower they had to scan the page, and those who, on the contrary, started from the bottom, and became more convinced as they progressed upward that their name would not appear at all. If those "bottom uppers" finally found their name, they would shout with jubilation that they had "beaten the Dean!" For, of course the Dean's signature appeared at the bottom of the page. This system must seem barbaric in our day when we announce grades by registration number, but it seemed normal to us, for grades were in any case not taken too seriously. The "gentleman's C" was certainly the main goal of the large majority of the "fraternity crowd" and in my more bohemian circles grades counted for even less. My best subject during these four years was certainly "mathematics" but this was of course "engineering mathematics" and really comprised only the usual calculus progression, differential, integral and advanced. No linear algebra or any sort of more rigorous introduction to real variables. My first teacher in the calculus was Prof. Williams. He was beautiful! Some professors taught in gowns, English style, and he was one of these. His gown was unbelievably old, chalk crusted and slightly torn. With his hair flying and his gown flapping Prof. Williams' lectures were not a model of clarity, but I found shining through them his love of the subject and also his general benevolent view of life and mankind. Williams had been a Rhodes scholar and I believe he came from a Quaker family. He naturally supported and fostered all waifs, foreigners, and other lost souls. Eventually it was he and his recommendation that got me admitted to Carnegie Tech as a graduate student, and he is now celebrated as one of the founders of Canadian mathematics. Kind though he was, he also had high standards, and I remember that he failed 75 percent of the calculus class one year! Who would dare to do that today at an American university?
Raoul Bott Of the many friends and of the loves of that period let me mention here only some of my other surrogate fathers; each in his own way made me perceive a possible path through the years to come. Chief amongst them was R.D. McLennan, Prof, of Philosophy, warden of Douglas Hall — the "male" residence of McGill — and really our local Socrates. He was a slow moving Scotsman with a boyish look, and a strong but quiet sense of humor. His classes were famous. He was a "Platonist" and practiced the Socratic method quite spontaneously with all of us. He was also a devout Presbyterian, and it was he who introduced me to Kierkegaard. During my teens I had become an almost militant atheist, but during these later years at McGill I found this view of the world more and more repugnant, and R-D, as we used to call him, was quite a master at sowing doubt about atheism. I had been brought up in the Catholic faith, and it was in this direction that I started to move again during these later years at McGill. My other role models, also pointing in this spiritual direction, were Karl Stern and Miguel Prados, both fascinating men, both psychiatrists and both Catholics, but with very dissimilar backgrounds and points of view. At the end of the academic year — in April 1945 — I graduated from McGill and received my iron "Engineering Ring" made from a fallen bridge for the Canadian engineering community. The impressive ceremony was written especially for these occasions by Rudyard Kipling. The next day I went to a recruiting office and volunteered as a private in the Canadian Army. Throughout the entire period I have been describing, the backdrop of our daily life was the far-off war. FVom the near despair of the early days, the Allied cause had now finally achieved victory in Europe. And throughout fate seemed to have shielded me from the war in the most uncanny way. After landing in England just two months before the war started, I had escaped internment in Britain as an "enemy alien" by the merest of flukes. (Namely, by being officially stateless, but with a Czechoslovakian passport. Without the passport I could not have entered Britain. But as a CSR citizen, I would have been subject to internment as an enemy alien because of the German occupation. The perplexed immigration officials finally decided to count me as a "Hungarian," a country which was neutral at that time, and so I avoided internment.) Then, next, I had escaped to Canada just a week before the great battle for Britain started. My duty vis-a-vis the war had therefore been troubling me all along. In fact I had tried to join up during my first year at McGill. But this was during that flamboyant freshman year, and I had volunteered for the "Secret Service!" It seemed plausible to me. I even had visions of being parachuted behind enemy lines. But the officials only laughed at me: "You are much too tall, and anyway unsuitable. You enter a room and everyone notices you." The Canadian, and later U.S. forces were also suspicious of us "enemy aliens." In fact they ironically interned most German Jews during these early years, and if one of these "enemy aliens" did get into the army, then they were often set to peeling potatoes and felt themselves to be "second class" soldiers. In any case, after this first attempt, I decided to stay at school until I got my degree, and in engineering we were actually encouraged to do so, because technically
Autobiographical Sketch trained personnel would be needed in the long war years ahead. After graduation then, and after the war in Europe was done, I seem to have decided that my time had come. In the army, during basic training, the next three months went by in a regimented dream; the ridiculous army rules and preoccupations set incongruously against the sparks of friendship, profane humor, and compassion, which so often pierced the constant companionship of one's comrades, and so made it bearable. On August 6 our sergeant took us aside, and lounging in the grass behind some bushes, we heard the unbelievable news of the atomic bomb and the tragic fate of Hiroshima. A few days later the soldiering dream was over. Again I had been rescued at the last moment. But on the other side of the ledger, the war — qua alibi — had also ceased. Now it was time to get on with the rest of one's life. Through the very kind intercession of the McGill Engineering Department, I somehow now managed to quit the army in time to enroll for a one-year Master's degree program at McGill that very fall, and even got a G.I. bill educational advance from the army of 40 dollars a month for the ensuing three months! I was sunburnt and in the best physical shape of my life thanks to this summer adventure, and I put it down to this slightly "military" look of mine that I also immediately landed a teaching job to pay for my living expenses and the modest tuition at McGill. This came about because of Prof. Gilson, the new and dynamic head of the mathematics department. After years of being considered one of the most eccentric and untractable professors at McGill, Gilson had become a high officer during the war, and, changing course entirely, had become the generalissimo of the mathematics department upon his return. He later went on to become Dean of Arts and Sciences at McGill and eventually became president of one of the other Canadian universities. In any case, in the fall of 1945 he needed calculus teachers desperately. He was clearly taken with me when I presented myself in crew cut and army shorts. There and then he promoted me from a potential teaching assistant, and put me in charge of a projected calculus class of more than 100 people, many of them returning veterans, some considerably older than I was. Mathematics had come to my rescue once more, although it certainly did not yet figure seriously as a career choice. At the time I was pretty certain that my interests were not so much in engineering, as in some sort of more fundamental research. My original project for a masters' dealt with wave guides but I found that the experiments just would not come together. Experimental work, the love of my childhood, just never quite "took" during my whole "engineering existence" at McGill. The art of "debugging" an experiment is a subtle one and I must have lost it in the transatlantic crossing. Finally in the spring of 1946 I produced a very mathematical master's thesis on "impedance matching," which the department accepted with some misgivings and about whose mathematical rigor I have doubts to this very day. And intellectually, the courses in mathematics with which I had loaded my program, were now clearly the ones I found most interesting. Gilson was a complex person. As one unkind friend put it: "He looked like
Raoul Bott one end of a horse and acted like the other." And indeed in many dealings with him, I can attest to the validity of both assertions. But in class he was one of the most brilliant teachers I have ever encountered. He was a first-rate applied mathematician of the Cambridge School, had been a "first wrangler" there, and used to lecture from his old school notebooks. In any case I now learned about Maxwell's equations in a quite new way, saw him deal with "shallow waves," and even learned the rudiments of relativity from him. At the same time, or possibly a bit later, the lectures of Prof. Wallace on quantum mechanics became one of my main intellectual preoccupations. Still I was at sea as to the general direction of my career much of the years 19451947. But I distinctly remember the occasion when this confusion was finally resolved! I had presented myself to the dean of the medical school at McGill to explore the possibility of becoming a medical doctor. Put bluntly, I wanted to know whether he could help pay for such a course. Dean Watson was another of a whole breed of fascinating characters we had for professors at McGill. He was a distinguished biochemist, but at the same time wrote detective stories under a pseudonym. He professed to be sympathetic to my cause. "We need technically trained physicians," he said. "But," he continued, "please tell me a little about yourself. What are your likes, your dislikes, etc." In the ensuing litany I soon realized that all my answers somehow seemed wrong. No, I never enjoyed disecting animals; No, I hated chemistry... Finally Prof. Watson leaned back in his chair and fixed me mercilessly: "Is it maybe," he said carefully, "that you want to do good to humanity?" I sat in silence, but he also did not give me much time before he continued emphatically "for they make the worst doctors." I got up, thanked him for the interview, and as I walked out the door I remember thinking to myself, "O.K. you so and so. If that is how it stands, then I will become a mathematician." Somehow he had given me the courage to follow my bent, to stake my life on it. There is only one other interview that can match this one in its impact on the rest of my life. It occurred a few months later, when on the advice of Prof. Williams I had applied to become a graduate student at the Carnegie Institute of Technology in Pittsburgh. For McGill itself was not very hospitable to the thought of having me switch to mathematics proper. They pointed out that my background was very sketchy, and insisted that I would have to get a Bachelor in Science first! So Prof. Williams had the inspiration to send me to Pittsburgh where Prof. John L. Synge, formerly the head of the department of mathematics at Toronto, had taken over to establish a new Doctor of Science program. Synge was an expert in mechanics, especially relativity theory, and he was also a personality of the first order. When I presented myself in his office I came face to face with a man with a black patch over one eye, a friendly but authorative manner, and some sort of filtering device in his nose! The air in Pittsburgh was foul in those days and Synge, individualist that he was, thought nothing of combatting the smog in any way he saw fit. Upon hearing my story, Synge immediately proposed a course of study
Autobiographical Sketch towards a master's degree in applied mathematics. This, despite my dubious background. He then looked at the requirements which his committee had just promulgated the week before. The course work alone would have taken three years! I groaned inwardly and he himself was taken aback by the strictness of his own decrees. For a while he peered into the mist which engulfed the window of his office and in which one could discern flames leaping up at the far off foundaries from time to time. Then suddenly he turned around with these golden words: "Let us look at the requirements for a Ph.D." There were hardly any. Two years residence, a qualifying exam, and of course a thesis. And so it was, once again, that I graduated rather earlier than I had any right to expect, in the spring of 1949. And then, miracles of miracles, later that year I found myself on the way to the Institute for Advanced Study in Princeton where the whole world of pure mathematics was awaiting me.
8 Reprinted from JOURNAL OF APPLIED PHYSICS Vol. 20, No. 8, 816, August 1949 Copyright 1949 by the American Institute of Physics
Impedance Synthesis without Use of Transformers
R. Bott and R.J. Duffin Department of Mathematics, Carnegie Institute of Technology, Pittsburgh, Pennsylvania December 13, 1948 Let Z(s) be termed a B(rune) function if: (1) it is a rational function; (2) it is real for real s; and (3) the real part of Z is positive when the real part of s is positive. In his significant thesis, O. Brune 1 shows that the drivingpoint impedance of a passive network is a B function of the complex frequency variable s. Conversely, he shows that any B function can be realized by some passive network and gives rules for constructing such a network. In this synthesis he is forced to employ transformers with perfect coupling. It is recognized by Brune and others that the introduction of perfect transformers is objectionable from an engineering point of view. Prior to Brune, R.M. Foster 2 has shown how to synthesize the driving-point impedance of networks containing no resistors by simple series-parallel combinations of inductors and capacitors. This note gives a similar synthesis of an arbitrary impedance by series-parallel combinations of inductors, resistors, and capacitors. A B function can be expressed as the ratio of two polynomials without common factor. Let the "rank" be the sum of degrees of the polynomials. Obviously any B function of rank O can be synthesized. Suppose, then, it has been shown that all B functions of rank lower than n can be synthesized, and let Z(s) be a B function of rank n. Brune gives four rules for carrying out a mathematical induction to a B function of lower rank: (a) If Z has a pole on the imaginary axis, then Z can be synthesized, by a parallel resonant element in series with an impedancce Z' of lower rank; Z = l/(cs + l/ts) + Z' where t~l, c > 0. (b) If Z has a zero on the imaginary axis, then Z can be synthesized by a series resonant element in parallel with an impedance Z' of lower rank; 1/Z = l / ( / s + Ilea) + l/Z' where (., c" 1 > 0. (c) If the real part of Z does not vanish on the imaginary axis, Z = r.+ Zo where r is a positive constant (to be interpreted as resistance) and ZQ is a B function of no greater rank than Z. Brune's fourth rule, (d), which employs the perfect transformer, we replace by the following procedure: 1 2
O. Brune, J. Math, and Phys. 10, 191-236 (1931). R.M. Foster, Bell Syst. Tech. J., 3, 259 (1924).
Impedance Synthesis
491
(d') If none of these reductions are possible, there exists a w > 0 such that Z(iw) is purely imaginary. First assume that Z(iw) = iwL with L > 0. We now make use of a key theorem discovered by P.I. Richards. 3 Let k be a positive number, and let H(S)
kZ(s) - sZ(k) kZ(k) - sZ{s) •
{ }
Then R{s) is a B function whose rank does not exceed the rank of Z(s). Richards states this theorem for A; = 1; the above form is an obvious modification, because Z(ks) is also a B function. Let k satisfy the equation L = Z(k)/k. This is clearly always possible, because the function on the right varies from oo to 0 as & varies from 0 to oo. With this choice of k, clearly R(iw) — 0. Solving (1) for Z gives Z(s) = (l/Z(k)R(s)
+ s/kZ{k))-1
+ (k/Z(k)s
+
= (1/ZiOO + Cs)-1 + (1/Ls + I/Z2)-1.
R{s)/Z{k))-1 (2)
Here Zx{s) = kLR(s),Z2(s) = kL/R(s),C = \/k2L. Since Zx is a B function with a zero on the imaginary axis, it can be synthesized. Likewise, Zi is a B function with a pole on the imarginary axis and can be synthesized. Z(s) is therefore synthesized by two networks in series. The first network consists of the impedance Z2 in parallel with an iductor L. In the case that Z{iw) — —iwL, similar considerations applied to the function 1/Z show that Z is synthesized by two networks in parallel. The synthesized network finally resulting has the configuration of a tree whose branches are ladder networks. Richards 4 has sought necessary and sufficient conditions for the drivingpoint impedance of resistor-transmission-line circuits by means of an ingenious transformation of the Brune theory. The perfect transformers, which are again found to be objectionable, may be dispensed with by the above procedure.
3 4
P.I. Richards, Duke Math. J., 14, 777-786 (1947). P.I. Richards, Proc. I. R. E., 36, 217-220 (1948).
PORTUGALIAE MATHEMATICS
Vol. 11 - Fasc. 1—1B52
TWO NEW COMBINATORIAL INVARIANTS FOR POLYHEDRA* BY RAOUL BOTT Institute for Advanced
Study,
Princeton
and University
of Michigan,
U. S. A.
In this note we introduce two new combinatorial invariants (see definition in section 1) for polyhedral complexes. The idea behind their construction is the following one: By removing the n-cells (their interiors) from a given finite n-complex n, we generate a family of subcomplexes it,. The invariants we introduce consist of certain expressions summed over the whole family jic.j. The form of the expressions to be averaged in this way is suggested by studying how the family J7rsJ changes when we go over to an elementary subdivision of %. (An elementary subdivision just splits an n-cell in two). The new invariants are therefore definitely «cooked up», and not much geometrical good can be expected of them. They are presented here, then, mainly as a curiosity. The most interesting feature of the R and S polynomials (as we call our invariants) of a complex jr, is that they are not invariants of the homotopy-type of JT\ However it is this very property which makes it unlikely that the topological invariance of the polynomials R and S will be decided before the «Hauptvermutung» is. 1.1. We recall the following definitions from [1]. A. A finite n-complex X is a finite set \x\ of elements ordered by a proper reflexive ordering relation < , together with two associated functions of the elements and element pairs whose values are integers: one, dim a;, the dimension of x, also denoted by means of an index as xp, and the other, [ar: x1], the incidence number of x and x' subject to the following conditions: Kl,
ar'< x -*• <j\m.x'^l dimx
* Received February 18, 1952.
36 K2, K3, K4,
RAOUL 1JOTT
[x:x'] = [x': x] [x: x'] =f= 0 -• x < x' or x'<x and |dim.zr — dimar' | = 1 For every pair x,x" whose dimension differs by two 2
K 5,
,[a?:ar'][jr':a;"J-=0.
dim X = sup dim x =-- n .
B. Let E m be the Cartesian m-space, E^cE" 1 a linear p dimensional subspace. By a p cell is meant a bounded convex region of some E p c E m . C. A polyhedral complex it in E m is a finite complex i r = | e | such that: a) a p dimensional element e" of n is a p cell in E r a ; b) the cells are disjoints ; c) If Cle"=\e'\e'<e\, then C t e ' = e". p The set \J^e is denoted by |ir| . D. The polyhedral complex it' is partition (subdivision) of it if a) every cell of r.' is contained in some cell of it; b) every cell of it is a finite union of cells of iz'. E. An elementary partition it' of it is defined as follows: A certain cell, say e>', is singled out. and is subdivided by passing an E p _ l through ep, into two convex cells e'1' and e"p and the convex p — \ cell e'*"1. The plane E p _ 1 subdivides the boundary ep of ep into a polyhedron itn'1. The replacement of C le" by « n _ 1 |J \e'p.e"p,e'p-1\ is a partition of it. If E" - 1 is such that e'' = it'"1, (in other words E p _ 1 does not refine the partition of ep in it) then the partition is elementary. Thus in an elementary partition e1' is replaced by \e''\e"p,e'p-x\. Concerning the elementary partitions LEFSHETZ proves the following theorem : : Let it be a polyhedron, it' a partition of it. Then there exists a polyhedron it': ivhich is obtained from both it and it' by an iteration of elementary partitions. THEOREM
1.2. A quantity associated with a polyedral complex will be called a combinatorial invariant if it remains unchanged under partitioning. It is a consequence of the theorem above that an object is an invariant under general partitioning if and only if it is an invariant under elementary partitioning.
TWO SEW COMBINATORIAL INVARIANTS FOR POLYHEDRA
37
Let n be a polyhedral M-complex. We set a(jr) = n ,h Betti Number of 7t. £ (it) = (n — l ) s l Betti number of it . Let the n cells of it be indexed from 1 to N , and let s denote subsets of the set of first integers I N (the empty set is legitimate!). \s\ shall denote the number of integers contained in s. By %s we will mean the closed subcomplex of Jt obtained from it by deleting precisely those n cells whose index is included in s. We are now in a position to define the two combinatorial invariants we are after. THEOREM
I.
Let. n be a polyhedral complex as above. Then the
polynomial
(where the summation is extended over all subsets of I N ) is a combinatorial invariant. P R O O F : Let it' be an elementary partition of it. If the cell partitioned is of dimension less than n all terms in the above formula remain unchanged. Assume therefore the H-cell e^ is partitioned into |/«
'"
e'n~x\
We split the subsets of IN into two complementary sets 2 Ix = A + B where s e A if and only if N c s . The subsets of IN + I we split into four classes 2'N+I = A'-j- B ' + C ' + D ' ,
where s ' e A ' s'eB' s'eC s'eD'
if if if if
and only and only and only and only
if if if if
N and N + l