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Home , Arthur Wightman, Edward Nelson, Eugene Dynkin, John Tate

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1995 Steele Prizes

Three Leroy P. Steele Prizes were presented at The text that follows contains, for each award, the awards banquet during the Summer Math- the committee’s citation, the recipient’s response fest in Burlington, Vermont, in early August. upon receiving the award, and a brief bio- These prizes were established in 1970 in honor graphical sketch of the recipient. of George David Birkhoff, William Fogg Osgood, and William Caspar Graustein Edward Nelson: 1995 Steele Prize for and are endowed under the Seminal Contribution to Research terms of a bequest from The 1995 Leroy P. Steele award for research of Leroy P. Steele. seminal importance goes to Professor Edward The Steele Prizes are Nelson of for the following ...for a research awarded in three categories: two papers in mathematical character- for a research paper of fun- ized by leaders of the as extremely innov- paper of damental and lasting impor- ative: tance, for expository writing, 1. “A quartic interaction in two dimensions” in fundamental and for cumulative influence Mathematical Theory of Elementary Particles, and lasting extending over a career, in- MIT Press, 1966, pages 69–73; cluding the education of doc- 2. “Construction of quantum fields from Markoff importance, toral students. The current fields” in Journal of Functional Analysis 12 award is $4,000 in each cate- (1973), 97–112. for expository gory. In these papers he showed for the first time The recipients of the 1995 how to use the powerful tools of probability writing, and for Steele Prizes are Edward Nel- theory to attack the hard analytic questions of son for seminal contribution constructive , controlling cumulative to research, Jean-Pierre Serre renormalizations with Lp estimates in the first influence for mathematical exposition, paper and, in the second, turning Euclidean and John T. Tate for lifetime quantum field theory into a subset of the the- achievement. ory of stochastic processes. The Steele Prizes are Citation awarded by the AMS Council The interaction of with relativistic acting through a selection quantum field theory is, in many respects, one committee whose members at the time of these of the signal mathematical developments of the selections were Eugenio Calabi (chair), Ingrid second half of this century. Edward Nelson was Daubechies, Eugene Dynkin, Robert P. Langlands, one of the pioneers in this development. From Barry Mazur, Paul Rabinowitz, Marina Ratner, the earliest attempts to turn quantum field the- Gary M. Seitz, and William P. Thurston. ory into rigorous mathematics, it had been clear

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that operator and distribution theory would play prominent roles. What Nelson real- ized, and implemented in these two fundamen- tal papers, is that probabilistic techniques could provide critical additional tools. In the first of the two papers recognized by the award, Nelson overcame the infinities associated with Wick- ordering renormalization in two-dimensional field theories by a combination of measure the- ory and Lp estimates of semigroups. The tech- niques that he introduced for establishing in two dimensions the stability of the quartic in- teraction were fundamental, strongly influenc- ing the further development by Glimm and Jaffe of rigorous quantum field theory in dimension three. They continue to be pertinent today in, for example, the theory of the nonlinear Schrödinger equation. Renormalization modifies a formal, nominally positive fourth power by the sub- traction of an infinite constant, so that the pos- itivity of the result was not at all clear, which was contested at the time the paper appeared. Nel- son resolved the controversy, and in so doing de- vised the mathematical tools later generalized to a much larger class of Hamiltonians. In the second paper recognized by the award, Nelson fired one of the first shots in what be- Edward Nelson came known as the Euclidean revolution. The an- alytic continuation of relativistic field theory to the difference between a Lagrangian and a Hamil- imaginary time transforms formally the tonian. Minkowskian field theory into a Euclidean the- In the first cited paper I put the field in a spa- ory. Nelson realized that this was not only a for- tial box and proved that certain operators were mal trick, but provided a mathematical inter- bounded. But it was James Glimm who then pretation of certain stochastic processes. The proved that the bound is in fact 1, a result es- concepts he introduced thus furnished a math- sential to removing the box. In the sequel to the ematically rigorous approach that combined the second paper, when I studied the free Markov operator formalism in Minkowski space with field, I omitted to refer to the work of Loren Pitt, the use of a Markov property symmetric with re- who first introduced this field and proved the spect to space and time. Markov property for it. He had sent me a Response preprint, but when I wrote the paper, I did not I was introduced to in a grad- consciously remember it—these things can hap- uate course taught by from galley pen. No one who knows Loren will be surprised proofs of Doob’s “Stochastic Processes”. Irving to hear that when I apologized to him, he was presented his own viewpoint in addition to very gracious indeed. Doob’s, and it was an exciting course. Once he One pleasant feature of receiving this prize drove me down to Urbana so we could talk with is that it reminds me of how much fun I had Doob. It was a memorable trip. Maintaining that working on those problems, almost as much fun the probability of an accident is directly pro- as I am having now in my work. My advice to any portional to the time spent on the road, Irving young approaching the age of drove in such a way as to minimize that time. fifty who wants to continue having fun doing Despite having Irving Segal as thesis adviser, mathematics is this: change field. I did not learn physics at the University of Now I come to the main point, which is to ex- Chicago. I took one course in the physics de- press my thanks to the AMS and the Selection partment but was defeated by the lab; I didn’t Committee for this Steele Prize. It was a great really know how to explain the 457 percent error surprise to me. (Notice the shade of difference in my result for the mechanical equivalent of between that statement and “The committee heat. But when I got to Princeton University, I at- made a very surprising choice.”) It is a great tended several of ’s courses and honor, it is great fun, and I am grateful. Thank pored over the papers of Richard Feynman and you. Also, thanks to the AMS for reserving a Kurt Symanzik, and after a while I began to learn room for us with a jacuzzi.

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Biographical Sketch by the author, and everything smoothly pol- Edward Nelson was born May 4, 1932, in Decatur, ished. It would be hard to make any significant Georgia. After first grade in , Italy, he re- improvement on his expositions; many are the turned to Georgia in September 1939 and moved everyday standard references in their areas, to in 1942, where children from Geor- both for working and graduate gia were put back a half grade and required to students. Serre brings his whole mathematical undergo speech therapy. After secondary school- personality to bear on the material of these ing at the Bronx High School of Science and the books; they are alive with the breath of real Liceo Scientifico Giovanni Verga in Rome, Nel- mathematics and are an example to all of how son enrolled at the , where to write for effect, clarity, and impact. One rea- he obtained a Ph.D. in 1955 with a thesis on son for choosing A Course in Arithmetic for Markov processes written under Irving Segal. the award is the basic nature of the subject. Nelson worked two years as a conscientious Every mathematics graduate student should objector in the Methodist Hospital of Gary, In- become thoroughly acquainted with at least diana, and then spent three years at the Institute the first quarter of the book as part of the al- for Advanced Study. Since 1959 he has been at gebra background, and with the first half if the Princeton University, where he served the math- chosen field of specialization is to be related ematics department for six years as director of to . What is remarkable is the concise- graduate studies and now as webmaster ness, the clarity, and the completeness of the (http://www.math.princeton.edu). topics treated. His wife of thirty-five years, Nancy Wong Nel- The second half of the book, also purely ex- son, died in 1988. Since 1990 he has been mar- pository and covering “classical” topics, is a ried to Sarah Jones Nelson. He has two children jewel of concise exposition of the link between and three grandchildren. the combinatorial aspects of elementary num- Nelson is a member of the American Academy ber theory and the methods of function theory of Arts and Sciences and doctor honoris causa of (zeta function, L-functions, and Eisenstein se- the Université Louis Pasteur in Strasbourg. His cur- ries): this is a beautiful encapsulation of what is rent research interests are logic and foundations. the glory of nineteenth-century arithmetic, pro- viding a background to the modern-day devel- Jean-Pierre Serre: 1995 Steele Prize for opments in geometry of numbers and number Mathematical Exposition theory. The 1995 Leroy P. Steele Prize for Mathematical The remarkable feature of this book is that Exposition is awarded to Professor Jean-Pierre the whole exposition is compressed into a little Serre of the Collège de France, , for his over one hundred pages, including all but the 1970 book Cours most obvious proofs and a thorough bibliogra- d’Arithmétique, phy of the more extensive coverage of the top- with its English ics treated. translation, pub- Response lished in 1973 by It is a nice surprise to receive a Steele Prize for Springer-Verlag, A my old little book Cours d’Arithmétique. Thank Course in Arith- you. metic. An old book indeed. Not only was it pub- Citation lished in 1970, but the actual writing took place It is difficult to de- much earlier: cide on a single • as an exposé at the 1961–1962 Cartan Semi- work by a mathe- nar on two papers of Milnor on quadratic matician of Jean- forms over Z and the homotopy type of 4- Pierre Serre’s manifolds, stature which is • as a set of lecture notes (École Normale most deserving of Supérieure, 1962) on the classification of qua- the Steele Prize. dratic forms over Q, Any one of Serre’s • as another such set of notes (E.N.S., 1964) on numerous other Dirichlet’s arithmetic progression theorem books might have and modular forms of level 1. served as the basis To get a book from these texts, only scissors of this award. Each and glue were needed (and a log table, in order of his books is to compute some mass formulae). Strangely beautifully written, enough, the different pieces fitted well together; with a great deal of I was especially pleased with the way algebraic and original material analytic arguments complemented each other. Jean-Pierre Serre

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The book first appeared in a handsome gebra, algebraic pocket-size format, very reasonably priced (about geometry, and three dollars). Still, there were some troubles, the number theory worst one being an erroneous computation of the during this time. Eisenstein series E2 (shame on me). The English His 1950 Prince- translation, A Course in Arithmetic, did not fare ton thesis,“Fourier better: the first edition had so many misprints Analysis in Num- that I dubbed it A Curse in Arithmetic. ber Fields and All this is past. New corrected editions have Hecke’s Zeta-Func- followed and by now the misprints are fewer. The tions”, initiated the real question is: will the book continue to be use- use of methods of ful? I hope so, and the Steele award seems a harmonic analysis good omen. Thanks, AMS! in the study of L- Biographical Sketch functions which Jean-Pierre Serre was born in Bages, France, on have importance in September 15, 1926. He studied at the École arithmetic, evalu- Normale Supérieure in Paris from 1945 to 1948 ated at the adelic and received his Ph.D. in mathematics from the points of various Sorbonne in 1951. Between 1948 and 1954, he reductive groups. A was attaché, chargé, and maître de recherches prodigious amount at the Centre National de la Recherche Scien- of later mathemat- tifique in Paris. After two years at the University ical activity has of Nancy, he was appointed professor and chair been inspired by of Algebra and Geometry at the Collège de France this initial contri- in 1956; he has been an honorary professor bution. there since 1994. Over the years, he has spent a Tate’s early work good deal of time in the , espe- was instrumental in cially at the Institute for Advanced Study at the development of the foundations of group co- Princeton and at . homology and . His famous Professor Serre has been elected to the Acad- collaboration with Artin, providing a Galois co- emies of Science of several nations: France homological exposition of ; his (1977), the Netherlands (1978), the United States 1952 paper,“The Higher Dimensional Cohomol- (1979), and Sweden (1980). He was named an ogy Groups of Class Field Theory” (for which he honorary member of the London Mathematical received the 1956 in Number Theory); Society in 1973, and an honorary fellow of the and his papers on principal homogeneous spaces Royal Society in 1974. He has received honorary for abelian varieties (1958 with Lang, and 1962) degrees from the Universities of Cambridge might be viewed as all leading the theory up to (1978), (1980), and Glasgow (1983). the point where its language is adequate for the He received the Fields Medal in 1954 and the formulation of Tate’s deep duality theorems (as Balzan Prize in 1985. first expressed in his 1962 International Con- The author of a dozen books and numerous gress of Mathematicians address in Stockholm, research papers, Professor Serre works in topol- “Duality Theorems in Galois Cohomology over ogy, analytic geometry, , Number Fields”). It would be hard to catalogue group theory, and number theory. all the later developments in the subject (and in nearby subjects) that are critically dependent John T. Tate: 1995 Steele Prize for upon this work. His construction of (what is now Lifetime Achievement known as) the Shafarevich-Tate group, and his The 1995 Leroy P. Steele Prize for Lifetime recognition of the crucial role that this group Achievement in Mathematics goes to Professor was destined to play in arithmetic, was, in itself, John Tate of the University of Texas in Austin. a fundamental insight. Citation In the years between 1962 and 1966, Tate John Tate was born in 1925, received his Ph.D. published papers (one with Sen, one with Lubin) from Princeton in 1950 and taught at Harvard related to local arithmetic. In 1966 he published University during the years 1954 through 1990, his very important article “p-Divisible Groups”, before moving to his current position as Sid W. which initiated the theory of those eponymous Richardson Chair of Mathematics at the Univer- groups and which was one of the first pene- sity of Texas at Austin. trating studies of interesting p-adic represen- Tate’s scientific accomplishments span four tations of Galois groups of local fields whose and a half decades. He has been deeply influential residual characteristic is p. This difficult chap- in many of the important developments in al- ter of number theory is still a crucial concern and

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remains the focus of intense activity; it would ory and algebra, and to help young people with not be an exaggeration to say that the basic similar interests get started along the same path. groundwork for much of this study is to be I myself as a young man had the great good for- found in Tate’s article “p-Divisible Groups”. tune to learn about algebraic numbers class This same period saw Tate formulating his fa- fields, and much more from E. Artin, who also mous conjectures, e.g., the about gave me a beautiful and important idea to work algebraic cycles, and saw the publication of his out as a Ph.D. thesis. I have always found it eas- article “ Endomorphisms of Abelian Varieties ier to learn from people than from books and over Finite Fields”, in which Tate manages to ac- papers and would like to thank the many fellow tually construct cycles given only cohomologi- mathematicians who over the years have taught cal information. me new things and suggested new ideas: David Beginning in the early 1970s, Tate published Mumford, , , J.-P. Serre, a series of papers in algebraic K-theory, more A. Grothendieck, and Barry Mazur, to name a few specifically in K2, and his 1976 paper, “Relations who have helped me the most. One of the best between K2 and Galois Cohomology”, is both el- ways to learn is to teach, and I want also to take egant and gets to the essential nature of K2 of this opportunity to thank my many students a number field. for the pleasure and stimulation working with In this same period, en passant, Tate’s “Rigid them has given me. A lifetime of mathematical Analytic Spaces” published in the Inventiones activity is a reward in itself, but it is nice to have (1971) initiated a completely different subject. recognition for it from peers. My warm thanks But it was no surprise that Tate might have in- to the Steele Prize committee for selecting my terests in that direction in view of his celebrated career from the many equally or more deserv- p-adic uniformization theory for elliptic curves ing ones for this award. and abelian varieties and his theory of what is Biographical Sketch now called the Tate curve. For a fine account of John T. Tate was born in in 1925. this, see Tate’s “A review of non-archimedean el- After three years in the U.S. Navy, he received liptic functions” published (pp. 162–184) in the his B.A. from Harvard University in 1946 and his volume Elliptic Curves, Modular Forms, and Fer- Ph.D. from Princeton University in 1950. He was mat’s Last Theorem (1995) by International Press. an instructor at Princeton from 1950 to 1954, The early 1980s saw Tate’s interest turning when he went to Harvard. After thirty-six years to the intriguing Conjectures of Stark on the at Harvard, he went to the University of Texas connection between L-functions at the point at Austin, where he is currently a professor and s =0 and logarithms of algebraic units (or de- the Sid Richardson Regents Chairholder. terminants of such logarithms). Tate pursued Professor Tate received a Sloan Foundation these conjectures also in the function field set- Fellowship (1957–1958) and a Guggenheim Fel- ting where one can understand the structure in- lowship (1965–1966). He has held visiting posi- volved with somewhat greater perspicuity and tions at , the University of make some progress. This work culminated in California at Berkeley, Institut des Hautes Études a book Tate wrote on the subject (1984). Scientifiques, Université de Paris at Orsay, Prince- Also in that decade, Tate was engaged in a ton University, and École Normale Supérieure. He close study of the Classical Conjectures of Birch was an invited speaker at the International Con- and Swinnerton-Dyer, various refinements of gress of Mathematicians in 1962 in Stockholm those conjectures, and especially p-adic ana- and again in 1970 in Nice. In 1973, he presented logues of them (and concomitantly, a construc- the AMS Colloquium Lectures. tion and study of the p-adic sigma function). A member of the National Academy of Sci- In the 1990s Tate has written papers on non- ences and of the Académie des Sciences, Paris, commutative ring theory and, more specifically, Professor Tate is the recipient of the 1956 AMS is currently engaged in the construction and Cole Prize in Number Theory. study of interesting Sklyanin algebras. And beyond all this published work, Tate’s en- gagement with mathematics, through his great number of students and his extensive (and widely circulated) correspondence, has been continuous and intense. All this adds up to a magnificent ca- reer well worthy, in the opinion of this commit- tee, of the 1995 Leroy P. Steele career award. Response It is an honor and a pleasure for me to get this award in recognition of my efforts over a life- time to discover new relationships in number the-

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