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1995 Steele Prizes steele.qxp 4/27/98 3:29 PM Page 1288 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 Princeton University for the following ...for a research awarded in three categories: two papers in mathematical physics character- for a research paper of fun- ized by leaders of the field 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 quantum field theory, 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 mathematics 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 1288 NOTICES OF THE AMS VOLUME 42, NUMBER 11 steele.qxp 4/27/98 3:29 PM Page 1289 that operator algebras 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 probability theory in a grad- consciously remember it—these things can hap- uate course taught by Irving Segal 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 mathematician 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 Arthur Wightman’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. NOVEMBER 1995 NOTICES OF THE AMS 1289 steele.qxp 4/27/98 3:30 PM Page 1290 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 Rome, 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 New York in 1942, where children from Geor- both for working mathematicians 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 University of Chicago, 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 algebra. 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.
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