1997 Steele Prizes

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1997 Steele Prizes

Three Leroy P. Steele Prizes were awarded at the Steele Prize for Mathematical Exposition: 103rd Annual Meeting of the AMS in January in Anthony W. Knapp San Diego. These prizes were established in 1970 Citation in honor of George David Birkhoff, William Fogg For his book Representation Theory of Semi- Osgood, and William Caspar Graustein and are simple Groups (An overview based on examples), endowed under the terms of a bequest from Princeton University Press, 1986, a beautifully Leroy P. Steele. written book which starts from scratch but takes The Steele Prizes are awarded in three cate- the reader far into a highly developed subject. gories: for expository writing, for a research The motivation, which is consistently and artfully paper of fundamental and lasting importance, provided as the general theory unfolds, is a and for cumulative influence extending over a model of exposition. In addition, Anthony Knapp career, including the education of doctoral stu- has written other major texts in more recent dents. The current award is $4,000 in each cat- years, all outstanding expositions of important egory. and difficult material. The recipients of the 1997 Steele Prizes are Anthony W. Knapp for Mathematical Exposi- Biographical Sketch tion, Mikhael Gromov for a Seminal Contribu- Anthony W. Knapp is the author of seven books. tion to Research, and Ralph S. Phillips for Life- His first book, Denumerable Markov Chains, time Achievement. was written jointly with John Kemeny and Lau- The Steele Prizes are awarded by the AMS rie Snell and appeared in 1966. He has written Council acting through a selection committee one book about elliptic curves, and the others whose members at the time of these selections are on Lie groups and representation theory, were Richard Askey, Ingrid Daubechies, Eugene the most recent one being Lie Groups beyond an Dynkin, Ciprian Foias, H. Blaine Lawson, Andrew Introduction, published in 1996. His book with J. Majda, Louis Nirenberg, Gary M. Seitz, and David Vogan entitled Cohomological Induction John T. Tate. and Unitary Representations was designated the The text that follows contains, for each award, best mathematics book in 1995 by the Profes- the committee’s citation, a brief biographical sional and Scholarly Publishing Division of the sketch of the recipient, and the recipient’s re- Association of American Publishers. sponse upon receiving the award. Knapp was born in 1941, was an undergrad- uate at Dartmouth, and received his Ph.D. from Princeton in 1965, with Salomon Bochner as the- sis advisor. He was a C. L. E. Moore Instructor at

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MIT for two years and joined the faculty of Cor- theorems and to provide nell University in 1967. Since 1986 he has been a guide to further read- professor of mathematics at SUNY at Stony ing so that a person could Brook. selectively go more He has been a member of the Institute for Ad- deeply into an aspect of vanced Study in Princeton for three one-year the subject at will. terms and has had visiting positions for one Several things made year at MIT; for one semester each at Princeton the writing of such a book University, Rice University, and Université Paris possible. One was the VII; and for shorter intervals at places in the readability of Harish- United States, Canada, France, Italy, Sweden, Chandra’s papers and the India, China, and Australia. He was an invited presence of several entry speaker at the International Congress of Math- points to their study. An- ematicians in Vancouver in 1974 and was a other was the encourage- Guggenheim Fellow in 1982–83. ment of editor Robert Response Langlands. A third was It is a great honor to be awarded the Steele Prize that the literature in the subject was in good order, for exposition and to have my book associated Anthony W. Knapp with the extraordinary books that have been the any mistakes not having subjects of this award in past years. I thank the spread into paper after Committee for its choice, and I thank the AMS paper. The opportunity to do a serious experi- for long recognizing that high-quality exposition ment with this writing style came with a se- has an important role to play in the advance of mester-length course at Université Paris VII in mathematics. Writing a book of this level and 1982. The handouts of notes for that course be- length takes large blocks of time and requires came a preliminary edition of the book, and the active support from one’s immediate family; I writing of the full text was complete two years thank my wife and two children for providing later. that support. As late as 1981, the field of representation theory, particularly the representation theory of semisimple groups, was notori- Steele Prize for a Seminal Contribution to ously difficult to enter. Tackling two thousand Research: Mikhael Gromov pages of Harish-Chandra was not for the faint- Citation hearted. One needed to learn from mentors in For his paper “Pseudo-holomorphic curves in order to see what was beautiful about the sub- symplectic manifolds”, Inventiones Math. 82 ject, to get through the background, and to find (1985), 307–347, which revolutionized the sub- out where the subject might be headed. About ject of symplectic geometry and topology and is that time, after having given several short series central to much current research activity, in- of lectures on aspects of representation theory cluding quantum cohomology and mirror sym- to nonexperts, I began to look for a way for metry. more mathematicians to gain some apprecia- Biographical Sketch tion for the field without help from a specialist. Gromov was born in 1943 in Boksitogorsk, Rus- That way was in fact already what I was doing sia. He received his Ph.D. in 1969 and his D.Sc. in my lecture series—explaining things often in in 1973 from the University of Leningrad. After the context of examples—and what I was some- holding positions at the University of Leningrad, times witnessing in the lectures of others. Many the State University of New York at Stony Brook, times with theorems about semisimple groups, and the Université de Paris, he moved to Insti- there is one example where one can see all the tut des Haute Études Scientifiques, where he is important ideas without being distracted by a permanent fellow (1982–). For five years he also technical details. I remember lectures by G. D. held the position of professor of mathematics Mostow, for example, where he would cut at the University of Maryland, College Park. He through technicalities right away by defining a is now a professor at the Courant Institute of semisimple Lie group to be a connected closed Mathematical Sciences. Gromov received the subgroup of real or complex matrices stable Moscow Mathematical Society Prize (1971), the under conjugate transpose and having finite AMS Oswald Veblen Prize in Geometry (1971), the center. Mostow’s definition does not cover all Elie Cartan Prize of the French Academy of Sci- cases, but it does cover enough cases to make a ences (1984), the Prix UAP (1989), and the Wolf start at appreciating the subject. It was a ques- Prize in Mathematics (1993). He also holds an tion of weaving such descriptions into a coher- honorary doctorate from the University of ent book. Despite the use of examples in this way, Geneva. He is a foreign member of the U.S. Na- I felt that it was important to state precise tional Academy of Sciences, the French Academy

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of Sciences, and the I still cannot forgive him for this. (Alas, preju- American Academy of dice does not pay in science.) McDuff started the Arts and Sciences. systematic hunt for them which goes on till present day. And what goes on today goes beyond Response these lines and the pen behind them. I saw the light when struggling with Pogo- Steele Prize for Lifetime Achievement: relov’s proof of rigidity Ralph S. Phillips of convex surfaces where he appeals to the Bers- Citation Vekua theory of quasi- Ralph Phillips is one of the outstanding ana- analytic functions. There lysts of our time. His early work was in functional was nothing seemingly analysis: his beautiful theorem on the relation complex-analytic in the between the spectrum of a semigroup and its in- linearized system written finitesimal generator is striking as well as very down by Pogorelov, and useful in the study of PDEs. His extension the- then it struck me that ory for dissipative linear operators predated the every first order elliptic interpolation approach to operator theory and Mikhael Gromov linear or quasilinear sys- robust control. He made major contributions to tem of two equations in acoustical scattering theory in his joint work two variables has the with Peter Lax, proving remarkable results on same principal symbol as local energy decay and the connections between Cauchy-Riemann and poles of the scattering matrix and the analytic then the solutions appear properties of the resolvent. He later extended this as (pseudo) holomorphic work to a spectral theory for the automorphic curves for the almost Laplace operator, relying on the Radon transform complex structure de- on horospheres to avoid Eisenstein series. In fined by the field of the the last fifteen years, Ralph Phillips has done bril- principal symbols. Now liant work, in collaboration with others, on spec- the surface rigidity triv- tral theory for the Laplacian on symmetric ially followed from posi- spaces, on the existence and stability of cusp tivity of the intersections forms for general noncompact quotients of the of holomorphic curves. hyperbolic plane, on the explicit construction of What fascinated me even sparse optimal expander graphs, and on the more was the familiar structure of families of isospectral sets in two web of algebraic curves in dimensions (the collection of drums that sound a surface emerging in its the same). full beauty in the softish Biographical Sketch environment of general Ralph S. Phillips was born on June 23, 1913, in (nonintegrable!) almost Oakland, California. He received his A.B. degree Ralph S. Phillips complex structures. (In- from the University of California, Los Angeles, tegrability had always in 1935 and his Ph.D. from the University of made me feel claustrophobic.) And my mind Michigan in 1939. He was an instructor at the was ready for the miracle; Donaldson’s ideas University of Washington (1940–41) and Har- were in the air. So I tried to replay Yang-Mills on vard University (1941–42) before becoming the my holomorphic curves (strings?) and reluc- leader of a research group at the Radiation Lab- tantly abandoned the idea, being convinced by oratory at the Massachusetts Institute of Tech- Pierre Deligne that the area of curves cannot be nology. In 1946 he became an assistant profes- controlled without a symplectic structure. Every- sor at New York University, and the following thing went smoothly with the symplectic struc- year he moved to the University of Southern ture, and I even came to understand the defini- California. He took a position as professor at tion of quasianalytic functions and of the UCLA in 1958 and in 1960 moved to Stanford nonlinear Riemann-mapping theorem of University, where he is currently a professor. Schapiro-Lavrentiev (albeit I am still unable to Professor Phillips was a Rackham Fellow at the read a single line of this style of analysis). University of Michigan while he was a doctoral I was happy to see my friends using holo- student there. He was a member of the Institute morphic curves immediately after birth: Eliash- for Advanced Study (1939–40 and 1951–52) and berg, Floer, McDuff. Eliashberg came across them was a Guggenheim Fellow (1954 and 1974). He independently in the contact framework but was was elected to the American Academy of Arts unable to publish (staying in the USSR). Floer has and Sciences in 1971. In 1977 he was the Robert morsified them by breaking the symmetry, and Grimmett Professor of Mathematics at Stanford.

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Response I am both elated and surprised to have been American Mathematical Society chosen as a recipient of this prize, and I am very grateful to all those who made it happen. Since Algebra and Algebraic Geometry this prize is for lifetime achievement, let me

briefly sketch my mathematical history. I started 234 ASTÉRISQUE Espaces out in functional analysis, and traces of this 1996 field can be found in all of my work. My pursuit Symétriques de of mathematics was interrupted for five years ESPACES SYMETRIQUES´ during World War II while I was at the MIT Ra- DE DRINFELD Drinfeld diation Lab. After this I worked my way back into Alain GENESTIER Alain Genestier mathematics by mastering Einar Hille’s book on D’aprés Drinfeld, l’espace symétrique functional analysis and semigroups. I wrote sev- p-adique d (ou plus exactement, d nr nr eral papers on semigroups of linear operators ou est l'Hensélisé strict SOCIÉTÉ MATHÉMATIQUE DE FRANCE OO O de l’anneauΩb de valuation discrête en and was invited by Hille to coauthor the revised O sonΩb pointb b fermé)b représente le problème edition of his book. Starting in 1961 I began a de modules des D-modules formels spéciaux O very fruitful twenty-year collaboration with Peter munis d’une rigidification convenable. Dans ce Lax on scattering theory, first on the acoustic travail, nous présenterons une autre approche de equation in Euclidean spaces and later on the ce résultat. Celle-ci ne sera valable que lorsque l’anneau de base est d’égale caractérisque, wave equation for automorphic functions in hy- O perbolic spaces. This automorphic function re- mais nous permettra d’obtenir une description locale du D-module formel universel. Toujours search led to a very productive ten-year collab- O dans le cas où l’anneau de base est d’égale O oration with Peter Sarnak on problems related caractéristique, nous nous intéresserons aussi to number theory and geometry. Finally I would au revêtement de Drinfeld d , pour lequel nous like to take advantage of this opportunity to construirons un analogue de l’accouplement present a list of what I consider to be the ten de Weil. Σ most insightful of my papers; they are not nec- Astérisque, Number 234; 1996; 124 pages; Softcover; List $27; essarily the most influential. Individual AMS member $24; Order code AST/234NA

References Introduction à la [1] (with S. Bochner) Absolutely convergent Fourier ex- Théorie de Hodge pansions for non-commutative normed rings, Ann. José Bertin, Jean-Pierre Demailly, INTRODUCTION À LA of Math. 43 (1942), 409–418. THÉORIE DE HODGE Luc Illusie, and Chris Peters

[2] Spectral theory for semigroups of linear operators, José Bertin, Jean-Pierre Demailly, Luc Illusie, Chris Peters Le présent ouvrage développe un Trans. Amer. Math. Soc. 71 (1951), 393–415. certain nombre d’éléments [3] The extension of dual subspaces invariant under fondamentaux de la théorie de Hodge. an algebra, Proc. Internat. Sympos. on Linear Panoramas et Synthèses Il est destiné principalement aux 1996 Spaces, Jerusalem Academic Press and Pergamon Numéro 3 étudiants et chercheurs non ´´ ´ Press, 1961, pp. 366–398. SOCIETE MATHEMATIQUE DE FRANCE spécialistes du sujet, qui souhaitent se [4] (with P. Lax) Scattering theory, Academic Press, familiariser en profondeur avec celui-ci et se New York, 1967. faire une idée de l´état actuel de la recherche. Le [5] (with P. Lax) The acoustic equation, J. Funct. Anal. texte comporte trois parties consacrées à des (1967), 37–83. aspects variés et complémentaires de la théorie: 2 [6] (with P. Lax) Decaying modes for the wave equa- aspects analytiques (méhodes L ), algébriques tion in the exterior of an obstacle, Comm. Pure (utilisation de la caractéristique p), et enﬁn Appl. Math. 22 (1969), 737–787. applications à la géométrie algébrique au travers [7] (with P. Lax) A local Paley-Wiener theorem for the de l’étude des variations de structure de Hodge et des conjectures de symétrie miroir pour les Radon transform of L2 functions in a non-Euclid- variétés de Calabi-Yau. ean setting, Comm. Pure Appl. Math. 35 (1982), 531–554. Panoramas et Synthèses, Number 3; 1996; 267 pages; Softcover; List $40; Individual AMS member $36; Order [8] (with P. Lax) Translation representations for au- code PASY/3NA tomorphic solutions of the wave equation in non- Titles in this series are published by the Société Euclidean spaces, I, II, III, and IV, Comm. Pure Mathématique de France and distributed by the AMS in Appl. Math. 37 (1984), 303–328; 37 (1984), the United States, Canada, and Mexico. Orders from other 779–813; 38 (1985), 179–207; 45 (1992), 179–201. countries should be sent to the SMF, Maison de la SMF, B.P. 67, 13274 Marseille cedex 09, France, or to Institut [9] (with P. Sarnak) On cusp forms for co-finite sub- Henri Poincaré, 11 rue Pierre et Marie Curie, 75231 Paris groups, Invent. Math. 80 (1985), 339–364. cedex 05, France. Members of the SMF receive a 30% [10] (with B. Osgood and P. Sarnak) Moduli space, discount from list. heights and isospectral sets of plane domains, Ann. All prices subject to change. Charges for delivery are $3.00 per order. For air delivery of Math. 129 (1989), 293–362. outside of the continental U. S., please include $6.50 per item. Prepayment required. Order from: American Mathematical Society, P. O. Box 5904, Boston, MA 02206-5904. For credit card orders, fax (401) 331-3842 or call toll free 800-321-4AMS (4267) in the U. S. and Canada, (401) 455-4000 worldwide. Or place your order through the AMS bookstore at http://www.ams.org/bookstore/. Residents of Canada, please include 7% GST.

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