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2002 Steele Prizes 2002 Steele Prizes The 2002 Leroy P. Steele Prizes were awarded at the Notices, pages 1216–20, or on the AMS website 108th Annual Meeting of the AMS in San Diego in at http://www.ams.org/prizes-awards/. January 2002. The 2002 Steele Prizes were awarded to YITZHAK The Steele Prizes were established in 1970 in KATZNELSON for Mathematical Exposition, to MARK honor of George David Birkhoff, William Fogg GORESKY and ROBERT MACPHERSON for a Seminal Con- Osgood, and William Caspar Graustein. Osgood tribution to Research, and to MICHAEL ARTIN and was president of the AMS during 1905–06, and ELIAS STEIN for Lifetime Achievement. The text that Birkhoff served in that capacity during 1925–26. The follows presents, for each awardee, the selection prizes are endowed under the terms of a bequest committee’s citation, a brief biographical sketch, from Leroy P. Steele. Up to three prizes are awarded and the awardee’s response upon receiving the prize. each year in the following categories: (1) Mathematical Exposition: for a book or substantial Mathematical Exposition: survey or expository-research paper; (2) Seminal Yitzhak Katznelson Contribution to Research: for a paper, whether re- Citation cent or not, that has proved to be of fundamental Although the subject of harmonic analysis has or lasting importance in its field, or a model of im- gone through great advances since the sixties, portant research; and (3) Lifetime Achievement: Fourier analysis is still its heart and soul. Yitzhak for the cumulative influence of the total mathe- Katznelson’s book on harmonic analysis has with- matical work of the recipient, high level of research stood the test of time. Written in the sixties and over a period of time, particular influence on the revised later in the seventies, it is one of those development of a field, and influence on mathe- “classic” Dover paperbacks that has made the sub- matics through Ph.D. students. Each Steele Prize ject of harmonic analysis accessible to generations carries a cash award of $5,000. of mathematicians at all levels. The Steele Prizes are awarded by the AMS Coun- The book strikes the right balance between the cil acting on the recommendation of a selection concrete and the abstract, and the author has wisely committee. For the 2002 prizes, the members of chosen the most appropriate topics for inclusion. the selection committee were: M. S. Baouendi, The clear and concise exposition and the presence Sun-Yung A. Chang, Michael G. Crandall, Constan- of a large number of exercises make it an ideal tine M. Dafermos, Daniel J. Kleitman, Hugh L. source for anyone who wants to learn the basics Montgomery, Barry Simon, S. R. S. Varadhan (chair), of the subject. and Herbert S. Wilf. Biographical Sketch The list of previous recipients of the Steele Prize Yitzhak Katznelson was born in Jerusalem in 1934. may be found in the November 2001 issue of the He graduated from the Hebrew University with a 466 NOTICES OF THE AMS VOLUME 49, NUMBER 4 Yitzhak Katznelson Mark Goresky Robert MacPherson master’s degree in 1956 and obtained the Dr. és Sci. most concrete terms and allow as much general- degree from the University of Paris in 1959. ity and abstraction as needed for development, After a year as a lecturer at the University of methods, and solutions. California, Berkeley, and a few more at the Hebrew University, Yale University, and Stanford University, Seminal Contribution to Research: he settled in Jerusalem in 1966. Until 1988 he Mark Goresky and Robert MacPherson taught at the Hebrew University, while making Citation extended visits to Stanford and Paris. He is now a In two closely related papers, “Intersection professor of mathematics at Stanford University. homology theory”, Topology 19 (1980), no. 2, Katznelson’s mathematical interests include 135–62 (IH1) and “Intersection homology. II”, Invent. harmonic analysis, ergodic theory (and in Math. 72 (1983), no. 1, 77–129 (IH2), Mark Goresky particular its applications to combinatorics), and Robert MacPherson made a great breakthrough and differentiable dynamics. by discovering how Poincaré duality, which had Response been regarded as a quintessentially manifold What a pleasant surprise! phenomenon, could be effectively extended to I am especially gratified by the committee’s ap- many singular spaces. Viewed topologically, the proval of “the balance between the concrete and the key difficulty had been that Poincaré duality reflects abstract,” which was one of my main concerns the transversality property that holds within a while teaching the course and while developing manifold but which fails in more general spaces. the notes into a book. IH1 introduced “intersection chain complexes”, How should one look at things, and in what gen- which are the subcomplexes of usual chain com- erality? If a statement and its proof apply equally plexes consisting of those chains which satisfy a in an abstract setup, should it be introduced in the transversality condition with respect to the natural most general or the most familiar terms? strata of a space. More precisely, by introducing a When I came to Paris in 1956 I heard a rumor kind of measure, called a “perversity”, of the that the old way of doing mathematics was being amount of variation from transversality a chain replaced by a new, “abstract” fashion which was the would be allowed, Goresky and MacPherson only proper way of doing things. The rumor was actually introduced a parametrized family of spread mostly by younger students—typically intersection chain complexes. Each of these yielded hugging a freshly-purchased volume of Bourbaki— a corresponding sequence of intersection homol- but seemed confirmed also by the way some ogy groups, and these theories intermediated courses were taught. between homology and cohomology. Starting with As late as 1962, Kahane and Salem found the need methods of local piecewise-linear transversality to apologize (undoubtedly tongue-in-cheek) in the that had been developed by investigations of preface to their exquisite book Ensembles Parfaits M. Cohen, E. Akin, D. Stone, and C. McCrory, IH1 et Séries Trigonométriques for dealing with subject showed that its intersection homology theories matter that might be considered too concrete. were related to each other by a version of Poincaré The balance I tried to strike in the book—and I duality; in particular, the intersection homology believe that I was strongly influenced by Kahane theory which was positioned midway between and Salem—was to set up the subject matter in the homology and cohomology satisfied, when defined, APRIL 2002 NOTICES OF THE AMS 467 a self-duality, as was familiar for manifolds. This functions on the points X(F) of X over F. This is immediately yielded a signature invariant for many the basic ingredient in the geometrization of singular varieties, and that, in turn, was used in IH1 representation theory which has had remarkable to yield, in analogy with the Thom-Milnor treatment successes in recent years. of piecewise linear manifolds, rational character- 3) Paul Segal used the methods of Goresky and istic classes for many triangulated singular varieties. MacPherson and a cobordism theory of singular However, these characteristic classes of singular varieties to show that their rational characteristic varieties naturally were elements in homology classes could in many cases be lifted, after invert- rather than cohomology groups, a distinction which ing 2, to a KO-homology class. Intersection chain for singular varieties was significant. sheaves were extensively used in various collabo- The continuation paper, IH2, reformulated this rations of Cappell, Shaneson, and Weinberger which theory in a natural and powerful sheaf language. extended results of classical Browder-Novikov- This language, suggested by Deligne, gave local Sullivan-Wall surgery theory of manifolds to yield formulations of a version of Poincaré duality for topological classifications of many singular varieties, singular spaces in terms of a Verdier duality of which developed new invariants for singular vari- sheaves. Furthermore, IH2 presented beautiful eties and their transformation groups, which gave axiomatic characterizations of its intersection methods of computing the characteristic classes of chain sheaves. These were all the more valuable as singular varieties, and which related these to knot the achievement of duality for nonsingular spaces invariants. came at the cost of giving up the familiar functo- 4) In investigations of the geometrical combi- rial and homotopy properties that characterized natorics of convex polytopes, the intersection usual homology theories; in particular, intersection homology groups of their associated toric varieties homology theory is not a “homology theory” in have become a fundamental tool. This began with the sense of homotopy theory. R. Stanley’s investigations of the face vectors of poly- IH1 and IH2 made possible investigations across topes. A calculation of the Goresky-MacPherson a great spectrum of mathematics which further characteristic classes of toric varieties was used by extended key classical manifold phenomena and Cappell and Shaneson in obtaining an Euler- methods to singular varieties and used these to MacLaurin formula with remainder for lattice sums solve well-known problems. While it is impossible in polytopes. Recent works of MacPherson and to list all of these, a few important ones in 1) dif- T. Braden on flags of faces of polytopes used results ferential geometry, 2) algebraic geometry and on the intersection chain sheaves of toric varieties. representation theory, 3) geometrical topology, The already astonishing range of research areas and 4) geometrical combinatorics will be indicated. influenced by this seminal work continues to grow. 1) An immediate question was the relation of in- Biographical Sketch: Mark Goresky tersection homology theory to an analytic theory Mark Goresky received his B.Sc. from the Univer- of L2 differential forms and L2 cohomology on sity of British Columbia in 1971 and attended grad- suitable singular varieties with metrics that uate school at Brown University.
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