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2004 Steele Prizes 2004 Steele Prizes The 2004 Leroy P. Steele Prizes were awarded at the The 2004 Steele Prizes were awarded to JOHN W. 110th Annual Meeting of the AMS in Phoenix in MILNOR for Mathematical Exposition, to LAWRENCE C. January 2004. EVANS and NICOLAI V. KRYLOV for a Seminal Contri- The Steele Prizes were established in 1970 in honor bution to Research, and to CATHLEEN SYNGE MORAWETZ of George David Birkhoff, William Fogg Osgood, and for Lifetime Achievement. The text that follows William Caspar Graustein. Osgood was president of presents, for each awardee, the selection commit- the AMS during 1905–06, and Birkhoff served in that tee’s citation, a brief biographical sketch, and the capacity during 1925–26. The prizes are endowed awardee’s response upon receiving the prize. under the terms of a bequest from Leroy P. Steele. Up to three prizes are awarded each year in the follow- Mathematical Exposition: John W. Milnor ing categories: (1) Lifetime Achievement: for the Citation cumulative influence of the total mathematical work The Leroy P. Steele Prize for Mathematical Exposi- of the recipient, high level of research over a period tion is awarded to John W. Milnor in recognition of time, particular influence on the development of of a lifetime of expository contributions ranging a field, and influence on mathematics through Ph.D. across a wide spectrum of disciplines including students; (2) Mathematical Exposition: for a book topology, symmetric bilinear forms, characteristic or substantial survey or expository-research paper; classes, Morse theory, game theory, algebraic K- (3) Seminal Contribution to Research (limited for theory, iterated rational maps…and the list goes on. 2004 to analysis): for a paper, whether recent or not, The phrase “sublime elegance” is rarely associated that has proved to be of fundamental or lasting im- with mathematical exposition, but it applies to all portance in its field or a model of important research. of Milnor’s writings, whether they be research or Each Steele Prize carries a cash award of $5,000. expository. Reading his books, one is struck with The Steele Prizes are awarded by the AMS Council the ease with which the subject is unfolding, and acting on the recommendation of a selection it only becomes apparent after reflection that this committee. For the 2004 prizes the members of ease is the mark of a master. Improvement of the selection committee were: M. Salah Baouendi, Milnor’s treatments often seems impossible. Andreas R. Blass, Sun-Yung Alice Chang, Michael G. A portion of Kauffman’s review of Symmetric Crandall (chair), Craig L. Huneke, Daniel J. Kleitman, Bilinear Forms by Milnor and Husemoller conveys Tsit-Yuen Lam, Robert D. MacPherson, and Lou P. the beauty evident in all of Milnor’s expository Van den Dries. work: “…Appendix 4, where this result is proved, The list of previous recipients of the Steele Prize is alone worth the price of the book. It contains may be found in the November 2003 issue of the Milnor’s proof of a Gauss sum formula (due to Notices, pages 1294–8, or on the World Wide Web, R. J. Milgram) that uses elegant combinatorics and http://www.ams.org/prizes-awards. Fourier analysis to produce an argument whose APRIL 2004 NOTICES OF THE AMS 421 John W. Milnor Lawrence C. Evans Nicolai V. Krylov corollaries include the divisibility theorem, the Response law of quadratic reciprocity and its equivalent It is a great pleasure to receive this award, and I in the language of forms over Z: the Weil reci- certainly want to thank the members of the Selec- procity theorem. The proof is short, beautiful, and tion Committee for their consideration. It is of mysterious.” course also a tribute to my many coauthors: let me Milnor’s many expository contributions to the mention Dale Husemoller, Larry Siebenmann, mathematical literature have influenced more than Jonathan Sondow, Mike Spivak, Jim Stasheff, and one generation of mathematicians. Moreover, the Robert Wells. examples that they provide have set a standard of I have always suspected that the key to the most clarity, elegance, and beauty for which every math- interesting exposition is the choice of a subject that ematician should strive. the author doesn’t understand too well. I have the Biographical Sketch unfortunate difficulty that it is almost impossible for me to understand a complicated argument unless John Milnor was born in Orange, New Jersey, in I try to write it down. Over the years I have run into 1931. He spent his undergraduate and graduate stu- a great many difficult bits of mathematics, and thus dent years at Princeton, working on knot theory I keep finding myself writing things down. (And also under the supervision of Ralph Fox, and also dab- rewriting, since I never get things right the first bling in game theory with his fellow students John few times. Years ago, I was the despair of secretaries Nash and Lloyd Shapley. However, like his mathe- who would produce beautifully typed manuscripts, matical grandfather, Solomon Lefschetz, he had only to have them repeatedly cut, pasted, and great difficulty sticking to one subject. Under the scribbled over. Computers have eliminated this inspiration of Norman Steenrod and later John particular problem, but it still makes life difficult Moore, he branched out into algebraic and differ- for coauthors.) ential topology. This led to problems in pure al- I am very happy to report that as mathematics gebra, including algebraic K-theory and the study keeps growing, there are more and more subjects of quadratic forms. More recently, conversations that I have to fight to understand. with William Thurston and Adrien Douady led to studies in real and complex dynamical systems, Seminal Contribution to Research: which have occupied him for the last twenty years. Lawrence C. Evans and Nicolai V. Krylov But he is still restless: one current activity is an Citation attempted exposition of problems of complexity The Steele Prize for Seminal Research is awarded in the life sciences. to Lawrence C. Evans and Nicolai V. Krylov for the After many years at Princeton at the university “Evans-Krylov theorem” as first established in the and also at the Institute for Advanced Study, and papers: after shorter stays at the University of California, Lawrence C. Evans, “Classical solutions of fully Los Angeles, and the Massachusetts Institute of nonlinear convex, second order elliptic equations”, Technology, Milnor moved to the State University Communications in Pure and Applied Mathematics of New York at Stony Brook, where he has been the 35 (1982), no. 3, 333–363; and director of the Institute for Mathematical Sciences N. V. Krylov, “Boundedly inhomogeneous ellip- since 1989. tic and parabolic equations”, Izvestiya Akad. Nauk 422 NOTICES OF THE AMS VOLUME 51, NUMBER 4 SSSR, ser. mat. 46 (1982), no. 3, 487–523; and trans- I was not especially afraid to look at so-called “fully lated in Mathematics of the USSR, Izvestiya 20 nonlinear” elliptic equations in the late 1970s and (1983), no. 3, 459–492. early 1980s. Fully nonlinear elliptic equations are of interest These are important PDEs, examples of which in many subjects, including the theory of controlled are the Monge-Ampère equation and Hamilton- diffusion processes and differential geometry. It is Jacobi-Bellman equations in stochastic optimal therefore of great interest to understand when these control theory. And they are really, really nonlin- equations have classical solutions. The first results ear. But their solutions satisfy maximum principles, of any generality exhibiting classical solutions of the and this was a clue. It turns out that (i) when the subclass of uniformly elliptic equations under suit- nonlinearity is convex, we can get “one-sided” con- able convexity conditions are due to the recipients trol on second derivatives; and that then (ii) the in the cited works. These authors, independently PDE itself provides a functional relationship among and with different arguments, established the Hölder the various second derivatives, yielding thereby continuity of second derivatives in the interior, via “two-sided” control. (Earlier Calabi had derived a priori estimates, a result now known as the Evans- third derivative bounds for the Monge-Ampère Krylov theorem. The Evans-Krylov theorem was equation, and Brezis and I had treated the very both a capstone on fundamental contributions of special case of the maximum of two linear elliptic the recipients and others and a harbinger of things operators.) to follow from the community. All success in mathematics turns largely upon While the Steele Prize for Seminal Research is persistence and luck; and while I can take some explicitly awarded for the named works, it is noted credit for the persistence, the luck was, well, luck— that both recipients have made a variety of distin- chiefly in that, quite unknown to me, one N. V. guished contributions to the theory of nonlinear Krylov in the Soviet Union had turned his attention partial differential equations. to these same problems at about the same time. Biographical Sketch: Lawrence C. Evans And Nick’s contributions to the subject have been Lawrence C. Evans was born November 1, 1949, in extraordinary, including not only the interior Hölder Atlanta, Georgia. He received his B.A. from Van- second derivative estimates, for which indepen- derbilt University in 1971 and his Ph.D. from the dent discovery we are being honored, but also his University of California, Los Angeles, in 1975; his previous, and great, work with Mikhail Safonov on advisor at UCLA was M. G. Crandall. Evans held Hölder bounds and the Harnack inequality for non- positions at the University of Kentucky from 1975 divergence structure second-order elliptic equa- to 1980, at the University of Maryland from 1980 tions with discontinuous coefficients.
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