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Borcherds, Gowers, Kontsevich, and Mcmullen Receive Fields Medals

Borcherds, Gowers, Kontsevich, and Mcmullen Receive Fields Medals

comm-fields.qxp 9/16/98 10:55 AM Page 1358

Borcherds, Gowers, Kontsevich, and McMullen Receive Fields Medals

On August 18, 1998, four Fields Medals were pre- , William P. Thurston, and Shing-Tung sented at the Opening Ceremonies of the Interna- Yau (1983); Simon K. Donaldson, , and tional Congress of (ICM) in . Michael H. Freedman (1986); , The four medalists are: RICHARD E. BORCHERDS, WILLIAM Vaughan F. R. Jones, , and Edward , , and CURTIS T. Witten (1990); , Pierre-Louis Lions, MCMULLEN. Jean-Christophe Yoccoz, and (1994). At the 1924 Congress in Toronto a resolution The committee choosing the 1998 - was adopted that at each ICM two gold medals ists consisted of: John Ball (Oxford University), should be awarded to recognize outstanding math- John Coates (Cambridge University), J. J. Duister- ematical achievement. J. D. Fields, a Canadian maat (University of Utrecht), Michael H. Freedman who was secretary of the 1924 Con- (Microsoft Research), Jürg Fröhlich (Eidgenössische gress, later donated funds establishing the medals, Technische Hochschule, Zürich), Robert MacPher- which were named in his honor. Consistent with son (Institute for Advanced Study, Princeton), Yuri Fields’s wish that the awards recognize both ex- Manin (chair, Max-Planck-Institut für Mathematik, isting work and the promise of future achieve- ), Kyoji Saito (University of Kyoto), and ment, the medals are awarded to young math- (City University of Hong Kong). ematicians, where “young” has traditionally been interpreted to mean no more than forty years of age in the year of the Congress. In 1966 it was Richard Borcherds was born on November 29, agreed that, in light of the great expansion of 1959, in Cape Town, South Africa. He received his mathematical research, up to four medals could be undergraduate and doctoral degrees from the Uni- awarded at each ICM. Today the Fields Medal is versity of Cambridge. He has held various positions widely recognized as the world’s highest honor in at Cambridge and at the , . Berkeley. Currently he is on leave from Berkeley and Previous recipients are: Lars V. Ahlfors and is a Royal Society Research Professor at Cambridge. (1936); and Atle In 1992 he received a European Mathematical So- Selberg (1950); and Jean-Pierre ciety Prize at the First European Congress of Math- Serre (1954); Klaus F. Roth and René Thom (1958); ematicians in . He was an Invited Speaker at Lars Hörmander and John W. Milnor (1962); Michael the ICM in Zürich in 1994. F. Atiyah, Paul J. Cohen, , As a student of John H. Conway, Borcherds and Stephen Smale (1966); Alan Baker, Heisuke began his research career in finite theory. Hironaka, Sergei P. Novikov, and John G. Thomp- He has distinguished himself not only by utilizing son (1970); and David B. Mumford techniques and ideas from outside of finite group (1974); Pierre R. Deligne, Charles L. Fefferman, theory but also by producing results that have had Grigorii A. Margulis, and Daniel G. Quillen (1978); an impact in other areas. In the classification of fi-

1358 NOTICES OF THE AMS VOLUME 45, NUMBER 10 comm-fields.qxp 9/16/98 10:55 AM Page 1359 Photographs courtesy of the ICM, Berlin, 1998. Richard Borcherds William Timothy Gowers Maxim Kontsevich

nite simple groups, one of the most mysterious ob- mensional subspaces has an jects found was the . There are var- unconditional basis. An un- ious conjectures that attempt to connect the mon- conditional basis provides a ster to other parts of mathematics. Borcherds useful coordinatization of invented the notion of a vertex algebra and used the , guaranteeing it to solve the Conway-Norton conjecture, which many “symmetries” (auto- concerns the of the monster ). group (this theory is sometimes called “monstrous Gowers also produced an moonshine”). He used these results to generate example of a product formulae for certain modular and auto- that is not isomorphic to morphic forms. The first such formulae were found any of its hyperplanes, in the one-dimensional case by Euler and Jacobi, thereby solving the famous and the conventional wisdom in algebraic geome- Banach hyperspace prob- try was that such product formulae could not exist lem. He proved a “di- in higher dimensions. Borcherds’s work is also im- chotomy theorem”, which portant in , as it lays rigorous groundwork says that every Banach for in two dimensions. space has either a subspace that has an unconditional William Timothy Gowers basis, and therefore many William Timothy Gowers was born on November symmetries, or is such that 20, 1963, in Marlborough, England. He received his all of its subspaces have only trivial symmetries. This undergraduate and doctoral degrees from the Uni- Curtis T. McMullen versity of Cambridge, where he was a student of work solves in the affirma- Belá Bollobás. Gowers was a Research Fellow at tive the homogeneous space Trinity College, Cambridge, before spending four problem, one of the central problems in Banach years at University College, London. He was then space theory, which asks whether a homogeneous appointed as a Lecturer at Cambridge and a Fel- Banach space is a Hilbert space. A hallmark of low of Trinity College. Currently he holds the Rouse Gowers’s work is the way in which it combines tech- Ball Chair of Mathematics at Cambridge. He was niques of analysis with combinatorial arguments. an Invited Speaker at the ICM in Zürich in 1994. In His work in and combinatorial num- 1996 he received a European Mathematical Soci- ber theory includes results about Szeméredi’s ety Prize at the Second European Congress of Math- lemma and an improved proof of Szeméredi’s the- ematics in Budapest. orem on arithmetic progressions. Gowers works in the areas of Banach space the- ory and combinatorics. His main achievements are Maxim Kontsevich his solutions to a number of famous problems Maxim Kontsevich was born on August 25, 1964, first stated in the 1930s by . Gow- in . He received his doctoral degree from ers and B. Maurey exhibited in 1991 a Banach space the , under the direction of having the property that none of its infinite-di- Don B. Zagier. After holding a professorship at

NOVEMBER 1998 NOTICES OF THE AMS 1359 comm-fields.qxp 9/16/98 10:55 AM Page 1360

the University of California, Berkeley, he moved to Pc is hyperbolic. He also brought new ideas and in- his present position as professor at the Institut des sights from dynamical systems to the geometriza- Hautes Études Scientifiques in Bûres-sûr-Yvette, tion program for three- formulated by . In 1992 Kontsevich received a European 1982 Fields Medalist . McMullen Mathematical Society Prize at the First European has also worked with Sullivan on a “dictionary” be- Congress of Mathematics in Paris. He was a Plenary tween the theory of iterations of rational maps of Speaker at the ICM in 1994 in Zürich. the Riemann sphere and that of Kleinian groups. Kontsevich first received international atten- —Allyn Jackson tion for his doctoral thesis, in which he proved a conjecture of . This conjecture says that the generating for intersection num- bers on the moduli spaces of algebraic curves sat- isfies the Korteweg-de Vries equation. Also draw- Wiles Receives Special Award ing on ideas of Witten, Kontsevich produced a vast At the ICM Open- generalization of the Gauss for ing Ceremonies, knots. He then used this generalization and a new Andrew J. Wiles of notion of “graph ” to generate Vas- Princeton Univer- siliev knot invariants as well as invariants for three- sity received a manifolds. Kontsevich produced the first math- one-time Special ematical definition of the “number” of rational Tribute from the curves on Calabi-Yau manifolds, such as three-di- International mensional quintics, and gave an explicit formula Mathematical for this number. This work was crucial for later work in the area of mirror symmetry. Most re- Union in recogni- cently Kontsevich has established that any Poisson tion of his work admits a formal quantization and has pro- that led to the vided an explicit formula for the flat case. proof of Fermat’s Last Theorem. Because he is over forty years Curtis T. McMullen old, Wiles was not considered eligible for a Curtis T. McMullen was born on May 21, 1958, in Fields Medal. Instead of a gold medal he re- Berkeley, California. He received his undergradu- ceived the “IMU Silver Plaque”, or, as number ate degree in 1980 from and his theorist Don B. Zagier called it, a “Quantized doctoral degree in 1985 from . Fields Medal”. Wiles’s award also differed from His thesis advisor was . McMullen the Fields Medals in that no lecture was pre- has held positions at the Massachusetts Institute sented about his work. Instead, the next day of Technology, the Mathematical Sciences Research Wiles himself gave a special lecture entitled Institute, the Institute for Advanced Study, Prince- “Twenty Years of ”. ton University, and the University of California, In 1993 Wiles announced that he had proved Berkeley. At present he is a professor of math- Fermat’s Last Theorem. The ground-breaking ematics at Harvard. In 1991 he received the Salem research he did in order to produce the proof Prize. seemed likely to secure him a Fields Medal at McMullen has produced important results in the ICM in Zürich in 1994 until a gap appeared several areas of mathematics, including the theory in the proof. The gap was not repaired until of computation, dynamical systems, and three- after the Zürich Congress. With the proof com- manifolds. In his doctoral thesis he used dynam- plete, the general consensus was that Wiles had ical systems techniques to solve completely the done work of Fields Medal quality. This view question of whether there exist generally conver- was reinforced by the 3,000 people assembled gent algorithms for finding the zeros of polyno- for the ICM Opening Ceremonies, who gave mials of degree three or greater. Newton’s method Wiles a thundering round of applause longer converges for almost all quadratic polynomials than that given to any of the other awardees. and almost all initial points. McMullen exhibited —A. J. an analogous algorithm for degree three polyno- mials and proved that no such algorithm exists for degree four and higher. He has also made impor- tant strides toward solving one of the central con- jectures in one-dimensional dynamics: Are the hy- perbolic maps of degree d dense in all maps of degree d? McMullen proved that, given Pc : C C, 2 → Pc = z + c, if c is in a connected component of the Mandelbrot set that intersects the real axis, then

1360 NOTICES OF THE AMS VOLUME 45, NUMBER 10

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