World Scientific Series in 20th Century Mathematics - Vol. 5 ffields J)/[edaUists' jTectures

Editors Sir Michael Atiyah Trinity College, Cambridge, UK

Daniel Iagolnitzer Service de Physique Theorique CEA-Saclay, France

Vfe World Scientific Wl Sinqapore»Singapore * New Jersey • LondonLondon* • Hong Kong

• SINGAPORE UNIVERSITY PRESS NATIONAL UNIVERSITY OF SINGAPORE CONTENTS

Preface v

Recipients of Fields Medals vi

1936 L. V. AHLFORS Autobiography 3 Commentary on: Zur Theorie der Uberlagerungsflächen (1935) 8 Quasiconformal Mappings, Teichmüller Spaces, and Kleinian Groups 10

1950 L. SCHWARTZ The Work of L. Schwartz by H. Bohr 25 Biographical Notice 31 Calcul Infinitesimal Stochastique 33

1958 K. F. ROTH The Work of K. F. Roth by H. Davenport 53 Biographical Notice 59 Rational Approximations to Algebraic Numbers 60

RTHOM The Work of R. Thom by H. Hopf 67 Autobiography 71

1962 L. HÖRMANDER Hörmander's Work On Linear Differential Operators by L. Gärding 77 Autobiography 83 Looking forward from ICM 1962 86

1966 M. F. ATIYAH L'oeuvre de Michael F. Atiyah by H. Cartan 105 Biography 113 The Index of Elliptic Operators 115

\ viü Fields Medallists' Lectures

S. SM ALE Sur les Travaux de Stephen Smale by R. Thom 129 Biographical Notice 135 A Survey of Some Recent Developments in Differential Topology 142

1970 A. BAKER The Work of Alan Baker by P. Turän 157 Biography 161 Effective Methods in the Theory of Numbers 162 Effective Methods in Diophantine Problems 171 Effective Methods in Diophantine Problems. II. Comments 183 Effective Methods in the Theory of Numbers/Diophantine Problems 190

S. NOVIKOV The Work of Serge Novikov by M. F. Atiyah 195 Biography 199 Röle of Integrable Models in the Development of Mathematics 202

1974 D. MUMFORD The Work of David Mumford by J. Täte 219 Autobiography 225 Pattern Theory: A Unifying Perspective 226

1978 G. A. MARGULIS The Work of Gregori Aleksandrovitch Margulis by J. Tits 263 Biographical Notice 271 Oppenheim Conjecture 272

1982 A. CONNES The Work of Alain Connes by H. Araki 329 Biographical Notice 339 Brisure de Symetrie Spontanee et Geometrie du Point de vue Spectral 340

W. P. THURSTON The Work of W. Thurston by C. T. C. Wall 373 Contents ix

1986 S. K. DONALDSON The Work of Simon Donaldson by M. F. Atiyah 377 Biographical Notice 383 Remarks on Gange Theory, Complex Geometry and 4-Manifold Topology 384

M. H. FREEDMAN The Work of M. H. Freedman by J. Milnor 405 Biographical Notice 411 Betti Number Estimates for Nilpotent Groups 413

1990 V. F. R. JONES The Work of Vaughan F. R. Jones by J. S. Birman 435 Biographical Notice 447 A Polynomial Invariant for Knots via von Neumann Algebras 448 Index for Subfactors 459

S. MORI The Work of Shigefumi Mori by H. Hironaka 487 Biographical Notice 495 Birational Classification of Algebraic Threefolds 496

E. WITTEN The Work of E. Witten by L. D. Faddeev 511 The Work of Edward Witten by M. F. Atiyah 514 Biographical Notice 521 Geometry and Quantum Field Theory 523

1994 J. BOURGAIN The Work of Jean Bourgain by L. Caffarelli 537 Biographical Notice 541 Hamiltonian Methods in Nonlinear Evolution Equations 542

P. L. LIONS The Work of Pierre-Louis Lions by S. R. S. Varadhan 555 Biographical Notice 560 On Some Recent Methods for Nonlinear Partial Differential Equations 563 Fields Medallists' Lectures

J. C. YOCCOZ Presentation de Jean-Christophe Yoccoz by A. Douady 581 Recent Developments in Dynamics 588

E. I. ZELMANOV The Work of Efim Zelmanov by W. Feit 611 Biographical Notice 621 On the Restricted Burnside Problem 623