Progress in Mathematics Volume 120

Series Editors J. Oesterle A. Weinstein First European Congress of Mathematics Paris, July 6-10, 1992 Volume II Invited Lectures (Part 2) A. Joseph F. Mignot F. Murat B. Prum R. Rentschler Editors

Birkhauser Verlag Basel . Boston . Berlin Editors: Fran<;ois Murat Anthony Joseph Laboratoire d'Analyse Numerique Laboratoire de MatMmatiques Universite Pierre et Marie Curie Fondamentales 4, place Jussieu Universite Pierre et Marie Curie 75252 Paris Cedex 05, France 4, place Jussieu 75252 Paris Cedex 05, France Bernard Prum and Laboratoire de Statistique Medicale The Weizmann Institute of Science Universite Paris V Rehovot 76100, Israel 45, rue des Saints Peres 75270 Paris Cedex 06, France Fulbert Mignot MatMmatiques Rudolf Rentschler Batiment 425 Laboratoire de MatMmatiques Universite de Paris-Sud Fondamentales 91405 Orsay Cedex, France Universite Pierre et Marie Curie 4, place Jussieu 75252 Paris Cedex 05, France

A CIP catalogue record for this book is available from the Library of Congress, Washington D. c., USA

Deutsche Bibliothek Cataloging-in-Publication Data

European Congress of Mathematics <1, 1992, Paris>: First European Congress of Mathematics: Paris, July 6--10, 1992/ A. Joseph ... ed. - Basel ; Boston ; Berlin : Birkhauser. NE: Joseph, Anthony [Hrsg.]; HST Vol. 2. Invited lectures. - Pt. 2. (Progress in mathematics ; Vol. 120)

NE:GT

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© 1994 Birkhauser Verlag, P.O. Box 133, CH-4010 Basel Softcover reprint of the hardcover 1st edition 1994 Printed on acid-free paper produced of chlorine-free pulp Layout, design, typesetting by TEXniques, Inc.

ISBN-13: 978-3-0348-9912-3 e-ISBN-13: 978-3-0348-9112-7 001: 10.1007/978-3-0348-9112-7

987654321 Table of Contents of Volume I Invited Lectures (Part 1)

Preface · xiii Lists of Committees · xv List of Sponsors · xvi Lists of Lectures and Round Tables · xix Opening Ceremony Discours de Fulbert Mignot xxiii Discours d'Henri Cartan . XXIV

Address of Friedrich Hirzebruch XXV Discours de Vincent Courtillot xxix Discours d'Hubert Curien xxxi Closing Ceremony

Address of Fulbert Mignot · XXXIV

Address of Friedrich Hirzebruch · XXXV Prizes Ceremony

Discours de Jacques Chirac · XXXVI

Discours de Max Karoubi · XXXIX Prizes Awarded by the City of Paris · . xli

Plenary Lectures

The Vassiliev Theory of Discriminants and Knots V.!. Arnold ...... 3 Transparent Proofs and Limits to Approximation Laszl6 Babai ...... 31 Poisson Algebraic Groups and Representations of Quantum Groups at Roots of 1 Corrado De Concini...... · 93 Gauge Theory and Four-Manifold Topology S. K. Donaldson ...... 121 Spectral Theory and Geometry Werner Muller ...... 153 Pattern Theory: A Unifying Perspective David Mumford...... 187 vi Table of Contents of Volume 1

Brownian Motion and Obstacles Alain-Sol Sznitman ...... 225 Geometric Quantization and Equivariant Cohomology Michele Vergne ...... 249

Parallel Lectures

The Power of Exponentiation in Arithmetic Zojia Adamowicz ...... 299 Subspace Arrangements Anders Bjorner. . . . 321 Optimal Recovery of Functions and Integrals Borislav Bojanov ...... 371 Existence globale et diffusion pour les modeles discrets de la cinetique des gaz Jean-Michel Bony...... 391 Sporadic Groups and String Theory Richard E. Borcherds ...... 411 A Harmonic Analysis Approach to Problems in Nonlinear Partial Differential Equations J. Bourgain ...... 423 (Some) Old and New Results on Algebraic Surfaces Fabrizio Catanese ...... 445 Evidence for a Cohomological Approach to Analytic Number Theory Christopher Deninger ...... 491

Cauchy~Riemann Operators, Self-Duality, and the Spectral Flow Stamatis Dostoglou and Dietmar A. Salamon 511

Index of Authors of Volumes I and II 547 Table of Contents of Volume II Invited Lectures (Part 2)

Parallel Lectures (continued)

Martingales, Arbitrage, and Portfolio Choice Darrell Duffie ...... 3 Mathematical Aspects of the Quantum Hall Effect Jurg Frohlich ...... 23 Analytic and Geometric Aspects of Variational Problems for Vector Valued Mappings M. Giaquinta...... 49 Harmonic Measures for Leafwise Elliptic Operators Along Foliations Ursula Hamenstiidt ...... 73 Feynman Diagrams and Low-Dimensional Topology Maxim Kontsevich ...... 97 KAM-Theory for Partial Differential Equations S.B. Kuksin ...... 123 Paradoxical Decompositions: A Survey of Recent Results Miklos Laczkovich...... 159 A Path-Valued Markov Process and its Connections with Partial Differential Equations Jean-Franr;ois Le Gall ...... 185 The Cyclotomic Trace in Algebraic K -Theory Ib Madsen ...... 213 Algebraic K-Theory and Galois Cohomology Alexander S. Merkurjev ...... 243 Values of L-Functions and p-adic Cohomology Jan Nekovrii' ...... 257 Mantles, Trains and Representations of Infinite Dimensional Groups Yurii A. Neretin ...... 293 The Evolutionary Dynamics of HIV Infections Marlin A. Nowak ...... 311 On the Enumeration of Algebraic Curves - from Circles to Instantons Ragni Piene ...... 327 viii Table of Contents of Volume II

Mathematical Aspects of Domain Decomposition Methods Alfio Quarteroni ...... 355 Paths in Graphs and Curves on Surfaces Alexander Schrijver ...... 381 Function Estimation and Functional Data Analysis Bernard W. Silverman...... 407 Algebra and Complexity V. Strassen ..... 429 Generalizations of Fuchsian and Kleinian Groups Pekka Tukia ...... 447 Properties of Embedded Lagrange Manifolds Claude Viterbo ...... 463 Alternative Entropies in Operator Algebras Dan Voiculescu ...... 475 Algebraic K-Theory and Functional Analysis Mariusz Wodzicki ...... 485 Values of Zeta Functions and Their Applications Don Zagier ...... 497

Index of Authors of Volumes I and II 513 Table of Contents of Volume III Round Tables

Preface v Round Table A: Mathematics and the General Public Jean-Pierre Kahane, Michele Chouchan . 1 Appendix: Replies to a Questionnaire Jean-Michel Kantor ...... 23 Round Table B: Women and Mathematics Eva Bayer-Fluckiger...... 37 Round Table C: Mathematics and Educational Policy Jacques Camus ...... 75 Introduction ...... 75 Mathematics in the School Curricula in Flemish Speaking Belgium W. Dewilde ...... 80 Mathematics in the French Educational System Jacques Camus ...... 89 The Role of Mathematics in Educational Policies in Germany Paul Bungartz, Rudolf Strasser ...... " 101 Italy: Mathematics in Higher Education Vinicio Villani ...... 110 Mathematics and Educational Policy (United Kingdom) Timothy Porter ...... 115 Round Table D: Let's Cultivate Mathematics! Yves Chevallard, Andre Rouchier 129 Introductory Remarks Yves Chevallard 129 Institutional Dimension in the Definition of Cultural Strategies in Mathematics Andre Rouchier ...... 131 Television Mathematics: Cultivation or Distortion? Celia Hoyles ...... 134 Mathematicians and the Need for Critical Self-Reflection Christine Keitel ...... 138 Mathematics in its Historical Context Lucia Grugnetti ...... 145 Ecology of Mathematical Knowledge: An Alternative Vision of the Popularization of Mathematics Juan D. Godino...... 150 x Table of Contents of Volume III

Round Table E: Mathematical Europe, Myth or Historical Reality? Catherine Goldstein, James Ritter ...... 157 The Creation of (the History of) Algebra Giovanna Cifoletti...... 161 The Perenniality of Programmes Henk Bos ...... 165 Mathematical Europe from Outside Europe Karine Chemla ...... 168 At the Beginning of our World: The Nineteenth Century Helene Gispert ...... 172 Centers and Peripheries Eduardo Ortiz 176 Conclusions 180

Round Table F: Philosophie des mathematiques : pourquoi ? comment ? H ourya Sinaceur ...... 183 Formes, operations, objets en mathematiques et en philosophie Gilles-Gaston Granger ...... ,...... 185 Le platonisme transcendantal et Ie probleme de l'objectivite Jean Petitot 192 Reflexions on Formalism and Reductionism in Logic and Computer Science Giuseppe Longo ...... 202 Pourquoi les mathematiques sont-elles difficiles ? Jacques Dubucs ...... 210 Le geste bien plutot que la regIe Gilles Chatelet 217 Discussion 223

Round Table G: Mathematiques et sciences sociales Pierre-Andre Chiappori, Roger Guesnerie ...... 229 Introduction au debat Roger Guesnerie ...... 229 On the Use of Mathematics in Social Sciences: Some Remarks Pierre-Andre Chiappori 232 La sociologie face it la mathematisation Pierre-Michel Menger ...... 238

Round Table H: Mathematics and Industry Julian C.R. Hunt, Helmut Neunzert ...... 257 Table of Contents of Volume III xi

Round Table I: Degree Harmonization and Student Exchange Programmes Christian Berg, Hans J. Munkholm, Ivan Netuka, David Salinger, Vladimir Soucek ...... 277 Round Table J: The Pythagoras Programme A European Policy for Mathematics Daniel Gabay...... 321

Round Table K: Collaboration with Developing Countries Pierre Berard...... 347 Appendix 1: Contributions made by Mathematical Societies 374 Appendix II: Individual Reports 381 Appendix III: Centers in Collaboration 394

Round Table L: Mathematical Libraries in Europe Pierre Barrat, Genevieve Sureau, Liliane Zweig 397 Introd uction ...... 398 Financing of the Libraries 400 The Situation of Libraries in Russia S. Kuksin ...... 400 The Situation of Libraries in Poland L. Goldstein ...... 401 The Financial Situation of Mathematical Libraries in Hungary A. Szendrei...... 403 Scientific Publishing and Problems of Documentation 406 Documentary Inflation and the Rising Cost of Journals C. Byrne ...... 407 New Technologies and Computerization 411 Computerization of Spanish Libraries P. Ibarrola ...... 411 The French Experience and Norms P. Barrat ...... 412 Appendix 1: Results of the Questionnaire 416 Appendix 2: Statistical Data ...... 419 Appendix 3: Manufacturing Costs C. Byrne ...... 423 Appendix 4: Electronic Publishing: A Few Remarks C. Byrne 424 Questionnaire 425

Round Table M: Mathematics and Economics Bernard Cornet...... 431 xii Table of Contents of Volume III

Round Table 0: Mathematiques et chimie Edgar Soulie ...... 451 Mathematiques et modelisation en dynamique chimique Patrick Hanusse...... 455 Mathematiques et science des materiaux Claude Bernard ...... 461 Knowledge Acquisition in Chemistry Joachim Johann Gasteiger ...... 466 Mathematical Methods for Determining Crystal Structure Gerard Bricogne...... 473 La theorie des groupes en chimie Michel Golfier ...... 477 Mathematiques et chimie quantique Marcel Allavena ...... 489 Mathematiques et chimie en recherche pharmaceutique Gilles Moreau...... 497 Two Global Methods for Molecular Geometry Optimization Susana Gomez, David Romero ...... 503 Geometrie fractale et equations differentielles d'ordre non entier en chimie physique Alain Le Mehaute, Frederic Heliodore . . . 510 Round Table P: Mathematics in Medicine and Biology Robert Hiorns, Bernard Prum ...... 515 Introduction 515 Medical Statistics: an Example from Diabetes Epidemiology Niels Keiding ...... 517 Point Process Approach of Life Histories: Causal Chains Elja Arjas ...... 520 Two Signal Cell Activation Explored with Simple Mathematics Jonathan Swinton ...... 528 A Theory of the Functional Organization of Biological Systems Applied to Neural Networks Gilbert Chauvet ...... 536 Stochastic Vistas to the Analysis of Physical Mapping of the Genome Petre Tautu ...... 550 Rhythms and Chaos in Biological Systems Albert Goldbeter. . . . . 559 Discussion and conclusions . 564

Index of Authors of Volume III 573 F. Hirzebruch, President of the European Mathematical Society at the meeting of the Round Table "Women and Mathematics" J.-P. Bourguignon H. Follmer, President of the President of the Societe . Scientific Committee Mathematique de France

G. Katona and L. Marki M. Karoubi. Founder J.-J. Olivier, F. Mignot, M. Martin-Deschamps

T. F. Banchoff and M. Dcmazure L. Schwartz J. Tits

J. Eells P. Malliavin Rear: Jiirg Frohlich, Alain-Sol Sznitman, Jean-Michel Bony Front: Ursula Hamenstadt, Darrell Duffie

Ragni Piene, Eduard J.N. Looijenga, Christopher Deninger, Fabrizio Catanese Vladimir 1. Arnold Sergej B. Kuksin

Jean-Christophe Yoccoz Bernard W. Silverman Volker Strassen

Dan Voiculescu Pekka Tukia Ragni Piene

Christopher Deninger Martin A. Nowak Jean-Franc:;ois Le Gall

C. Cercignani and Pierre-Louis Lions Ib Madsen

Miklos Laczkovich Alain-Sol Sznitman Corrado De Concini

Werner Miiller Maxim Kontsevich (standing) with Don Zagier in the first row Zofia Adamowicz

Yurii Neretin Question time The Figa-Talamancas

1. M. Gelfand (center) and T. Alekseevna (left) with Birkhiiuser team A. Kostant and E. Beschler T. Luczak, S. Miiller, M. Kontsevich, A. Goncharov, J. Franke, C. Voisin, J. Chirac, M. Karoubi, F. Hirzebruch, F. Labourie, R. E. Borchcrds, G. Tardos

J. Chirac, Mayor of Paris, congratulating V. Svenik with F. Hirzcbruch in the background 1. M. Gelfand and V. 1. Arnold

Waiting for the champagne