Mathematical Research Today and Tomorrow Viewpoints of Seven Fields
The Symposium on the Current State and Prospects of Mathematics was held in Barcelona from June 13 to June 18, 1991. Seven invited Fields medalists gavetalks on the development of their respective research fields. The contents of all lectures were collect
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zurich F. Takens, Groningen
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Mathematical Research Today and Tomorrow Viewpoints of Seven Fields Medalists
Lectures given at the Institut d'Estudis Catalans, Barcelona, Spain, June 1991 Editors: C. Casacuberta, M. Castellet
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Editors Carles Casacuberta Departament d'Algebra i Geometria Universitat de Barcelona Gran Via de les Corts Catalanes, 585 E-08007 Barcelona, Spain Manuel Castellet Departament de Maternatiques Universitat Autonorna de Barcelona E-08193 Bellaterra, Spain
Cover drawing by Joan Sola. Photo of Sergei Novikov by Jordi Morera; courtesy of £1 Temps. Photos of Alain Connes, Gerd Faltings, Vaughan Jones, Stephen Smale and Rene Thorn by Salvador Esparbe; courtesy of the Science Museum of Barcelona.
Mathematics Subject Classification (1991): 00-02, 00B20
ISBN 3-540-56011-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-56011-4 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer- Verlag Berlin Heidelberg 1992 Printed in Germany Typesetting: Camera-ready by author/editor 46/3140-543210 - Printed on acid-free paper
Preface
As Beno Eckmann said in his Inaugural Lecture for the academic year 1978-79 at the ETH Zurich, "Nowadays, mathematics plays three different roles: It is an important and essential tool in our modern scientific and technological world, it is a model to seek the objective truth in a perfectly defined context, and, moreover, forms part of the basis of our cultural tradition." From 13th to 18th of June 1991, the Institut d 'Estudis Catalans -an academic and cultural body whose aim is higher scientific research and, in particular, research into all elements of Catalan culture- presented to the mathematical community the Symposium on the Current State and Prospects of Mathematics, whose main goal was to combine the above three aspects. Among the different cultural events preceding the 1992 Barcelona Olympic Games, mathematics was expected to play -and in fact has played- an outstanding role. Thanks to the financial support coming not only from the Olimpiada Cultural but also from the CIRIT and DGICYT, the Centre de Recerca Matemaiica (CRM) was able to gather seven Fields Medalists in Barcelona, who presented the most recent advances and debated the present and future of mathematical research. "Mathematics is a truly inter
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