Hilbert's Tenth Problem

Mathematical Events of the Twentieth Century Mathematical Events of the Twentieth Century Edited by A. A. Bolibruch, Yu.S. Osipov, and Ya.G. Sinai 123 PHASIS † A. A. Bolibruch Ya . G . S i n a i University of Princeton Yu. S. Osipov Department of Mathematics Academy of Sciences Washington Road Leninsky Pr. 14 08544-1000 Princeton, USA 117901 Moscow, Russia e-mail: [email protected] e-mail: [email protected] Editorial Board: V.I. Arnold A. A. Bolibruch (Vice-Chair) A. M. Vershik Yu. I. Manin Yu. S. Osipov (Chair) Ya. G. Sinai (Vice-Chair) V. M . Ti k h o m i rov L. D. Faddeev V.B. Philippov (Secretary) Originally published in Russian as “Matematicheskie sobytiya XX veka” by PHASIS, Moscow, Russia 2003 (ISBN 5-7036-0074-X). Library of Congress Control Number: 2005931922 Mathematics Subject Classification (2000): 01A60 ISBN-10 3-540-23235-4 Springer Berlin Heidelberg New York ISBN-13 978-3-540-23235-3 Springer Berlin Heidelberg New York 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, reuse of illustrations, recitation, broad- casting, reproduction on microfilm 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. Violations are liable to prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg and PHASIS Moscow 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant pro- tective laws and regulations and therefore free for general use. Set for publishing by PHASIS using a special LATEX macro package Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper 46/3142/YL - 5 4 3 2 1 0 Preface Russian mathematics (later Soviet mathematics, and Russian mathematics once again) occupies a special place in twentieth-century mathematics. In addition to its well-known achievements, Russian mathematics established a unique style of research based on the existence of prominent mathematical schools. These schools were headed by recognized leaders, who became famous due to their talents and outstanding contributions to science. The present collection is intended primarily to gather in one book the tes- timonies of the participants in the development of mathematics over the past century. In their articles the authors have expressed their own points of view on the events that took place. The editors have not felt that they had a right to make any changes, other than stylistic ones, or to add any of their own commentary to the text. Naturally, the points of view of the authors should not be construed as those of the editors. The list of mathematicians invited to participate in the present edition was quite long. Unfortunately, some of the authors for various reasons did not accept our invitation, and regretfully a number of areas of research are not fully represented here. Nevertheless, the material that has been assembled is of great value not only in the scientific sense, but also in its historical context. We wish to express our gratitude to all the authors who contributed. We hope that this collection will induce other authors to write their memoirs of mathematical events of the twentieth century which they witnessed and par- ticipated in. This will open a possibility for a continuation of the present edition. We are very glad to thank Owen de Lange for his help in the preparation of the English edition of this book. The Editors P. S. A. A. Bolibruch played a very important role in preparing this book. His untimely death was a great shock and unrepairable loss for all people who were fortunate to work with him. Heidelberg, 2005 Moscow, 2005 Contents Preface..................................... V D. V.Anosov Dynamical Systems in the 1960s: The Hyperbolic Revolution . 1 V. I. A rn old From Hilbert’s Superposition Problem to Dynamical Systems . 19 A. A. Bolibruch Inverse Monodromy Problems of the Analytic Theory of Differential Equations.................................... 49 L. D. Faddeev What Modern Mathematical Physics Is Supposed to Be . 75 R. V.Gamkrelidze DiscoveryoftheMaximumPrinciple .................... 85 Yu.S. Il’yashenko The Qualitative Theory of Differential Equations in the Plane . 101 P.S. Krasnoshchekov Computerization...Let’sBeCareful.................... 133 V.A. Marchenko The Generalized Shift, Transformation Operators, and Inverse Problems . 145 V. P. Maslov Mathematics and the Trajectories of Typhoons . 163 Yu. V.Matiyasevich Hilbert’s Tenth Problem: Diophantine Equations in the Twentieth Century 185 V.D. Milman Observations on the Movement of People and Ideas in Twentieth-Century Mathematics.................................. 215 VIII Contents E. F.Mishchenko About Aleksandrov, Pontryagin and Their Scientific Schools . 243 Yu. V.Nesterenko Hilbert’sSeventhProblem .......................... 269 S. M. Nikol’skii The Great Kolmogorov . 283 A. N. Parshin Numbers as Functions: The Development of an Idea in the Moscow School of Algebraic Geometry . 297 A. A. Razborov ? The P=NP-Problem: A View from the 1990s . 331 L. P.Shil’nikov Homoclinic Trajectories: From Poincare´ to the Present . 347 A. N. Shiryaev From “Disorder” to Nonlinear Filtering and Martingale Theory . 371 Ya. G. Sinai How Mathematicians and Physicists Found Each Other in the Theory of Dynamical Systems and in Statistical Mechanics . 399 V.M. Tikhomirov Approximation Theory in the Twentieth Century . 409 A. M. Vershik The Life and Fate of Functional Analysis in the Twentieth Century . 437 A. G. Vitushkin HalfaCenturyAsOneDay ......................... 449 V.S. Vladimirov Nikolai Nikolaevich Bogolyubov — Mathematician by the Grace of God 475 V.I. Yudovich Global Solvability Versus Collapse in the Dynamics of an Incompressible Fluid ...................................... 501 NameIndex .................................. 529 D. V.Anosov Dynamical Systems in the 1960s: The Hyperbolic Revolution Translated by R. Cooke Probably everyone would agree that the theory of dynamical systems underwent a profound change during the 1960s, that its general features as then formed have been retained to the present time despite significant changes in content, and that this theory turned into a separate discipline at that time, having been up to that point a branch of the theory of ordinary differential equations. Here I intend to discuss certain events of this period, after which I shall say a few words about what came after. The theory of dynamical systems consists of three branches, corresponding to the general character of the objects and questions it considers: differential dynamics — the theory of smooth dynamical systems, the ergodic theory — the theory of metric dynamical systems (in the sense of measure theory), and topological dynamics — the theory of topological (continuous but not differentiable) dynamical systems. 1 1 These explanations of the subject matter of each of these three parts, being rather general and brief, are not entirely accurate. Thus, in reality the first part also encompasses one-dimensional dynamics, including real dynamics, while the mappings considered in the last part are nonsmooth and even have discontinuities. Such failure of the subject matter of real one-dimensional dynamics to comply with the norms of the theory of smooth dynamical systems is compensated by their conceptual proximity. The picture is different for symplectic systems. They have not lost their 2 D. V.Anosov The rise of dynamical systems as a separate discipline was due mostly to two events in differential dynamics: the development of KAM theory (its seeds were planted earlier, but did not immediately receive wide development) and the “hyperbolic revolution.” The chronology of all this is framed by the penetration of ideas of probabilistic origin into ergodic theory: the entropic theory arose (and immediately began to develop vigorously) in the mid-1950s. The Ornstein 2 theory (which within the limits of its applicability sometimes establishes an even closer connection of dynamical systems with the most ordinary random processes such as coin tossing than does the entropic theory, while at other times it points up subtle differences between them) arose in the 1970s. It should be mentioned that two topics vary with particular vigor in the theory of dynamical systems. 3 In their “purest” form they occur in differential dynamics as quasi-periodicity, for which a certain regularity is characteristic, and as hyperbolicity, which is connected with those phenomena that are descrip- tively named “quasi-randomness,” “stochasticity,” or “chaos” (see the article of Yoccoz [1]). The use of spectral concepts and methods in ergodic theory, which began as early as the 1930s, can be regarded as a sort of implementation (in a suitable context) of the first topic. 4 The second topic found an adequate expres- sion in ergodic theory just in the 1950s and 1970s, when the entropic theory and the Ornstein theory arose. Kolmogorov, whose name provided the first letter in the acronym KAM, was the only one at that time who made an equally large contribution to the study of both regular and chaotic motions

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