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ARNOLD: Real Analysis a Comprehensive Course in Analysis, Part 1 ARNOLD: Real Analysis A Comprehensive Course in Analysis, Part 1 Barry Simon Boris A. Khesin Serge L. Tabachnikov Editors http://dx.doi.org/10.1090/mbk/086 ARNOLD: AMERICAN MATHEMATICAL SOCIETY Photograph courtesy of Svetlana Tretyakova Photograph courtesy of Svetlana Vladimir Igorevich Arnold June 12, 1937–June 3, 2010 ARNOLD: Boris A. Khesin Serge L. Tabachnikov Editors AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island Translation of Chapter 7 “About Vladimir Abramovich Rokhlin” and Chapter 21 “Several Thoughts About Arnold” provided by Valentina Altman. 2010 Mathematics Subject Classification. Primary 01A65; Secondary 01A70, 01A75. For additional information and updates on this book, visit www.ams.org/bookpages/mbk-86 Library of Congress Cataloging-in-Publication Data Arnold: swimming against the tide / Boris Khesin, Serge Tabachnikov, editors. pages cm. ISBN 978-1-4704-1699-7 (alk. paper) 1. Arnold, V. I. (Vladimir Igorevich), 1937–2010. 2. Mathematicians–Russia–Biography. 3. Mathematicians–Soviet Union–Biography. 4. Mathematical analysis. 5. Differential equations. I. Khesin, Boris A. II. Tabachnikov, Serge. QA8.6.A76 2014 510.92–dc23 2014021165 [B] Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are now being handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Translation rights and licensed reprint requests should be sent to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2014 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 191817161514 Epigraph Development of mathematics resembles a fast revolution of a wheel: sprinkles of water are flying in all directions. Fashion – it is the stream that leaves the main trajectory in the tan- gential direction. These streams of epigone works attract most attention, and they constitute the main mass, but they inevitably disappear after a while because they parted with the wheel. To remain on the wheel, one must apply the effort in the direction perpendicular to the main stream. —V. I. Arnold, translated from “Arnold in His Own Words,” an interview with the mathematician originally published in Kvant Magazine, 1990 and republished in the Notices of the American Mathematical Society, 2012. v Contents Epigraph v Preface ix Permissions and Acknowledgments xiii Part 1. By Arnold Chapter 1. Arnold in His Own Words V. I. Arnold 3 Chapter 2. From Hilbert’s Superposition Problem to Dynamical Systems V. I. Arnold 11 Chapter 3. Recollections Jurgen¨ Moser 31 Chapter 4. Polymathematics: Is Mathematics a Single Science or aSetofArts? V. I. Arnold 35 Chapter 5. A Mathematical Trivium V. I. Arnold 47 Chapter 6. Comments on “A Mathematical Trivium” Boris Khesin and Serge Tabachnikov 57 Chapter 7. About Vladimir Abramovich Rokhlin V. I. Arnold 67 Photographs of V. I. Arnold 1940s–1970s 79 1980s–1990s 87 The 2000s 99 Part 2. About Arnold Chapter 8. To Whom It May Concern Alexander Givental 113 vii viii CONTENTS Chapter 9. Remembering Vladimir Arnold: Early Years Yakov Sinai 123 Chapter 10. Vladimir I. Arnold Steve Smale 127 Chapter 11. Memories of Vladimir Arnold Michael Berry 129 Chapter 12. Dima Arnold in My Life Dmitry Fuchs 133 Chapter 13. V. I. Arnold, As I Have Seen Him Yulij Ilyashenko 141 Chapter 14. My Encounters with Vladimir Igorevich Arnold Yakov Eliashberg 147 Chapter 15. On V. I. Arnold and Hydrodynamics Boris Khesin 151 Chapter 16. Arnold’s Seminar, First Years Askold Khovanskii and Alexander Varchenko 157 Chapter 17. Topology in Arnold’s Work Victor Vassiliev 165 Chapter 18. Arnold and Symplectic Geometry Helmut Hofer 173 Chapter 19. Some Recollections of Vladimir Igorevich Mikhail Sevryuk 179 Chapter 20. Remembering V. I. Arnold Leonid Polterovich 183 Chapter 21. Several Thoughts about Arnold A. Vershik 187 Chapter 22. Vladimir Igorevich Arnold: A View from the Rear Bench Sergei Yakovenko 197 Vladimir Arnold, an eminent mathematician of our time, is known both for his mathematical results, which are many and prominent, and for his strong opinions, often expressed in an uncompromising and provoking manner. His dictum that “Mathematics is a part of physics where experiments are cheap” is well known. This book consists of two parts: selected articles by and an interview with Vladimir Arnold, and a collection of articles about him written by his friends, colleagues, and students. The book is generously illustrated by a large collection of photographs, some never before published. The book presents many a facet of this extraordinary mathematician and man, from his mathematical discoveries to his daredevil outdoor adventures. For additional information and updates on this book, visit www.ams.org/bookpages/mbk-86 AMS on the Web www.ams.org MBK/86.
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