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 Preface Vladimir Igorevich Arnold is one of the most influential mathematicians of our era. Arnold launched several mathematical domains (such as modern geometric mechanics, symplectic topology, and topological fluid dynamics) and contributed, in a fundamental way, to the foundations and methods in many subjects, from ordinary differential equations and celestial mechanics to singularity theory and real algebraic geometry. Even a quick look at a (certainly incomplete) list of notions and results named after Arnold is telling: • KAM (Kolmogorov–Arnold–Moser) theory • The Arnold conjectures in symplectic topology • The Hilbert–Arnold problem for the number of zeros of abelian integrals • Arnold’s inequality, comparison, and complexification method in real al- gebraic geometry • Arnold–Kolmogorov solution of Hilbert’s 13th problem • Arnold’s spectral sequence in singularity theory • Arnold diffusion • The Euler–Poincar´e–Arnold equations for geodesics on Lie groups • Arnold’s stability criterion in hydrodynamics • ABC (Arnold–Beltrami–Childress) flows in fluid dynamics • Arnold–Korkina dynamo • Arnold’s cat map • The Liouville–Arnold theorem in integrable systems • Arnold’s continued fractions • Arnold’s interpretation of the Maslov index • Arnold’s relation in cohomology of braid groups • Arnold tongues in bifurcation theory • The Jordan–Arnold normal forms for families of matrices • Arnold’s invariants of plane curves Arnold wrote several hundreds of papers, and many books, including 10 univer- sity textbooks. He is known for his lucid writing style which combines mathematical rigor with physical and geometric intuition. Arnold’s books Ordinary Differential Equations and Mathematical Methods of Classical Mechanics have become math- ematical bestsellers and integral parts of the mathematical education throughout the world. Here is a brief biography and a list of distinctions of V. I. Arnold.1 V. I. Arnold was born on June 12, 1937, in Odessa, USSR. The family lived in Moscow, and Arnold graduated from the Moscow school #59. Later in life, on 1Adapted from Arnold’s own CV, the Preface to his “Collected Works”, Springer, 2009, and the website of the MCCME. ix xPREFACE numerous occasions, he warmly recalled his mathematics teacher Ivan Vassilievich Morozkin. From 1954–1959, he was a student at the Department of Mechanics and Mathematics of the Moscow State University. His M.Sc. diploma work was entitled “On mappings of a circle to itself.” The degree of a “candidate of physical-mathematical sciences,” an analogue of the Ph.D. degree in the West, was conferred on him in 1961 by the Keldysh Applied Mathematics Institute, Moscow; his thesis advisor was A.N. Kolmogorov. Arnold’s thesis described the representation of continuous functions of three variables as superpositions of continuous functions of two variables, thus completing the solution of Hilbert’s 13th problem. Arnold obtained this result back in 1957, being a third year undergraduate student (by then, A.N. Kolmogorov had shown that continuous functions of more variables could be represented as superpositions of continuous functions of only three variables). The degree of “doctor of physical-mathematical sciences,” an analogue of the Habilitation degree, was awarded to him in 1963 by the Keldysh Applied Mathe- matics Institute, Moscow. (The same Institute where Arnold completed his thesis on the stability of Hamiltonian systems, which subsequently became a part of what is now known as KAM theory.) After graduating from Moscow State University in 1961, Arnold worked there until 1986. He then worked at the Steklov Mathematical Institute and later at the Paris Dauphine University. Arnold became a corresponding member of the USSR Academy of Sciences in 1986 and a full member in 1990. He was an honorary member of the London Mathematical Society (1976), a member of the National Academy of Sciences of the Unites States (1983), the French Academy (1984), the American Academy Arts and Sciences (1987), the Royal Society (1988), the Accademia dei Lincei (1989), the American Philosophical Society (1990), the Russian Academy of Natural Sciences (1991), and the European Academy of Sciences (1991). Arnold received a degree of Doctor Honoris Causa from the following universi- ties: P. et M. Curie, Paris (1979), Warwick (1988), Utrecht (1991), Bologna (1991), Madrid (1994), and Toronto (1997). Arnold served as a vice-president of the Inter- national Mathematical Union from 1995–1998. Arnold was a recipient of