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: A M E R I C A N M AT H E M AT I C A L S O C I E T Y va o k a y et Tr a n a etl v S of y es urt o c h p a gr o ot h P Vladimir Igorevich Arnold June 12, 1937–June 3, 2010 ARNOLD: Boris A. Khesin Serge L. Tabachnikov Editors A M E R I C A N M AT H E M AT I C A L S O C I E T Y Providence, Rhode Island Translationof 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 informationand updates on this book, visit www.ams.org/bookpages/mbk-86 Library of Congress Cataloging-in-Publication Data Arnold: swimmingagainstthetide/BorisKhesin,SergeTabachnikov,editors. pagescm. ISBN978-1-4704-1699-7(alk.paper) 1. Arnol(cid:2)d, V. I. (Vladimir Igorevich), 1937–2010. 2. Mathematicians–Russia–Biography. 3. Mathematicians–Soviet Union–Biography. 4. Mathematical analysis. 5. Differential equations. I.Khesin,BorisA.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 useinteachingorresearch. Permissionisgrantedtoquotebriefpassagesfromthispublicationin reviews,providedthecustomaryacknowledgmentofthesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication 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(cid:2) service. Formoreinformation,pleasevisit: http://www.ams.org/rightslink. Translationrightsandlicensedreprintrequestsshouldbesenttoreprint-permission@ams.org. Excludedfromtheseprovisionsismaterialforwhichtheauthorholdscopyright. Insuchcases, requestsforpermissiontoreuseorreprintmaterialshouldbeaddresseddirectlytotheauthor(s). Copyrightownershipisindicatedonthecopyrightpage,oronthelowerright-handcornerofthe firstpageofeacharticlewithinproceedingsvolumes. (cid:2)c 2014bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://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,translatedfrom“ArnoldinHisOwnWords,”aninterviewwith 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 Ju¨rgen Moser 31 Chapter 4. Polymathematics: Is Mathematics a Single Science or a Set of Arts? 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 Arnoldis 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 ordinarydifferentialequationsandcelestialmechanicstosingularitytheoryandreal 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 Arnoldwroteseveralhundredsofpapers,andmanybooks,including10univer- sitytextbooks. Heisknownforhislucidwritingstylewhichcombinesmathematical 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 1AdaptedfromArnold’sownCV,thePrefacetohis“CollectedWorks”,Springer,2009,and thewebsiteoftheMCCME. ix