Arnold: Swimming Against the Tide

213 Pages·2014·4.68 MB·English

by  Khesin

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.
Translation[e-mail].
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