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24 Edwar d Bierstone , Bori s Khesin , Askol d Khovanskii , an d Jerrol d E . Marsden , Editors, Th e Arnoldfest : Proceeding s o f a conferenc e i n honou r o f V. I . Arnol d fo r hi s sixtieth birthday , 199 9 23 Myrn a H . Wooders , Editor , Topic s i n mathematical economic s an d gam e theory . Essay s in hono r o f Robert J . Aumann , 199 9 22 Graciel a Chichilnisky , Editor , Topolog y an d markets , 199 9 21 Shigu i Ruan , Gai l S . K . Wolkowicz , an d Jianhon g Wu , Editors , Differentia l equations wit h application s t o biology , 199 8 20 Pete r A . Fillmor e an d Jame s A . Mingo , Editors , Operato r algebra s an d thei r applications II , 199 8 19 Willia m G . Dwyer , Stephe n Halperin , Richar d Kane , Stanle y O . Kochman , Mark E . Mahowald , an d Pau l S . Selic k (Editor-in-Chief) , Editors , Stabl e an d unstable homotopy , 199 8 18 Pano s M . Pardalo s an d Henr y Wolkowicz , Editors , Topic s i n semidefinit e an d interior-point methods , 199 8 17 Joachi m J . R . Cunt z an d Masou d Khalkhali , Editors , Cycli c cohomolog y an d noncommutative geometry , 199 7 16 Victo r P . Snaith , Editor , Algebrai c K-theory , 199 7 15 Stephe n P . Braham , Jac k D . Gegenberg , an d Rober t J . McKellar , Editors , Sixt h Canadian conferenc e o n genera l relativit y an d relativisti c astrophysics , 199 7 14 Moura d E . H . Ismail , Davi d R . Masson , an d Miza n Rahman , Editors , Specia l functions, g-serie s an d relate d topics , 199 7 13 Pete r A . Fillmor e an d Jame s A . Mingo , Editors , Operato r algebra s an d thei r applications, 199 7 12 Dan-Virgi l Voiculescu , Editor , Fre e probabilit y theory , 199 7 11 Collee n D . Cutle r an d Danie l T . Kaplan , Editors , Nonlinea r dynamic s an d tim e series: Buildin g a bridg e betwee n th e natura l an d statistica l sciences , 199 7 10 Jerrol d E . Marsden , Georg e W . Patrick , an d Willia m F . Shadwick , Editors , Integration algorithm s an d classica l mechanics , 199 6 9 W . H . Kliemann , W . F . Langford , an d N . S . Namachchivaya , Editors , Nonlinea r dynamics an d stochasti c mechanics , 199 6 8 Larr y M . Bate s an d Davi d L . Rod , Editors , Conservativ e system s an d quantu m chaos, 199 6 7 Willia m F . Shadwick , Perinkula m Sambamurth y Krishnaprasad , an d Tudo r Stefan Ratiu , Editors , Mechanic s day , 199 6 6 Ann a T . Lawnicza k an d Raymon d Kapral , Editors , Patter n formatio n an d lattic e gas automata , 199 6 5 Joh n Chadam , Marti n Golubitsky , Willia m Langford , an d Bria n Wetton , Editors, Patter n formation : Symmetr y method s an d applications , 199 6 4 Willia m F . Langfor d an d Wayn e Nagata , Editors , Norma l form s an d homoclini c chaos, 199 5 3 Anthon y Bloch , Editor , Hamiltonia n an d gradien t flows, algorithm s an d control , 199 4 2 K . A . Morris , Editor , Contro l o f flexibl e structures , 199 3 1 Michae l J . Enos , Editor , Dynamic s an d contro l o f mechanica l systems : Th e fallin g ca t and relate d problems , 199 3 This page intentionally left blank The Arnoldfes t Proceedings o f a Conferenc e in Honour o f V I . Arnold for his Sixtiet h Birthda y This page intentionally left blank http://dx.doi.org/10.1090/fic/024

FIELDS INSTITUT E COMMUNICATIONS

THE FIELD S INSTITUT E FO R RESEARC H I N MATHEMATICA L SCIENCE S

The Arnoldfes t Proceedings o f a Conferenc e in Honour o f V I . Arnold for his Sixtiet h Birthda y

Edward Bierston e Boris Khesi n Askold Khovanski i Jerrold E . Marsde n Editors

American Mathematical Societ y Providence, Rhod e Islan d The Field s Institut e for Researc h i n Mathematical Science s

The Field s Institute i s named i n honour o f the Canadia n mathematicia n Joh n Charle s Fields (1863-1932) . Field s wa s a visionar y wh o receive d man y honour s fo r hi s scientifi c work, includin g election to the Royal Society o f Canada i n 190 9 and t o the Royal Society o f London i n 1913 . Amon g othe r accomplishment s i n th e servic e o f th e internationa l math - ematics community , Field s wa s responsibl e fo r establishin g th e world' s mos t prestigiou s prize fo r mathematic s research—th e Field s Medal . The Field s Institute fo r Researc h i n Mathematical Science s is supported b y grants fro m the Ontario Ministr y o f Education an d Trainin g and th e Natural Science s and Engineerin g Research Counci l o f Canada . Th e Institut e i s sponsore d b y McMaste r University , th e University o f Toronto, th e Universit y o f Waterloo, an d Yor k Universit y an d ha s affiliate d universities i n Ontari o an d acros s Canada .

1991 Mathematics Subject Classification. Primar y 01A65 , 14-06 , 34-06 , 57-06 , 58-06 , 70-06, 76-06 .

Library o f Congres s Cataloging-in-Publicatio n Dat a The Arnoldfes t : proceeding s o f a conferenc e i n honou r o f V. I . Arnold fo r hi s sixtieth birthda y / Edward Bierstone.. . [e t al.] , editors. p. cm . — (Field s Institut e communications , ISS N 1069-526 5 ; v. 24 ) Includes bibliographica l references . ISBN 0-8218-0945- 8 (alk . paper ) 1. Singularitie s (Mathematics)—Congresses . 2 . Symplecti c manifolds—Congresses . I. Arnol'd, V . I . (Vladimi r Igorevich) , 1937 - II . Bierstone , Edward , 1946 - III . Series . QA614.58.A78 199 9 510—dc21 99-04577 8 CIP

Copying an d reprinting . Materia l i n thi s boo k ma y b e reproduce d b y an y mean s fo r edu - cational an d scientifi c purpose s withou t fe e o r permissio n wit h th e exceptio n o f reproduction b y services that collec t fee s fo r deliver y o f documents an d provide d tha t th e customar y acknowledg - ment o f the sourc e i s given . Thi s consen t doe s no t exten d t o othe r kind s o f copyin g fo r genera l distribution, fo r advertisin g o r promotiona l purposes , o r fo r resale . Request s fo r permissio n fo r commercial us e o f materia l shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematical Society , P . O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o be mad e b y e-mai l t o reprint-permissionQams.org . Excluded fro m thes e provisions i s material i n article s fo r whic h the autho r hold s copyright. I n such cases , requests fo r permissio n to us e or reprint shoul d b e addressed directl y t o the author(s) . (Copyright ownershi p i s indicated i n the notic e i n the lowe r right-hand corne r o f the first pag e o f each article. ) © 199 9 b y the America n Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o the Unite d State s Government . Printed i n the Unite d State s o f America . @ Th e pape r use d i n this boo k i s acid-free an d fall s within th e guideline s established t o ensur e permanenc e an d durability . This publicatio n wa s prepared b y The Field s Institute . Visit th e AM S hom e pag e a t URL : http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 0 4 0 3 0 2 0 1 0 0 9 9 Contents

Preface x i

Program o f the Conferenc e xii i

Photographs xv i

Prom Hilbert' s Superpositio n Proble m t o Dynamica l System s 1 V. I . ARNOL D

Recollections 1 9 JURGEN MOSE R

Symplectization, Complexificatio n an d Mathematica l Trinitie s 2 3 V. I . ARNOL D

Topological Problem s i n Wav e Propagation Theor y an d Topologica l Economy Principl e i n Algebraic Geometr y 3 9 V. I . ARNOL D

Geometry an d Contro l o f Three-Wave Interaction s 5 5 MARK S . ALBER , GREGOR Y G . LUTHER , JERROL D E . MARSDE N and JONATHA N M . ROBBIN S

Standard Basi s Alon g a Samue l Stratum, an d Implici t Differentiatio n 8 1 EDWARD BIERSTON E an d PIERR E D . MILMA N

A Globa l Weighte d Versio n o f Bezout's Theore m 11 5 JAMES DAMO N

Real Enrique s Surface s Withou t Rea l Points an d Enriques-Einstein-Hitchin 4-Manifold s 13 1 ALEXANDER DEGTYARE V an d VIATCHESLA V KHARLAMO V

On the Inde x o f a Vector Fiel d a t a n Isolate d Singularit y 14 1 W. EBELIN G an d S . M . GUSEIN-ZAD E

The Exponentia l Ma p o n £> * 15 3 DAVID G . EBI N an d GERAR D MISIOLE K

Zeldovich's Neutro n Sta r an d th e Predictio n o f Magnetic Frot h 16 5 MICHAEL H . FREEDMA N Contents

Arnold Conjectur e an d Gromov-Witte n Invarian t fo r Genera l Symplectic Manifold s 17 3 KENJI FUKAY A an d KAOR U ON O

Multiplicity o f a Zer o o f an Analyti c Functio n o n a Trajector y of a Vector Fiel d 19 1 ANDREI GABRIELO V

Singularity Theor y an d Symplecti c Topolog y 20 1 ALEXANDER B . GIVENTA L

On Enumeration o f Meromorphic Function s o n the Lin e 20 9 V. V . GORYUNO V an d S . K . LAND O

Pseudoholomorphic Curve s an d Dynamic s 22 5 H. HOFE R an d E . ZEHNDE R

Bifurcation o f Planar an d Spatia l Polycycles : Arnold' s Progra m and It s Developmen t 24 1 Yu. S . ILYASHENK O an d V . Yu . KALOSHI N

Singularity Whic h Ha s N o M-Smoothing 27 3 V. M . KHARLAMOV , S . YU . OREVKO V an d E . I . SHUSTI N

Symplectic Geometr y o n Modul i Space s o f Holomorphic Bundle s over Comple x Surface s 31 1 BORIS KHESI N an d ALEXE I ROSL Y

Newton Polyhedra , a Ne w Formula fo r Mixe d Volume , Product o f Roots o f a Syste m o f Equations 32 5 A. KHOVANSKI I

Interactions o f Andronov-Hopf an d Bogdanov-Taken s Bifurcation s 36 5 WILLIAM F . LANGFOR D an d KAIJU N ZHA N

Solutions o f the qKZ B Equation i n Tenso r Product s o f Finit e Dimensional Module s Ove r the Ellipti c Quantu m Grou p E T^sl2 38 5 E. MUKHI N an d A . VARCHENK O

Schrodinger Operator s o n Graph s an d Symplecti c Geometr y 39 7 S. P . NOVIKO V

On the Dominan t Fourie r Mode s i n the Serie s Associated wit h Separatrix Splittin g fo r a n A-Prior i Stable , Thre e Degree-of-Freedom Hamiltonia n Syste m 41 5 MICHAEL RUDNE V an d STEPHE N WIGGIN S Contents

Homology o f i-Connected Graph s an d Invariant s o f Knots, Plan e Arrangements, etc . 45 1 V. A . VASSILIE V

On Arnold' s Variationa l Principle s i n Fluid Mechanic s 47 1 V. A . VLADIMIRO V an d K . I . ILI N

On Function s an d Curve s Define d b y Ordinar y Differentia l Equation s 49 7 SERGEI YAKOVENK O

Global Finitenes s Propertie s o f Analytic Familie s an d Algebr a o f Thei r Taylor Coefficient s 52 7 Y. YOMDI N This page intentionally left blank Preface

The Arnoldfes t - a conference t o celebrate the work o f Vladimir Arnol d o n th e occasion o f hi s sixtiet h birthda y - too k plac e a t th e Field s Institut e i n Toront o from Jun e 1 5 to 21 , 1997 . Th e highligh t o f th e meetin g wa s Arnold' s ow n talk s in th e Institute' s Distinguishe d Lecture r program . Thi s volum e contain s note s o f Arnold's lecture s an d invite d paper s b y participants i n the Arnoldfest . The Arnoldfes t an d a Worksho p o n Symplecti c Geometr y hel d th e followin g week wer e the final majo r activitie s durin g a Program o n Singularit y Theor y an d Geometry a t th e Field s Institute fro m Januar y t o June , 1997 . Th e contributors t o this volum e includ e mai n speaker s i n th e Arnoldfes t an d th e Worksho p o n Sym - plectic Geometry, mathematicians an d scientists who had been invited to speak bu t were unabl e t o attend , an d participant s i n th e Singularit y Theor y an d Geometr y Program. The ver y broa d spectru m o f the article s her e reflect s th e scop e o f Arnold's in - terests. W e could not easil y classify the articles into groups, so that, apar t fro m th e notes o f Arnold' s thre e lectures , th e paper s ar e liste d i n alphabetica l order . Th e one exceptio n i s Jurge n Moser' s historica l Recollections o n Arnold' s firs t lecture , that w e have presented immediatel y followin g th e note s o f the latter. Arnold' s lec - tures were as provocative a s usual, and stimulated som e strong discussion ! Moser' s Recollections wer e a welcome contribution . Arnold wa s awarde d a n honorar y degree , Docto r o f Science , honoris causa by the Universit y o f Toront o a t a convocatio n ceremon y o n Jun e 9 , 199 7 (thre e day s before hi s birthday) . W e hav e reproduce d som e photo s take n durin g bot h th e convocation ceremon y an d th e Arnoldfest . W e hav e als o include d th e progra m o f lectures i n the Arnoldfest . A participant i n the Arnoldfes t coul d no t avoi d bein g struck b y the manifesta - tions not onl y o f esteem but als o of warm affectio n fo r Arnol d b y his colleagues an d former students. Thes e feelings were communicated i n part b y stories about Arnol d told b y some o f the lecturers. I n most cases , the speakers chose not to include thes e stories i n thei r articles , bu t w e hop e tha t thi s volum e communicate s som e o f th e spirit o f the Arnoldfest . Among th e feature s tha t mad e th e conferenc e a succes s wer e it s ver y activ e poster sessions , the related activitie s i n the Singularit y Theor y an d Geometr y Pro - gram, an d a n informa l worksho p o n Applications o f Arnold Stability . The Field s Institut e provide d excellen t facilitie s an d a marvellou s ambienc e for th e Arnoldfest . W e woul d lik e to than k Debbi e Iscoe , Kare n Walke r an d Ale - sia Zuccal a fo r thei r amiabl e hel p i n organizin g th e conferenc e an d i n preparin g this volume . Fundin g fo r th e Arnoldfes t an d th e Progra m o n Singularit y Theor y and Geometr y wa s generousl y provide d b y th e Field s Institut e (and , throug h it , xi Xll Preface the Natura l Science s an d Engineerin g Researc h Counci l o f Canada , an d th e On - tario Ministr y o f Education). W e are grateful fo r financial assistanc e als o fro m th e University o f Toronto , fro m th e Californi a Institut e o f Technolog y fo r som e U.S . participants, an d fro m othe r institution s tha t helpe d suppor t participants .

Edward Bierstone , Bori s Khesin, Askol d Khovanski i an d Jerr y Marsde n June, 199 9

Organizing Committee s

The Arnoldfest , Conferenc e i n Honou r o f Vladimi r Arnold , Jun e 15-21 , 1997. Jerrol d Marsden , California Institute of Technology (Chair), Edwar d Bier - stone, University of Toronto, Joh n Chadam , University of Pittsburgh, Askol d Kho - vanskii, University of Toronto, Rober t Moody , University of Alberta, Tudo r Ratiu , University of California at Santa Cruz, Willia m Shadwick , Alexande r Varchenko , University of North Carolina at Chapel Hill, Victor Vassiliev , Steklov Mathematical Institute, Moscow. Singularity Theor y an d Geometry , January-June , 1997 . Edwar d Bierstone , Askold Khovanskii , Pierr e Milman , University of Toronto, Alexande r Nabutovsky , University of Toronto, Mar k Spivakovsky , University of Toronto. Workshop o n Symplecti c Geometry , Jun e 23-27 , 1997 . Yako v Eliashberg , Stanford University, Bori s Khesin , University of Toronto. Informal Worksho p o n Application s o f Arnold Stability , Jun e 13-14 , 1997 . Ted Shepherd , University of Toronto, Vladimi r Vladimirov , Hong Kong University of Science and Technology. Conference i n Honour o f Vladimir Arnol d The Field s Institut e June 15-21 , 199 7 Program

Monday, Jun e 16 , 199 7

A. Varchenko, Universit y o f North Carolin a a t Chape l Hil l Elliptic Quantum Groups and the Bethe Ansatz

M. E. Kazarian , Steklo v Mathematica l Institut e Functions on the Circle: Singularities and Invariants

Y. Eliashberg, Stanfor d Universit y Invariants of Legendrian Knots via Contact Homology

M. Golubitsky , Universit y o f Housto n Animal Gaits and Coupled Cell Systems

J. E . Marsden , Californi a Institut e o f Technolog y The Dynamics and Geometry of Nonholonomic Mechanics

FIELDS INSTITUTE DISTINGUISHE D LECTUR E SERIE S V. I. Arnold , Steklo v Mathematica l Institut e an d Universit e de Paris-Dauphin e From the Hilbert Superposition Problem to Dynamical Systems

Tuesday, Jun e 17 , 199 7

R. V . Moody , Universit y o f Albert a Self similarity and Diffraction in Quasicrystals

David Ebin , Stat e Universit y o f New Yor k at Ston y Broo k Motion of Perfect Fluids with Surface Tension

John Harnad , Concordi a University/CRM , Universit e d e Montrea l Loop Groups, R-Matrices and Liouville-Arnold Integrations

P. Holmes , Princeto n Universit y Constrained Euler Buckling

J. W . Milnor , SUN Y at Ston y Broo k Complex Dynamics Wednesday, Jun e 18 , 199 7

E. Bierstone , Universit y o f Toront o Resolution of Singularities

R. d e la Llave , Universit y o f Texa s Renormalization and Smooth Conjugacy

S. Wiggins, Californi a Institut e o f Technolog y Arnold Diffusion: Issues and Recent Results

V. A . Vassiliev, Steklo v Mathematical Institut e Homology of i-Connected Graphs and Invariants of Knots, Plane Arrangements, etc.

V. I. Yudovich, Rosto v Stat e Universit y Dynamical Systems with Cosymmetries and Invariant Pfaffian Forms

FIELDS INSTITUTE DISTINGUISHE D LECTUR E V. I. Arnol d

Poly mathematics: Symplectization, Complexification, and So On

Thursday, Jun e 19 , 199 7

K. Moffatt , Isaa c Newto n Institut e Relaxation Under Topographical Constraints V. Vladimirov, Th e Hon g Kong Universit y o f Scienc e an d Technolog y Hydrodynamic Equilibrium States: Stability

S. Tremaine, Universit y o f Toront o The Stability of the Solar System: An Astronomer's View

T. Ratiu, Universit y o f Californi a a t Sant a Cru z Current affiliation : Ecol e Pol y technique Federal e d e Lausanne , Switzerland Reduction by Stages

S. Smale, Cit y Universit y o f Hong Kon g Great Problems

Friday, Jun e 20 , 199 7

P. Slodowy , Universita t Hambur g On Simple Singularities and Lie Groups

D. Fuchs, Universit y o f Californi a a t Davi s Deformations of Lie Algebras Program xv

A. Khovanskii, Universit y o f Toront o Newton Polyhedra, a New Formula for Mixed Volume, Product of Roots of a System of Equations

Y. S . Ilyashenko, Mosco w State Universit y Bifurcation of Planar and Spatial Poly cycles: Arnold's Program and its Development

B. Khesin , Universit y o f Toront o Holomorphic Linking Number and Gauge Groups on Complex Manifolds

FIELDS INSTITUTE DISTINGUISHE D LECTUR E V. I. Arnol d Topologically Necessary Wavefront Singularities, the Sturm-Hurwitz Theorem on Fourier Series and Higher Derivative Extensions of Morse (Conley-Zehnder- Chekanov- Chaperon-Floer) Theory

Saturday, Jun e 21 , 199 7

A. Givental, Universit y o f Californi a a t Berkele y Singularity Theory and Symplectic Topology

A. Vershik, Russia n Academ y o f Science s Three Dimensional Combinatorics: Statistical Approach

Y. Yomdin, Weizman n Institut e o f Science s Algebraic Differential Equations: Geometry of Solutions and of the First Return Mapping and Algebra of Taylor Coefficients

O. Bogoyavlenskij, Queens ' Universit y The Lie Algebraic Criteria for the Arnold and Komogorov Non-Degeneracy of the Integrable Hamiltonian Systems

A. Neishtadt, Spac e Researc h Institute , Russi a On Stability Loss Delay for Dynamical Bifurcations

5 llilf::»;::^| i ; J . Jfp ';,,Jif l

From lef t t o right : Askol d Khovanskii ; Honou r ary degre e Elya an d Vladimi r Arnol d recipients, Si r Martin Ree s an d Vladimi r Arnold ; Wend y Cecil-Cockwell, Chairma n o f the Governin g Counci l

13 o

Arnoldfest Speaker s Arnoldfest Participant s

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