Selected Title s i n Thi s Subserie s

34 A . KhovanskiT , A . Varchenko , an d V . Vassil iev , Editors , Topic s i n Singularit y Theory (TRANS2/180 )

33 V . M . Buchstabe r an d S . P . Novikov , Editors , Solitons , Geometry , an d Topology : O n

the Crossroa d (TRANS2/179 )

32 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors , Topics i n Statistica l an d Theoretica l Physic s (F . A . Berezi n Memoria l Volume )

(TRANS2/177)

31 E . V . Shikin , Editor , Som e Question s o f Differentia l Geometr y i n th e Larg e

(TRANS2/176)

30 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors ,

Contemporary Mathematica l Physic s (F . A . Berezi n Memoria l Volume ) (TRANS2/175 )

29 A . A . Bol ibruch , A . S . Merkur'ev , an d N . Yu . N e t s v e t a e v , Editors , Mathematic s

in St . Petersbur g (TRANS2/174 )

28 V . Kharlamov , A . Korchagin , G . PolotovskiT , an d O . Viro , Editors , Topolog y o f

Real Algebrai c Varietie s an d Relate d Topic s (TRANS2/173 )

27 K . Nomizu , Editor , Selecte d Paper s o n Numbe r Theor y an d Algebrai c Geometr y

(TRANS2/172)

26 L . A . Bunimovich , B . M . Gurevich , an d Ya . B . Pes in , Editors , Sinai' s Mosco w

Seminar o n Dynamica l System s (TRANS2/171 )

25 S . P . Novikov , Editor , Topic s i n Topolog y an d Mathematica l Physic s (TRANS2/170 )

24 S . G . Gindiki n an d E . B . Vinberg , Editors , Li e Group s an d Li e Algebras : E . B .

Dynkin's Semina r (TRANS2/169 )

23 Yu . I lyashenk o an d S . Yakovenko , Editors , Concernin g th e Hilber t 16t h Proble m

(TRANS2/165)

22 N . N . Uraltseva , Editor , Nonlinea r Evolutio n Equation s (TRANS2/164 )

Published Earlie r a s Advance s i n Sovie t Mathematic s 21 V . I . Arnold , Editor , Singularitie s an d Bifurcations , 1994

20 R . L . Dobrushin , Editor , Probabilit y Contribution s t o Statistica l Mechanics , 199 4

19 V . A . Marchenko , Editor , Spectra l Operato r Theor y an d Relate d Topics , 199 4

18 Ole g Viro , Editor , Topolog y o f Manifold s an d Varieties , 199 4

17 D m i t r y Fuchs , Editor , Unconventiona l Li e Algebras , 199 3

16 Serge i Gelfan d an d Simo n Gindikin , Editors , I . M . Gelfan d Seminar , Part s 1 and 2 ,

1993

15 A . T . Fomenko , Editor , Minima l Surfaces , 199 3

14 Yu . S . Il 'yashenko , Editor , Nonlinea r Stoke s Phenomena , 199 2

13 V . P . Maslo v an d S . N . SamborskiY , Editors , Idempoten t Analysis , 199 2

12 R . Z . KhasminskiT , Editor , Topic s i n Nonparametri c Estimation , 199 2

11 B . Ya . Levin , Editor , Entir e an d Subharmoni c Functions , 199 2

10 A . V . Babi n an d M . I . Vishik , Editors , Propertie s o f Globa l Attractor s o f Partia l

Differential Equations , 199 2

9 A . M . Vershik , Editor , Representatio n Theor y an d Dynamica l Systems , 199 2

8 E . B . Vinberg , Editor , Li e Groups , Thei r Discret e Subgroups , an d Invarian t Theory ,

1992

7 M . Sh . B irman , Editor , Estimate s an d Asymptotic s fo r Discret e Spectr a o f Integra l an d

Differential Equations , 199 1

6 A . T . Fomenko , Editor , Topologica l Classificatio n o f Integrabl e Systems . 199 1

5 R . A . Minlos , Editor , Many-Particl e Hamiltonians : Spectr a an d Scattering . 199 1

4 A . A . Suslin , Editor , Algebrai c /C-Theory . 199 1

3 Ya . G . SinaT , Editor , Dynamica l System s an d Statistica l Mechanics . 199 1

2 A . A . Kirillov , Editor , Topic s i n Representatio n Theory , 199 1

1 V . I . Arnold , Editor , Theor y o f Singularitie s an d it s Applications . 199 0

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Topics in Singularit y Theory V. I. Arnold's 60t h Anniversar y Collection

m

V. I . Arnol d

American Mathematical Societ y

TRANSLATIONS Series 2 • Volum e 18 0

Advances in the Mathematical Sciences — 34

(Formerly Advances in Soviet Mathematics)

Topics in Singularit y Theory V. I . Arnold's 60t h Anniversar y Collection

A. Khovanski i A. Varchenk o V. Vassiliev Editors

AgEWy.

American Mathematica l Societ y Providence, Rhod e Islan d

http://dx.doi.org/10.1090/trans2/180

A D V A N C E S I N T H E M A T H E M A T I C A L S C I E N C E S E D I T O R I A L C O M M I T T E E

V. I . A R N O L D

S. G . G I N D I K I N

V . P . M A S L O V

Transla t ion edite d b y A . B . Sossinsk y

1991 Mathematics Subject Classification. P r i m a r y 57-XX , 58-XX ; Secondar y 00B30 .

ABSTRACT. Thi s volum e contain s a selectio n o f origina l paper s prepared , o n th e occasio n o f Vladimir Igorevic h Arnold' s sixtiet h anniversary , b y hi s forme r student s an d othe r participant s i n his Mosco w seminar .

The semina r ha s bee n a weekl y even t sinc e th e mid-1960's , an d it s participant s hav e bee n inspired b y Arnold' s creativ e idea s an d b y hi s universa l approac h t o mathematics .

These paper s reflec t th e wid e range o f Arnold's interest s an d scientifi c contributions : Singularit y Theory, Symplecti c an d Contac t Geometry , Mathematica l Physics , an d Dynamica l Systems .

The spiri t o f thi s collectio n o f paper s follow s Arnold' s vie w o f mathematics : geometrical , ric h in applications , an d connectin g differen t area s o f mathematic s an d theoretica l physics .

Library o f Congres s Car d Numbe r 91-64074 1 ISBN 0-8218-0807- 9

ISSN 0065-929 0

C o p y i n g a n d r e p r i n t i n g . Materia l i n this boo k ma y be reproduced b y any means fo r educationa l and scientifi c purpose s withou t fe e o r permissio n wit h th e exceptio n o f reproductio n b y service s that collec t fee s fo r deliver y o f documents an d provide d tha t th e customar y acknowledgmen t o f th e source i s given. Thi s consen t doe s no t exten d t o othe r kind s o f copying fo r genera l distribution , fo r advertising o r promotiona l purposes , o r fo r resale . Request s fo r permissio n fo r commercia l us e o f material shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematica l Society , P. O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o b e mad e b y e-mai l t o reprint-permissionOams.org.

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Contents

Preface i x

On the geometry o f caustic s J. W . BRUC E AN D V. M . ZAKALYUKI N 1

Lagrangian embedding s an d Lagrangian cobordis m Yu. V . CHEKANO V 1 3

Polynomial invariant s o f Legendrian link s and plane front s S. CHMUTO V AN D V. GORYUNO V 2 5

Solutions o f the elliptic qKZ B equation s an d Bethe ansat z I G. FELDER, V. TARASOV, AND A. VARCHENKO 45

Singular deformation s o f Lie algebras. Example : Deformation s o f the Lie algebra L\ ALICE FIALOWSK I AN D DMITRY FUCH S 7 7

On imaginar y plan e curve s an d spi n quotient s o f comple x surface s b y complex conjugatio n SERGEI FINASHI N AN D EUGENI I SHUSTI N 9 3

Stationary phas e integrals , quantu m tod a lattices , fla g manifold s an d the mirror conjectur e ALEXANDER GIVENTA L 10 3

On a problem o f B. Teissier S. M . GUSEIN-ZAD E 11 7

Embedding theorem s fo r loca l maps , slow-fas t system s an d bifurcation fro m Morse-Smale t o Smale-William s YU. ILYASHENK O 12 7

Topological invariant s o f fiber singularitie s MAXIM E . KAZARYA N 14 1

Informal complexificatio n an d Poisson structure s o n moduli space s BORIS A . KHESI N 14 7

Consistent partition s o f polytopes an d polynomial measure s A. KHOVANSKI I 15 7

On primitiv e element s i n the bialgebra o f chord diagram s S. K . LAND O 16 7

viii C O N T E N T S

Spaces o f meromorphic function s o n Rieman n surface s S. M . NATANZO N 17 5

Hamiltonian loop s an d Arnold' s principl e LEONID POLTEROVIC H 18 1

Singularities i n th e presenc e o f symmetrie s INNA SCHERBA K 18 9

Discrete version s o f the four-verte x theore m V. D . SEDYK H 19 7

Excitation o f ellipti c norma l mode s o f invarian t tor i i n Hamiltonia n system s M. B . SEVRYU K 20 9

Ramified covering s o f 5 2 wit h on e degenerat e branchin g poin t an d enumeration o f edge-ordere d graph s B. SHAPIRO , M . SHAPIRO , AN D A . VAINSHTEI N 21 9

On zero s o f the Schwarzia n derivativ e SERGE TABACHNIKO V 22 9

Stratified Picard-Lefschet z theor y wit h twiste d coefficient s VICTOR A . VASSILIE V 24 1

Preface

"I make bol d t o sa y tha t a s regard s the developmen t o f the younge r

generation, I have exerte d a greate r influenc e tha n th e entir e

ministry o f education , despit e the obviou s inequalit y o f means. "

(attributed by V.I. Arnold to A.S. Pushkin)

One of the leadin g mathematicians o f today, Vladimi r Arnol d turn s 6 0 on Jun e 12, 1997 . Thi s volum e wa s writte n speciall y fo r th e occasio n b y Arnold' s student s and othe r participant s i n hi s seminar . Th e semina r ha s bee n a weekl y even t sinc e the mid-sixties , an d it s participant s hav e bee n inspire d b y Arnold' s creativ e idea s and b y hi s universa l approac h t o mathematics .

Arnold i s one o f the founder s o f KAM theory , whic h allow s one to giv e a qual -itative descriptio n o f th e globa l behaviou r o f curve s determine d b y Hamiltonia n equations tha t ar e clos e t o integrabl e ones . Completel y integrabl e system s ar e dy -namical system s fo r whic h ther e i s a sufficien t numbe r o f conservatio n law s (firs t integrals). Th e trajectorie s o f suc h system s li e o n tor i tha t ar e th e commo n leve l surfaces o f the firs t integral s o f the system .

According to KAM theory, given a small perturbation o f the system, most of the invariant tor i do not vanish , bu t jus t underg o a small perturbation, an d mos t o f the points mov e alon g thes e deforme d tori . KA M theor y i s applicabl e t o th e proble m of stabilit y o f th e sola r system , t o th e stud y o f th e magneti c line s i n "tokamak " type systems . I t ha s le d t o a whole ne w branc h o f mathematics an d mechanics .

Arnold's result s o n hydrodynami c instabilit y show , i n particular , tha t a long -range dynamic forecas t i s impossible i n practice. Furthe r developmen t o f this circl e of ideas ha s le d t o th e revolutionar y ide a o f strange attractors .

Arnold i s th e founde r o f th e theor y o f singularitie s an d perestroika s (meta -morphoses) o f caustic s o n wavefronts , whic h lead s t o th e solutio n o f problem s i n symplectic geometry whose study was initiated b y Huygens, Newton, an d Leibniz in the framework o f geometric optics . Th e metho d o f study o f caustics and wavefront s is base d o n thei r connection , discovere d b y Arnold , wit h th e geometr y o f regula r polyhedra an d crystallographi c symmetr y groups . Th e proble m o f reducin g geo -metric object s (functions , vecto r fields , differentia l forms , etc. ) t o norma l for m b y

ix

x P R E F A C E

using diffeomorphism s ha s bee n studie d b y man y mathematicians , includin g New -ton an d Poincare . Bu t Arnol d invente d th e stronges t method—spectra l sequence s of quasihomogeneous fields—applicable i n situation s wher e classica l method s fail .

Mathematicians hav e alway s bee n intereste d i n plan e algebrai c curves . Coni c sections wer e wel l know n t o th e ancien t Greeks . Newto n determine d th e curve s of degree three , whil e thos e o f degree s fou r an d fiv e wer e describe d later . Bu t th e description o f th e curve s o f highe r degre e remaine d elusive . Hilber t fel t tha t no t all a priori possibl e qualitativ e picture s o f algebraic curve s ar e realizable , an d tha t there shoul d b e som e mysteriou s restrictions . Therefor e h e include d th e proble m of topological classificatio n o f curves o f a fixed degree amon g hi s famou s problems . Despite som e progres s (Petrovskii , Gudkov) , th e situatio n remaine d mysteriou s until Arnol d foun d a completel y unexpecte d connectio n betwee n thi s proble m an d 4-dimensional topology . A s a resul t o f th e influenc e o f Arnold' s wor k i n thi s area , a ne w branc h o f rea l algebrai c geometr y wa s born .

Even a s a student , Arnol d complete d th e solutio n o f Hilbert' s thirteent h prob -lem, whic h wa s begu n b y hi s teacher , Kolmogorov . Late r on , h e returne d t o thi s subject fro m a completel y differen t poin t o f view . Hi s wor k o n th e superpositio n of algebrai c function s le d t o th e discover y o f a topologica l varian t o f differentia l Galois theory , t o th e complet e calculatio n o f the cohomolog y o f brai d groups , an d to the creation o f the theory o f arrangements. Hi s work on the cohomolog y o f braid groups initiate d th e topologica l theor y o f discriminants , i.e. , subset s o f functio n spaces consistin g o f function s wit h certai n singularities . Th e recen t developmen t of thi s theor y ha s led , i n particular , t o finite-type invariant s o f knot s an d plan e curves.

In 1967 , Arnold gave his now classic interpretation o f the Maslov index, thereb y introducing i t int o moder n topology . Subsequentl y thi s inde x aros e i n th e theor y of Lagrangian an d Legendria n cobordism s tha t h e founded .

Poincare discovere d shortl y befor e hi s death tha t a n area-preservin g diffeomor -phism of the annulus which rotates the boundary circle s in opposite directions mus t have a t leas t tw o fixed points . Poincar e di d no t prov e hi s las t geometri c theorem . A proo f wa s give n b y Birkhoff , bu t furthe r developmen t i n th e field stoppe d unti l Arnold mad e th e nex t crucia l step . H e guesse d tha t th e symplectomorphism s o f symplectic manifold s mus t hav e man y fixed point s (n o les s tha n th e su m o f th e Betti numbers) . Th e Arnol d hypothesi s le d t o substantia l progres s i n variationa l calculus, the founding o f symplectic topology, the discovery of quantum cohomolog y and Floe r cohomology .

Arnold i s a n individua l wit h a uniqu e wa y o f lookin g a t th e world , a ma n bursting wit h ideas , projects , an d hypotheses . Eac h semeste r Arnol d begin s hi s seminar wit h a lectur e i n whic h h e present s a s problem s t o b e solve d hi s ow n creative ideas , empirica l fact s h e ha s discovered , an d unexpecte d connection s t o which hi s intuitio n ha s le d him . Everythin g Arnol d touche s turn s t o gold . Th e mathematical problem s h e pose s ar e magical ; the y becom e widel y disseminate d and ove r tim e tur n int o ne w work s an d theories .

One ca n alway s rel y o n Vladimi r Igorevich . Throughou t th e lon g perio d o f our acquaintanc e ther e wa s no t a single instanc e whe n Arnol d promise d somethin g that h e di d no t carr y ou t o r whe n h e di d no t se e a projec t throug h t o completion . Indeed, ho w ca n on e accomplis h s o much withou t possessin g enormou s energ y an d focus?

P R E F A C E x i

Aside fro m everythin g else , Vladimi r Igorevic h i s abl e t o inspir e hi s entir e seminar t o a vigorou s work-out . O n a winte r Sunda y eac h year , h e invite s th e participants fo r a sk i trip . H e leads th e pac k ove r sixt y kilometers , dresse d onl y i n his swimmin g trunks , removin g th e latte r t o swi m a t ever y opportunit y presente d by a hol e i n th e ice . Indeed , skiing , swimming , an d bikin g ar e Arnold' s elemen t and, accordin g t o Vladimi r Igorevich , hi s mos t significan t discoverie s wer e mad e during hi s endles s peregrinations .

Arnold possesse s a n unusuall y broa d rang e o f interests . Suffic e i t t o mentio n that h e ha s recentl y complete d a n articl e o n hi s remarkabl e ide a tha t Pushkin' s epigraph t o Eugene Onegin ha s a n origi n i n th e Frenc h nove l Les Liaisons Dan-gereuses. Arnold' s book on the history o f natural philosophy , Huygens and Barrow, Newton and Hooke: Pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals, i s mor e interestin g tha n an y detectiv e story . An d wh o is bette r a t playin g charade s tha n Vladimi r Igorevich ? Wh o i s a bette r guid e t o the haunt s o f Pari s o r t o secre t place s wher e wil d mushroom s an d berrie s lur k o r to hidde n ponds—unknow n eve n t o th e locals—wher e i t i s so wonderful t o swim .

Arnold i s a super b teacher . H e ha s create d a stron g mathematica l school . H e is alway s surrounde d b y students . H e know s ho w t o discove r i n th e mos t unlikel y places ne w and beautifu l problem s tha t gra b th e interes t o f young researchers . Hi s recent wor k o n invariant s o f plane curve s i s a beautifu l example .

The whol e worl d studie s fro m Arnold' s books . Hi s textbook s Mathematical Methods of Classical Mechanics an d Ordinary Differential Equations hav e becom e integral part s o f mathematica l education .

We were fortunate . W e studied unde r Arnold . W e moved i n hi s orbi t an d ha d the opportunit y t o discus s wit h hi m everythin g unde r th e sun . Fo r ever y on e o f u s this wa s a rare gift , a grea t goo d fortun e i n ou r lives .

A fe w word s abou t th e conten t o f the volume . A larg e portio n o f article s i s i n Symplecti c an d Contac t Topolog y an d (ver y

closely related ) Projectiv e Geometry . Chekanov introduce s th e cobordis m ring s o f Lagrang e embeddings , paralle l t o

Arnold's ring s o f Lagrang e immersions . Usin g Gromov' s theor y o f J-holomorphi c curves, he proves stron g relation s betwee n thes e theories . H e also construct s man y Lagrange embedde d tor i i n symplecti c M 2n, whic h ar e Lagrang e isotopi c an d hav e the sam e Liouvill e cocycle , bu t ar e no t Hamiltonia n isotopic .

Polterovich introduces and studies a natural character of the fundamental grou p of th e spac e o f Hamiltonia n diffeomorphism s o f an y compac t symplecti c manifold ; his constructio n "i s made" o f the symplecti c actio n an d th e Maslo v index .

The works of Tabachnikov an d Sedyk h develop the four-vert ex theorem (assert -ing tha t th e curvatur e o f any close d plan e curv e ha s a t leas t fou r extrem a points) : Sedykh establishe s a simila r fac t i n a discret e settin g (fo r plan e polygons) , an d Tabachnikov give s a ne w proo f o f th e assertio n (firs t prove d b y E . Ghys ) tha t th e Schwarzian derivativ e o f an y diffeomorphis m o f MP 1 ha s a t leas t fou r zeroes .

Bruce an d Zakalyuki n stud y th e varietie s projectivel y dua l t o th e caustic s o f geometric optics , an d prov e tha t i n th e "generic " situatio n th e Nas h transfor m o f the causti c (i.e. , th e closur e o f the se t o f tangen t plane s a t it s nonsingula r points ) is a smoot h manifold .

The wor k o f Chmuto v an d Goryuno v deal s wit h th e topolog y o f Legendria n links. The y prov e ne w estimate s o n th e Bennequi n number s o f suc h link s (i.e. ,

xii PREFAC E

their self-linkin g number s wit h respec t t o th e natura l framing ) an d develo p th e analogues o f the Kauffma n an d HOMFL Y polynomial s fo r suc h links .

Another larg e par t o f th e volum e consist s o f work s i n mor e o r les s classica l Singularity Theory , especiall y i n it s topologica l aspects .

Kazaryan uses universal complexes of classes of multisingularities t o express the first Cher n clas s of the circl e fibre bundle ove r a manifold i n terms o f the minimu m points o f th e restrictio n t o th e fibres o f a generi c functio n define d o n th e spac e of the bundle .

Gusein-Zade prove s tha t a ger m o f a functio n / o f tw o rea l variable s wit h isolated singularitie s ha s a deformatio n withou t critica l point s i f an d onl y i f it s zero-level curv e { / = 0 } consists o f one branch .

Newton polyhedr a connec t algebrai c geometr y wit h th e geometr y o f conve x polyhedra. Th e wor k o f Khovansk h contain s a ne w resul t i n combinatoric s an d finite-additive measur e theor y o n conve x polyhedra suggeste d b y thi s connection .

Scherbak studie s the function germ s invariant unde r th e group Z | o f reflection s in severa l coordinat e hyperplanes .

In Vassiliev' s articl e Pham' s generalize d Picard-Lefschet z formulas , whic h ex -press th e monodrom y actio n o n th e homolog y group s (wit h twiste d coefficients ) of complement s o f algebrai c varietie s dependin g o n parameters , ar e extende d t o a large clas s o f loca l degeneration s o f these varieties .

The topics of the next five papers ar e related to differen t version s of topologica l field theories .

Givental studie s th e quantu m cohomolog y o f the manifol d o f complete flags i n Cn; i n particular , h e give s a proo f o f a generalizatio n o f th e mirro r conjectur e fo r this manifold .

In the paper by Felder, Tarasov , and Varchenko, the authors give integral repre-sentations for solutions of the system of elliptic quantum Knizhnik-Zamolodchikov -Bernard (qKZB ) differenc e equation s associate d wit h 5/2 . Th e qKZ B equation s ar e a quantu m deformatio n o f the KZ B differential equation s obeye d b y the correlatio n functions o f the Wess-Zumino-Witten mode l of conformal field theory o n tori . Th e main them e o f thi s pape r i s a geometri c constructio n o f tensor product s o f evalua -tion Verm a module s ove r th e ellipti c quantu m grou p E TiV(sl2)- A s a n applicatio n of the geometri c construction , th e author s giv e a n explci t descriptio n o f the Beth e ansatz eigenfunction s o f the commutin g syste m o f difference operators .

Fialowski an d Fuch s stud y th e versa l deformation s o f the Li e algebra o f forma l vector fields i n R l wit h trivia l 1-jet .

Khesin studie s modul i space s o f integrable (0,1)-connection s o n comple x man -ifolds, thu s developin g a comple x analogu e o f Chern-Simon s theory .

Lando proves tha t th e algebr a o f chord diagram s arisin g i n the theor y o f finite-type kno t invariants , i s a Hop f algebr a alread y ove r Z . (Kontsevic h establishe d such a structur e onl y ove r th e comple x numbers. )

Natanzon an d B . Shapiro-M . Shapiro-Vainshtei n exten d recen t investigation s by Arnol d o n th e topologica l classificatio n o f comple x trigonometri c polynomi -als. Natanzo n establishe s th e structur e o f th e modul i spac e o f meromorphi c func -tions wit h a fixed topologica l typ e o n a Rieman n surface , an d Shapiro-Shapiro -Vainshtein presen t a n explici t calculatio n o f th e numbe r o f nonequivalen t mero -morphic function s o n Rieman n surface s wit h som e restrictions o n the order s o f th e poles.

PREFACE xii i

Finashin an d Shusti n construc t man y example s o f complex projectiv e surface s X, define d ove r R, havin g prescribe d singula r points . Thi s constructio n provide s in particula r th e first know n example s o f simpl y connecte d Spin-manifold s X/con j with positiv e signature .

Finally, iTyashenk o an d Sevryu k contribut e t o tw o othe r area s o f Arnold' s activity: bifurcatio n theor y fo r dynamica l system s an d KA M theory .

We ver y muc h hop e tha t thi s editio n wil l reflec t a t leas t a smal l portio n o f Arnold's interest s an d o f the knowledg e an d enthusias m tha t hav e been hi s gift s t o us.

We wish hi m man y mor e surprisin g discoverie s an d man y ne w students . We thank th e AM S an d S . Gindiki n fo r assistanc e i n publishing thi s volume .

A. Khovanskii , A . Varchenko , V . Vassilie v

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Selected Title s i n Thi s Serie s

180 A . Khovanskii , A . Varchenko , an d V . Vassil iev , Editors , Topic s i n Singularit y Theory

179 V . M . Buchstabe r an d S . P . Novikov , Editors , Solitons , Geometry , an d Topology :

On th e Crossroa d

178 V . Kreinovic h an d G . Mints , Editors , Problem s o f Reducin g th e Exhaustiv e Searc h

177 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Edi tors ,

Topics i n Statistica l an d Theoretica l Physic s (F . A . Berezi n Memoria l Volume )

176 E . V . Shikin , Editor , Som e Question s o f Differentia l Geometr y i n th e Larg e

175 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Edi tors ,

Contemporary Mathematica l Physic s (F . A . Berezi n Memoria l Volume )

174 A . A . Bol ibruch , A . S . Merkur'ev , an d N . Yu . N e t s v e t a e v , Editors , Mathematic s

in St . Petersbur g

173 V . Kharlamov , A . Korchagin , G . PolotovskiT , an d O . Viro , Editors , Topolog y o f

Real Algebrai c Varietie s an d Relate d Topic s

172 K . N o m i z u , Editor , Selecte d Paper s o n Numbe r Theor y an d Algebrai c Geometr y

171 L . A . Bunimovich , B . M . Gurevich , an d Ya . B . Pes in , Editors , Sinai' s Mosco w

Seminar o n Dynamica l System s

170 S . P . Novikov , Editor , Topic s i n Topolog y an d Mathematica l Physic s

169 S . G . Gindiki n an d E . B . Vinberg , Edi tors , Li e Group s an d Li e Algebras : E . B .

Dynkin's Semina r

168 V . V . Kozlov , Editor , Dynamica l System s i n Classica l Mechanic s

167 V . V . Lychagin , Editor , Th e Interpla y betwee n Differentia l Geometr y an d Differentia l

Equations

166 O . A . Ladyzhenskaya , Editor , Proceeding s o f the St . Petersbur g Mathematica l Society ,

Volume II I

165 Yu . I lyashenk o an d S . Yakovenko , Editors , Concernin g th e Hilber t 16t h Proble m

164 N . N . Uraltseva , Editor , Nonlinea r Evolutio n Equation s

163 L . A . Bokut' , M . Hazewinkel , an d Yu . G . Reshetnyak , Editors , Thir d Siberia n

School "Algebr a an d Analysis "

162 S . G . Gindikin , Editor , Applie d Problem s o f Rado n Transfor m

161 K . N o m i z u , Editor , Selecte d Paper s o n Analysis , Probability , an d Statistic s

160 K . N o m i z u , Editor , Selecte d Paper s o n Numbe r Theory , Algebrai c Geometry , an d

Differential Geometr y

159 O . A . Ladyzhenskaya , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume I I

158 A . K . Ke lmans , Editor , Selecte d Topic s i n Discret e Mathematics : Proceeding s o f th e

Moscow Discret e Mathematic s Semina r 1972-199 0

157 M . Sh . B irman , Editor , Wav e Propagation . Scatterin g Theor y

156 V . N . Geras imov , N . G . Nes terenko , an d A . I . Valitskas , Thre e Paper s o n

Algebras an d Thei r Representation s

155 O . A . Ladyzhenskay a an d A . M . Vershik , Editors , Proceeding s o f th e St .

Petersburg Mathematica l Society , Volum e I

154 V . A . A r t a m o n o v e t ai . , Selecte d Paper s i n /C-Theor y

153 S . G . Gindikin , Editor , Singularit y Theor y an d Som e Problem s o f Functiona l Analysi s

152 H . Draskovicov a e t al. , Ordere d Set s an d Lattice s I I

151 I . A . Aleksandrov , L . A . Bokut' , an d Yu . G . Reshetnyak , Editors , Secon d

Siberian Winte r Schoo l "Algebr a an d Analysis "

150 S . G . Gindikin , Editor , Spectra l Theor y o f Operator s

149 V . S . AfraTmovic h e t al. , Thirtee n Paper s i n Algebra , Functiona l Analysis , Topology ,

and Probability , Translate d fro m th e Russia n

148 A . D . Aleksandrov , O . V . B e l e g r a d e k , L . A . Bokut' , a n d Yu . L . E r s h o v , Editors. Firs t Siberia n Winte r Schoo l "Algebr a an d Analysis "

(See th e AM S catalo g fo r earlie r titles )