tribute to vladimir arnold - national chiao tung universitysmchang/news/2012_v_arnold.pdf ·...

Tribute toVladimir ArnoldBoris Khesin and Serge Tabachnikov,

Coordinating Editors

Vladimir Arnold, an eminent mathematician ofour time, passed away on June 3, 2010, nine daysbefore his seventy-third birthday. This article,along with one in the next issue of the Notices,touches on his outstanding personality and hisgreat contribution to mathematics.

A word about spelling: we use “Arnold”, asopposed to “Arnol’d”; the latter is closer to the Rus-sian pronunciation, but Vladimir Arnold preferredthe former (it is used in numerous translations ofhis books into English), and we use it throughout.

Arnold in His Own WordsIn 1990 the second author interviewed V. Arnoldfor a Russian magazine Kvant (Quantum). Thereadership of this monthly magazine for physicsand mathematics consisted mostly of high schoolstudents, high school teachers, and undergraduatestudents; the magazine had a circulation of about200,000. As far as we know, the interview was nevertranslated into English. We translate excerpts fromthis interview;1 the footnotes are ours.

Q: How did you become a mathematician? Whatwas the role played by your family, school, math-ematical circles, Olympiads? Please tell us aboutyour teachers.

A: I always hated learning by rote. For thatreason, my elementary school teacher told myparents that a moron, like myself, would nevermanage to master the multiplication table.

Boris Khesin is professor of mathematics at the University

of Toronto. His email address is [email protected].

edu.

Serge Tabachnikov is professor of mathematics at Penn-

sylvania State University. His email address is tabachni@

math.psu.edu.

1Full text is available in Russian on the website of Kvantmagazine (July 1990), http://kvant.mirror1.mccme.

ru/.

DOI: http://dx.doi.org/10.1090/noti810

My first mathematical revelation was when Imet my first real teacher of mathematics, IvanVassilievich Morozkin. I remember the problemabout two old ladies who started simultaneously

Vladimir Igorevich Arnold

from two towns to-ward each other, metat noon, and whoreached the oppo-site towns at 4 p.m.and 9 p.m., respec-tively. The questionwaswhentheystartedtheir trip.

We didn’t have al-gebra yet. I inventedan “arithmetic” solu-tion (based on a scal-ing—or similarity—argument) and expe-rienced a joy of dis-covery; the desire toexperience this joy

again was what made me a mathematician.A. A. Lyapunov organized at his home “Chil-dren Learned Society”. The curriculum includedmathematics and physics, along with chemistryand biology, including genetics that was just re-cently banned2 (a son of one of our best geneticistswas my classmate; in a questionnaire, he wrote:“my mother is a stay-at-home mom; my father isa stay-at-home dad”).

Q: You have been actively working in mathe-matics for over thirty years. Has the attitude ofsociety towards mathematics and mathematicianschanged?

A: The attitude of society (not only in the USSR)to fundamental science in general, and to mathe-matics in particular, is well describedby I. A. Krylov

2In 1948 genetics was officially declared “a bourgeois

pseudoscience” in the former Soviet Union.

378 Notices of the AMS Volume 59, Number 3

Vladimir Arnold, circa 1985.

in the fable “The hog under the oak”.3 In the 1930sand 1940s, mathematics suffered in this countryless than other sciences. It is well known that Viètewas a cryptographer in the service of Henry IV ofFrance. Since then, certain areas of mathematicshave been supported by all governments, and evenBeria4 cared about preservation of mathematicalculture in this country.

In the last thirty years the prestige of mathe-matics has declined in all countries. I think thatmathematicians are partially to be blamed as well(foremost, Hilbert and Bourbaki), particularly theones who proclaimed that the goal of their sciencewas investigation of all corollaries of arbitrarysystems of axioms.

Q: Does the concept of fashion apply to mathe-matics?

A: Development of mathematics resembles afast revolution of a wheel: sprinkles of water areflying in all directions. Fashion—it is the streamthat leaves the main trajectory in the tangentialdirection. These streams of epigone works attractthe most attention, and they constitute the mainmass, but they inevitably disappear after a whilebecause they parted with the wheel. To remainon the wheel, one must apply the effort in thedirection perpendicular to the main stream.

A mathematician finds it hard to agree thatthe introduction of a new term not supported by

3See A. Givental and E. Wilson-Egolf’s (slightly modern-

ized) translation of this early nineteenth-century Russian

fable at the end of this interview.4The monstrous chief of Stalin’s secret police.

new theorems constitutes substantial progress.However, the success of “cybernetics”, “fractals”,“synergetics”, “catastrophe theory”, and “strangeattractors” illustrates the fruitfulness of wordcreation as a scientific method.

Q: Mathematics is a very old and important part

of human culture. What is your opinion about the

place of mathematics in cultural heritage?

A: The word “mathematics” means scienceabout truth. It seems to me that modern science(i.e., theoretical physics along with mathematics) isa new religion, a cult of truth, founded by Newtonthree hundred years ago.

Q: When you prove a theorem, do you “create”

or “discover” it?

A: I certainly have a feeling that I am discoveringsomething that existed before me.

Q: You spend much time popularizing math-

ematics. What is your opinion about populariza-

tion? Please name merits and demerits of this hard

genre.

A: One of the very first popularizers, M. Faraday,arrived at the conclusion that “Lectures whichreally teach will never be popular; lectures whichare popular will never teach.” This Faraday effect iseasy to explain: according to N. Bohr, clearness andtruth are in a quantum complementarity relation.

Q: Many readers of Kvant aspire to become

mathematicians. Are there “indications” and “con-

traindications” to becoming a mathematician, or

can anyone interested in the subject become one?

Is it necessary for a mathematician-to-be to success-

fully participate in mathematical Olympiads?

A: When 90-year-old Hadamard was tellingA. N. Kolmogorov about his participation inConcours Général (roughly corresponding to ourOlympiads), he was still very excited: Hadamardwon only the second prize, while the studentwho had won the first prize also became amathematician, but a much weaker one!

Some Olympiad winners later achieve nothing,and many outstanding mathematicians had nosuccess in Olympiads at all.

Mathematicians differ dramatically by their timescale: some are very good tackling 15-minuteproblems, some are good with the problems thatrequire an hour, a day, a week, the problemsthat take a month, a year, decades of thinking.A. N. Kolmogorov considered his “ceiling” to betwo weeks of concentrated thinking.

Success in an Olympiad largely depends onone’s sprinter qualities, whereas serious mathe-matical research requires long distance endurance(B. N. Delaunay used to say, “A good theorem takesnot 5 hours, as in an Olympiad, but 5,000 hours”).

There are contraindications to becoming a re-search mathematician. The main one is lack oflove of mathematics.

March 2012 Notices of the AMS 379

Teaching at Moscow State University, 1983.

But mathematical talents can be very diverse:geometrical and intuitive, algebraic and compu-tational, logical and deductive, natural scientificand inductive. And all kinds are useful. It seemsto me that one’s difficulties with the multipli-cation table or a formal definition of half-planeshould not obstruct one’s way to mathematics.An extremely important condition for seriousmathematical research is good health.

Q: Tell us about the role of sport in your life.A: When a problem resists a solution, I jump

on my cross-country skis. After forty kilometers asolution (or at least an idea for a solution) alwayscomes. Under scrutiny, an error is often found.But this is a new difficulty that is overcome in thesame way.

The Hog Under the Oak

A Hog under a mighty OakHad glutted tons of tasty acorns, then, supine,

Napped in its shade; but when awoke,He, with persistence and the snoot of real swine,

The giant’s roots began to undermine."The tree is hurt when they’re exposed.”

A Raven on a branch arose."It may dry up and perish—don’t you care?”

"Not in the least!” The Hog raised up its head.“Why would the prospect make me scared?

The tree is useless; be it deadTwo hundred fifty years, I won’t regret a second.Nutritious acorns—only that’s what’s reckoned!”—“Ungrateful pig!” the tree exclaimed with scorn.

“Had you been fit to turn your mug aroundYou’d have a chance to figure outWhere your beloved fruit is born.”

Likewise, an ignoramus in defianceIs scolding scientists and science,And all preprints at lanl_dot_gov,

Oblivious of his partaking fruit thereof.

Arnold’s Doctoral StudentsThe list below includes those who defended theirPh.D. theses under Arnold’s guidance. We have toadmit that it was difficult to compile. Along with

straightforward cases when Arnold supervised thethesis and was listed as the person’s Ph.D. advisor,there were many other situations. For example,in Moscow State University before perestroika, aPh.D. advisor for a foreigner had to be a memberof the Communist Party, so in such cases therewas a different nominal Ph.D. advisor while Arnoldwas supervising the student’s work. In other casesthere were two co-advisors or there was a differ-ent advisor of the Ph.D. thesis, while the persondefended the Doctor of Science degree (the secondscientific degree in Russia) under Arnold’s super-vision. In these “difficult cases” the inclusion inthe list below is based on “self-definition” as anArnold student rather than on a formality. Wetried to make the list as complete and precise aspossible, but we apologize in advance for possibleomissions: there were many more people whosework Arnold influenced greatly and who mightfeel they belong to Arnold’s school.

Names are listed chronologically according tothe defense years, which are given in parentheses.Many former Arnold’s students defended the sec-ond degree, the Doctor of Science or Habilitation,but we marked it only in the cases where the firstdegree was not under Arnold’s supervision.

Edward G. Belaga (1965)Andrei M. Leontovich (1967)Yulij S. Ilyashenko (1969) (1994, DSci)Anatoly G. Kushnirenko (1970)Askold G. Khovanskii (1973)Nikolai N. Nekhoroshev (1973)Alexander S. Pyartli (1974)Alexander N. Varchenko (1974)Sabir M. Gusein-Zade (1975)Alexander N. Shoshitaishvili (1975)Rifkat I. Bogdanov (1976)Lyudmila N. Bryzgalova (1977)Vladimir M. Zakalyukin (1977)Emil Horozov (1978)Oleg V. Lyashko (1980)Olga A. Platonova (1981)Victor V. Goryunov (1982)Vladimir N. Karpushkin (1982)Vyacheslav D. Sedykh (1982)Victor A. Vassiliev (1982)Aleksey A. Davydov (1983)Elena E. Landis (1983)Vadim I. Matov (1983)Sergei K. Lando (1986)Inna G. Scherbak (1986)Oleg P. Scherbak (1986)Victor I. Bakhtin (1987)Alexander B. Givental (1987)Mikhail B. Sevryuk (1988)Anatoly I. Neishtadt (1976) (1989, DSci)Ilya A. Bogaevsky (1990)Boris A. Khesin (1990)Vladimir P. Kostov (1990)


Boris Z. Shapiro (1990)Maxim E. Kazarian (1991)Ernesto Rosales-Gonzalez (1991)Oleg G. Galkin (1992)Michael Z. Shapiro (1992)Alexander Kh. Rakhimov (1995)Francesca Aicardi (1996)Yuri V. Chekanov (1997)Emmanuel Ferrand (1997)Petr E. Pushkar (1998)Jacques-Olivier Moussafir (2000)Mauricio Garay (2001)Fabien Napolitano (2001)Ricardo Uribe-Vargas (2001)Mikhail B. Mishustin (2002)Adriana Ortiz-Rodriguez (2002)Gianmarco Capitanio (2004)Oleg N. Karpenkov (2005)Alexander M. Lukatsky (1975) (2006, DSci)

Alexander Givental

To Whom It May ConcernNo estь i Boжii sud ...

M. �. Lermontov, “Smertь Poзta”6

Posthumousmemoirs seemtohave the unintendedeffect of reducing the person’s life to a collection ofstories. For most of us it would probably be a justand welcome outcome, but for Vladimir Arnold,I think, it would not. He tried and managed totell us many different things about mathematics,education, and beyond, and in many cases we’vebeen rather slow listening or thinking, so I believewe will be returning again and again not only toour memories of him but to his own words aswell. What is found below is not a memoir, but arecommendation letter, albeit a weak one, for hedid not get the prize, and yet hopefully useful asan interim attempt to overview his mathematicalheritage.

January 25, 2005Dear Members of [the name of the committee],

You have requested my commentaries on the workof Vladimir Arnold. Writing them is an honorableand pleasurable task for me.

In the essence the task is easy:Yes, Vladimir Arnold fully merits [the name of

the prize] since his achievements are of extraordi-

nary depth and influence.His work indeed resolves fundamental prob-

lems, and introduces unifying principles, and

opens up major new areas, and (at least in some of

Alexander Givental is professor of mathematics at the

University of California, Berkeley. His email address is

[email protected], there is God’s Court, too. . ., M. Yu. Lermontov,“Death of Poet”.

At Dubna, 2006.

these areas) it introduces powerful new techniquestoo.

On the other hand, writing this letter is noteasy, mainly because the ways Arnold’s work con-tributes to our knowledge are numerous and go farbeyond my personal comprehension. As Arnold’sstudent, I am quite familiar with those aspects ofhis work which inspired my own research. Outsidethese areas, hopefully, I will be able to conveythe conventional wisdom about Arnold’s most fa-mous achievements. Yet this leaves out the oceanof numerous, possibly less famous but extremelyinfluential, contributions, of which I have onlypartial knowledge and understanding. So, I willhave to be selective here and will mention just ahandful of examples which I am better familiarwith and which for this reason may look chosenrandomly.

Perhaps the most legendary, so to speak, ofArnold’s contributions is his work on small

denominators,7 followed by the discovery ofArnold’s diffusion,8 and known now as part of theKolmogorov–Arnold–Moser theory. Among otherthings, this work contains an explanation (or, de-pending on the attitude, a proof, and a highlytechnical one) of stability of the solar planetarysystem. Even more importantly, the KAM theoryprovides a very deep insight into the real-world dy-namics (perhaps one of the few such insights so far,one more being stability of Anosov’s systems) and

7Small denominators III. Problems of stability of motionin classical and celestial mechanics, Uspekhi Mat. Nauk

18 (1963), no. 6, 91–192, following Small denominatorsI. Mappings of a circle onto itself, Izvestia AN SSSR, Ser.

Mat. 25 (1961), 21–86, Small denominators II. Proof of atheorem of A. N. Kolmogorov on the preservation of con-ditionally periodic motions under a small perturbationof the Hamiltonian, Uspekhi Mat. Nauk 18 (1963), no. 5,

13–40, and a series of announcements in DAN SSSR.8Instability of dynamical systems with many degrees offreedom, DAN SSSR 156 (1964), 9–12.


is widely regarded as one of the major discoveriesof twentieth-century mathematical physics.

Symplectic geometry has established itself asa universal geometric language of Hamiltonianmechanics, calculus of variations, quantization,representation theory and microlocal analysis ofdifferential equations. One of the first mathe-maticians who understood the unifying nature ofsymplectic geometry was Vladimir Arnold, and hiswork played a key role in establishing this status ofsymplectic geometry. In particular, his monographMathematical Methods of Classical Mechanics9 hasbecome a standard textbook, but thirty years agoit indicated a paradigm shift in a favorite subjectof physicists and engineers. The traditional “an-alytical” or “theoretical” mechanics got suddenlytransformed into an active region of modern math-ematics populated with Riemannian metrics, Liealgebras, differential forms, fundamental groups,and symplectic manifolds.

Just as much as symplectic geometry is merelya language, symplectic topology is a profoundproblem. Many of the best results of such power-ful mathematicians as Conley, Zehnder, Gromov,Floer, Hofer, Eliashberg, Polterovich, McDuff, Sala-mon, Fukaya, Seidel, and a number of othersbelong to this area. It would not be too much ofan overstatement to say that symplectic topologyhas developed from attempts to solve a singleproblem: to prove the Arnold conjecture aboutHamiltonian fixed points and Lagrangian intersec-tions.10 While the conjecture has been essentiallyproved11 and many new problems and ramifica-tions discovered, the theory in a sense continuesto explore various facets of that same topologicalrigidity property of phase spaces of Hamiltonianmechanics that goes back to Poincaré and Birkhoffand whose symplectic nature was first recognizedby Arnold in his 1965 notes in Comptes Rendus.

Arnold’s work in Riemannian geometry ofinfinite-dimensional Lie groups had almost asmuch of a revolutionizing effect on hydrodynam-ics as his work in small denominators produced inclassicalmechanics. Inparticular,Arnold’s seminal

9Nauka, Moscow, 1974.10First stated in Sur une propriété topologique des ap-plications globalement canoniques de la mécaniqueclassique, C. R. Acad. Sci. Paris 261 (1965), 3719–3722,

and reiterated in a few places, including an appendix to

Math Methods….11By Hofer (1986) for Lagrangian intersections and by

Fukaya–Ono (1996) for Hamiltonian diffeomorphisms,

while “essentially” refers to the fact that the conjectures

the way Arnold phrased them in terms of critical point of

functions rather than (co)homology, and especially in the

case of possibly degenerate fixed or intersection points,

still remain open (and correct just as likely as not, but

with no counterexamples in view).

Between the lectures at Arnoldfest, 1997.

paper in Annales de L’Institut Fourier12 draws onhis observation that flows of incompressible fluidscan be interpreted as geodesics of right-invariantmetrics on the groups of volume-preserving dif-feomorphisms. Technically speaking, the aim ofthe paper is to show that most of the sectionalcurvatures of the area-preserving diffeomorphismgroup of the standard 2-torus are negative andthus the geodesics on the group typically divergeexponentially. From time to time this result makesthe news as a “mathematical proof of impossibilityof long-term weather forecasts”. More importantly,the work had set Euler’s equations on coadjointorbits as a blueprint and redirected the attentionin many models of continuum mechanics towardsymmetries, conservation laws, relative equilib-ria, symplectic reduction, topological methods (inworks of Marsden, Ratiu, Weinstein, Moffat, andFreedman among many others).13

Due to the ideas of Thom and Pham andfundamental results of Mather and Malgrange,singularity theory became one of the most activefields of the seventies and eighties, apparentlywith two leading centers: Brieskorn’s seminarin Bonn and Arnold’s seminar in Moscow. Thetheory of critical points of functions and itsapplications to classification of singularities ofcaustics, wave fronts and short-wave asymptoticsin geometrical optics as well as their relationshipwith the ADE-classification are perhaps the mostfamous (among uncountably many other) results

12Sur la géométrie différentielle des groupes de Lie de di-mension infinie et ses applications à l’hydrodynamiquedes fluides parfaits, Ann. Inst. Fourier 16:1 (1966), 319–

361, based on a series of earlier announcements in C. R.

Acad. Sci. Paris.13As summarized in the monograph Topological Meth-ods in Hydrodynamics, Springer-Verlag, 1998, by Arnold

and Khesin.


V. Arnold, 1968.

of Arnold in singularity theory.14 Arnold’s rolein this area went, however, far beyond his ownpapers.

Imagine a seminar of about thirty partici-pants: undergraduates writing their first researchpapers, graduate students working on their dis-sertation problems, postgraduates employed else-where as software engineers but unwilling togive up their dream of pursuing mathematicseven if only as a hobby, several experts—Fuchs,Dolgachev, Gabrielov, Gusein-Zade, Khovansky,Kushnirenko, Tyurin, Varchenko, Vassiliev—andthe leader, Arnold—beginning each semester byformulating a bunch of new problems, giving talksor listening to talks, generating and generouslysharing new ideas and conjectures, editing his stu-dents’ papers, and ultimately remaining the onlyperson in his seminar who would keep in mindeveryone else’s works-in-progress and understandtheir relationships. Obviously, a lion’s share ofhis students’ achievements (and among the quitefamous ones are the theory of Newton polyhedraby Khovansky and Kushnirenko or Varchenko’s re-sults on asymptotical mixed Hodge structures andsemicontinuity of Steenbrink spectra) is due to hishelp, typically in the form of working conjectures,but every so often through his direct participation(for, with the exception of surveys and obituaries,Arnold would refuse to publish joint papers—wewill learn later why).

Moreover, under Arnold’s influence, the elitebranch of topology and algebraic geometry study-ing singular real and complex hypersurfaces wastransformed into a powerful tool of applied

14Normal forms of functions near degenerate criti-cal points, the Weyl groups Ak, Dk, Ek and Lagrangiansingularities, Funct. Anal. Appl. 6, no. 4 (1972), 3–25;

see also Arnold’s inspiring paper in Proceedings of the

ICM-74 Vancouver and the textbooks Singularities of Dif-ferential Maps, Vols. I and II, by Arnold, Gusein-Zade and

Varchenko.

mathematics dealing with degenerations of allkinds of mathematical objects (metamorphosesof wave fronts and caustics, evolutes, evolventsand envelopes of plane curves, phase diagramsin thermodynamics and convex hulls, accessibilityregions in control theory, differential forms andPfaff equations, symplectic and contact struc-tures, solutions of Hamilton-Jacobi equations,the Hamilton-Jacobi equations themselves, theboundaries between various domains in func-tional spaces of all such equations, etc.) andmerging with the theory of bifurcations (of equi-libria, limit cycles, or more complicated attractorsin ODEs and dynamical systems). Arnold had de-veloped a unique intuition and expertise in thesubject, so that when physicists and engineerswould come to him asking what kind of catastro-phes they should expect in their favorite problems,he would be able to guess the answers in smalldimensions right on the spot. In this regard, thesituation would resemble experimental physics orchemistry, where personal expertise is often moreimportant than formally registered knowledge.

Having described several (frankly, quite obvi-ous) broad areas of mathematics reshaped byArnold’s seminal contributions, I would like toturn now to some more specific classical problemswhich attracted his attention over a long timespan.

The affirmative solution of the 13th Hilbertproblem (understood as a question about super-positions of continuous functions) given by Arnoldin his early (essentially undergraduate) work15 wasthe beginning of his interest in the “genuine” (andstill open) Hilbert’s problem: Can the root ofthe general degree 7 polynomial considered asan algebraic function of its coefficients be writ-ten as a superposition of algebraic functions of 2variables? The negative16 solution to the more gen-eral question about polynomials of degree n wasgiven by Arnold in 1970 for n = 2r .17 The resultwas generalized by V. Lin. Furthermore, Arnold’sapproach, based on his previous study of coho-mology of braid groups, later gave rise to Smale’sconcept of topological complexity of algorithmsand Vassiliev’s results on this subject. Even moreimportantly, Arnold’s study of braid groups viatopologyofconfigurationspaces18 wasgeneralizedby Brieskorn to E. Artin’s braid groups associatedwith reflection groups. The latter inspired Orlik–Solomon’s theory of hyperplane arrangements,

15On the representations of functions of several vari-ables as a superposition of functions of a smallernumber of variables, Mat. Prosveshchenie (1958), 41–46.16That is, positive in Hilbert’s sense.17Topological invariants of algebraic functions. II, Funct.

Anal. Appl. 4 (1970), no. 2, 1–9.18The cohomology ring of the group of dyed braids, Mat.

Zametki 5 (1969), 227–231.


K. Saito–Terao’s study of free divisors, Gelfand’sapproach to hypergeometric functions, Aomoto’swork on Yang-Baxter equations, and Varchenko–Schekhtman’s hypergeometric “Bethe ansatz” forsolutions of Knizhnik–Zamolodchikov equationsin conformal field theory.

Arnold’s result19 on the 16th Hilbert prob-lem, Part I, about disposition of ovals of realplane algebraic curves, was immediately improvedby Rokhlin (who applied Arnold’s method butused more powerful tools from the topologyof 4-manifolds). This led Rokhlin to his proofof a famous conjecture of Gudkov (who cor-rected Hilbert’s expectations in the problem),inspired many new developments (due to Viroand Kharlamov among others), and is consid-ered a crucial breakthrough in the history of realalgebraic geometry.

Among other things, the paper of Arnold out-lines an explicit diffeomorphism between S4 andthe quotient of CP 2 by complex conjugation.20 Thefact was rediscovered by Kuiper in 1974 and isknown as Kuiper’s theorem [31]. Arnold’s argu-ment, based on hyperbolicity of the discriminantin the space of Hermitian forms, was recentlyrevived in a far-reaching paper by M. Atiyah andJ. Berndt [19].

Another work of Arnold in the same field21 uni-fied the Petrovsky-Oleinik inequalities concerningtopology of real hypersurfaces (or their comple-ments) and brought mixed Hodge structures (justintroduced by Steenbrink into complex singularitytheory) into real algebraic geometry.

Arnold’s interest in the 16th Hilbert problem,Part II, on the number of limit cycles of polynomialODE systems on the plane has been an open-endedsearch for simplifying formulations. One such for-mulation22 (about the maximal number of limitcycles born under a nonconservative perturbationof a Hamiltonian system and equivalent to theproblem about the number of zeroes of Abelianintegrals over a family of real algebraic ovals) gen-erated extensive research. The results here includethe general deep finiteness theorems of Khovan-sky and Varchenko, Arnold’s conjecture aboutnonoscillatory behavior of the Abelian integrals,his geometrization of higher-dimensional Sturm

19The situation of ovals of real algebraic plane curves,the involutions of four-dimensional smooth manifolds,and the arithmetic of integral quadratic forms, Funct.

Anal. Appl. 5 (1971), no. 3, 1–9.20Details were published much later in The branched cov-ering CP2 → S4, hyperbolicity and projective topology,Sibirsk. Mat. Zh. 29 (1988), no. 5, 36–47.21The index of a singular point of a vector field,the Petrovsky-Oleinik inequalities, and mixed Hodgestructures, Funct. Anal. Appl. 12 (1978), no. 1, 1–14.22V. I. Arnold, O. A. Oleinik, Topology of real algebraicvarieties, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1979),

no. 6, 7–17.

V. Arnold and J. Moser at the Euler Institute,St. Petersburg, 1991.

theory of (non)oscillations in linear Hamiltoniansystems,23 various attempts to prove this conjec-ture (including a series of papers by Petrov-Tan’kinon Abelian integrals over elliptic curves, my ownapplication of Sturm’s theory to nonoscillation ofhyperelliptic integrals, and more recent estimatesof Grigoriev, Novikov–Yakovenko), and furtherwork by Horozov, Khovansky, Ilyashenko and oth-ers. Yet another modification of the problem (adiscrete one-dimensional analogue) suggested byArnold led to a beautiful and nontrivial theoremof Yakobson in the theory of dynamical systems[39].

The classical problem in the theory of Dio-phantine approximations of inventing the higher-dimensional analogue of continued fractions hasbeen approached by many authors, with a paradox-ical outcome: there are many relatively straightfor-ward and relatively successful generalizations, butnone as unique and satisfactory as the elementarycontinued fraction theory. Arnold’s approach tothis problem24 is based on his discovery of a rela-tionship between graded algebras and Klein’s sails(i.e., convex hulls of integer points inside simplicialconvex cones in Euclidean spaces). Arnold’s prob-lems and conjectures on the subject have led to theresults of E. Korkina and G. Lauchaud generaliz-ing Lagrange’s theorem (which identifies quadraticirrationalities with eventually periodic continuedfractions) and to the work of Kontsevich–Sukhovgeneralizing Gauss’s dynamical system and itsergodic properties. Thus the Klein-Arnold gener-alization, while not straightforward, appears to

23Sturm theorems and symplectic geometry, Funct. Anal.

Appl. 19 (1985), no. 4, 1–10.24A-graded algebras and continued fractions, Com-

mun. Pure Appl. Math. 49 (1989), 993–1000; Higher-dimensional continued fractions, Regul. Chaotic Dyn. 3

(1998), 10–17; and going back to Statistics of integralconvex polyhedra, Funct. Anal. Appl. 14 (1980), no. 1,

1–3; and to the theory of Newton polyhedra.


be just as unique and satisfactory as its classicalprototype.

The above examples show how Arnold’s interestin specific problems helped to transform them intocentral areas of modern research. There are otherclassical results which, according to Arnold’s in-tuition, are scheduled to generate such new areas,but to my understanding have not yet achieved thestatus of important mathematical theories in spiteof interesting work done by Arnold himself andsome others. But who knows? To mention one: theFour-Vertex Theorem, according to Arnold, is theseed of a new (yet unknown) branch of topology(in the same sense as the Last Poincaré Theoremwas the seed of symplectic topology). Another ex-ample: a field-theoretic analogue of Sturm theory,broadly understood as a study of topology of zerolevels (and their complements) of eigenfunctionsof selfadjoint linear partial differential operators.

Perhaps with the notable exceptions of KAM-theory and singularity theory, where Arnold’scontributions are marked not only by fresh ideasbut also by technical breakthroughs (e.g., a heavy-duty tool in singularity theory—his spectral se-quence),25 a more typical path for Arnold wouldbe to invent a bold new problem, attack its firstnontrivial cases with his bare hands, and thenleave developing an advanced machinery to hisfollowers. I have already mentioned how the the-ory of hyperplane arrangements emerged in thisfashion. Here are some other examples of this sortwhere Arnold’s work starts a new area.

In 1980 Arnold invented the concepts of La-grangian and Legendrian cobordisms and stud-ied them for curves using his theory of bifurcationsof wave fronts and caustics.26 The general ho-motopy theory formulation was then given byYa. Eliashberg, and the corresponding “Thomrings” computed in an award-winning treatiseby M. Audin [20]. A geometric realization of La-grange and Legendre characteristic numbers asthe enumerative theory of singularities of globalcaustics and wave fronts was given by V. Vassiliev[38]. The method developed for this task, namelyassociating a spectral sequence to a stratificationof functional spaces of maps according to typesof singularities, was later applied by Vassiliev sev-eral more times, of which his work on Vassilievinvariants of knots is the most famous one.

Arnold’s definition27 of the asymptotic Hopfinvariant as the average self-linking number of

25A spectral sequence for the reduction of functions tonormal forms, Funct. Anal. Appl. 9 (1975), 81–82.26Lagrange and Legendre cobordisms. I, II, Funct. Anal.

Appl. 14 (1980), no. 3, 1–13; no. 4, 8–17.27See The asymptotic Hopf invariant and its applications,Selected translations. Selecta Math. Soviet. 5 (1986), no. 4,

327–345, which is the translation of a 1973 paper and

one of Arnold’s most frequently quoted works.

Lecturing in Toronto, 1997.

trajectories of a volume-preserving flow on asimply connected 3-fold and his “ergodic” theoremabout coincidence of the invariant with Moffatt’shelicity gave the start to many improvements,generalizations, and applications of topologicalmethods in hydro- and magneto-dynamics due toM. H. Freedman et al., É. Ghys, B. Khesin, K. Moffatt,and many others.28

As one can find out, say, on MathSciNet, Arnoldis one of the most prolific mathematicians ofour time. His high productivity is partly dueto his fearless curiosity and enormous appetitefor new problems.29 Paired with his taste andintuition, these qualities often bring unexpectedfruit, sometimes in the areas quite remote fromthe domain of his direct expertise. Here are someexamples.

Arnold’s observation30 on the pairs of triplesof numbers computed by I. Dolgachev andA. Gabrielov and characterizing respectivelyuniformization and monodromy of 14 exceptional

unimodal singularities of surfaces (in Arnold’s clas-sification) is known now under the name Arnold’s

Strange Duality. In 1977, due to Pinkham andDolgachev–Nikulin, the phenomenon received abeautiful explanation in terms of geometry ofK3-surfaces. As became clear in the early nineties,Arnold’s Strange Duality was the first, and highlynontrivial, manifestation of Mirror Symmetry: aprofound conjecture discovered by string theo-rists and claiming a sort of equivalence betweensymplectic topology and complex geometry (orsingularity theory).

28See a review in Chapter III of V. I. Arnold, B. A. Khesin,

Topological methods in hydrodynamics, Applied Math.

Sciences, vol. 125, Springer-Verlag, NY, 1998.29See the unusual book Arnold’s Problems, Springer-

Verlag, Berlin; PHASIS, Moscow, 2004.30See Critical points of smooth functions, Proceed-

ings of the International Congress of Mathematicians

(Vancouver, B. C., 1974), Vol. 1, pp. 19–39.


Arnold’s work in pseudo-periodic geometry31

encouraged A. Zorich to begin a systematic studyof dynamics on Riemann surfaces defined bylevels of closed 1-forms, which led to a number ofremarkable results of Kontsevich–Zorich [29] andothers related to ergodic theory on Teichmüllerspaces and conformal field theory, and of Eskin–Okounkov [24] in the Hurwitz problem of countingramified covers over elliptic curves.

Arnold seems to be the first to suggest32 thatmonodromy (say of Milnor fibers or of flag va-rieties) can be realized by symplectomorphisms.The idea, picked up by M. Kontsevich and S. Don-aldson, was upgraded to the monodromy action onthe Fukaya category (consisting of all Lagrangiansubmanifolds in the fibers and of their Floer com-plexes). This construction is now an importantingredient of the Mirror Symmetry philosophyand gave rise to the remarkable results of M.Khovanov and P. Seidel about faithfulness of suchHamiltonian representations of braid groups [27].

The celebrated Witten’s conjecture proved byM. Kontsevich in 1991 characterizes intersectiontheory on Deligne–Mumford moduli spaces ofRiemann surfaces in terms of KdV-hierarchy ofintegrable systems. A refreshingly new proofof this result was recently given by Okounkov–Pandharipande. A key ingredient in their argumentis an elementary construction of Arnold from hiswork on enumerative geometry of trigonometricpolynomials.33

Among many concepts owing Arnold their exis-tence, let me mention two of general mathematicalstature which do not carry his name.

One is the Maslov index, which proved to beimportant in geometry, calculus of variations,numbers theory, representation theory, quanti-zation, index theory of differential operators,and whose topological origin was explained byArnold.34

The other one is the geometric notion of inte-

grability in Hamiltonian systems. There is a lot of

31Topological and ergodic properties of closed 1-formswith incommensurable periods, Funct. Anal. Appl. 25

(1991), no. 2, 1–12.32See Some remarks on symplectic monodromy of Milnorfibrations, The Floer memorial volume, 99–103, Progr.

Math., 133, Birkhäuser, Basel, 1995.33Topological classification of complex trigonometricpolynomials and the combinatorics of graphs with anidentical number of vertices and edges, Funct. Anal.

Appl. 30 (1996), no. 1, 1–17.34In his paper On a characteristic class entering intoconditions of quantization, Funct. Anal. Appl. 1 (1967),

1–14.

controversy over which of the known integrabilitymechanisms is most fundamental, but there is aconsensus that integrability means a complete setof Poisson-commuting first integrals.

This definition and “Liouville’s Theorem” ongeometric consequences of the integrability prop-erty (namely, foliation of the phase space byLagrangian tori) are in fact Arnold’s originalinventions.

Similar to the case with integrable systems,there are other examples of important develop-ments which have become so common knowledgethat Arnold’s seminal role eventually became in-visible. Let me round up these comments with apeculiar example of this sort.

The joint 1962 paper of Arnold and Sinai35

proves structural stability of hyperbolic linear dif-feomorphisms of the 2-torus. Their idea, pickedup by Anosov, was extended to his famous gen-eral stability theory of Anosov systems [2]. Yet,according to Arnold, the paper is rarely quoted,for the proof contained a mistake (although eachauthor’s contribution was correct, so that neitherone could alone be held responsible). By the way,Arnold cites this episode as the reason why herefrained from writing joint research papers.

To reiterate what I said at the beginning,Vladimir Arnold has made outstanding contri-butions to many areas of pure mathematics andits applications, including those I described aboveand those I missed: classical and celestial mechan-ics, cosmology and hydrodynamics, dynamicalsystems and bifurcation theory, ordinary and par-tial differential equations, algebraic and geometrictopology, number theory and combinatorics, realand complex algebraic geometry, symplectic andcontact geometry and topology, and perhaps someothers. I can think of few mathematicians whosework and personality would influence the scien-tific community at a comparable scale. And beyondthis community, Arnold is a highly visible (andpossibly controversial) figure, the subject of sev-eral interviews, of a recent documentary movie,and even of the night sky show, where one canwatch an asteroid, Vladarnolda, named after him.

I am sure there are other mathematicians whoalso deserve [the name of the prize], but awardingit to Vladimir Arnold will hardly be perceived byanyone as a mistake.

35Arnold, V. I., Sinai, Ya. G., On small perturbations of theautomorphisms of a torus, Dokl. Akad. Nauk SSSR 144

(1962), 695–698.


Ya. Sinai and V. Arnold, photo by J. Moser,1963.

Yakov Sinai

Remembering Vladimir Arnold: Early Years

De mortuis veritas37

My grandparents and Arnold’s grandparents werevery close friends since the beginning of the twen-tieth century. Both families lived in Odessa, whichwas a big city in the southern part of Russia andnow is a part of Ukraine. At that time, Odessa wasa center of Jewish intellectual life, which producedmany scientists, musicians, writers, and other sig-nificant figures.

My maternal grandfather, V. F. Kagan, was awell-known geometer who worked on the founda-tions of geometry. During World War I, he gavethe very first lecture course in Russia on thespecial relativity theory. At various times his lec-tures were attended by future famous physicistsL. I. Mandelshtam, I. E. Tamm, and N. D. Papaleksi.In the 1920s all these people moved to Moscow.

L. I. Mandelshtam was a brother of Arnold’smaternal grandmother. He was the founder andthe leader of a major school of theoretical physicsthat included A. A. Andronov, G. S. Landsberg, andM. A. Leontovich, among others. A. A. Andronovis known to the mathematical community forhis famous paper “Robust systems”, coauthoredwith L. S. Pontryagin, which laid the foundationsof the theory of structural stability of dynami-cal systems. A. A. Andronov was the leader of agroup of physicists and mathematicians workingin Nizhny Novgorod, formerly Gorky, on nonlinearoscillations. M. A. Leontovich was one of the lead-ing physicists in the Soviet Union. In the 1930s

Yakov Sinai is professor of mathematics at Prince-

ton University. His email address is sinai@math.

princeton.edu.37About those who have died, only the truth.

he coauthored with A. N. Kolmogorov the well-known paper on the Wiener sausage. I. E. Tammwas a Nobel Prize winner in physics in the fifties.N. D. Papaleksi was a great expert on nonlinearoptics.

V. I. Arnold was born in Odessa, where hismother had come for a brief visit with her family.She returned to Moscow soon after her son’s birth.When Arnold was growing up, the news that hisfamily had a young prodigy soon became widelyknown. In those days, when we were both in highschool, we did not really know each other. Onone occasion, Arnold visited my grandfather toborrow a mathematics book, but I was not there atthe time. We met for the first time when we wereboth students at the mathematics department ofthe Moscow State University; he was walking bywith Professor A. G. Vitushkin, who ran a fresh-man seminar on real analysis, and Arnold wasone of the most active participants. When Arnoldwas a third-year undergraduate student, he wasinspired by A. N. Kolmogorov to work on super-position of functions of several variables and therelated Hilbert’s thirteenth problem. Eventuallythis work became Arnold’s Ph.D. thesis. When Ivisited the University of Cambridge recently, Iwas very pleased to learn that one of the mainlecture courses there was dedicated to Arnold’sand Kolmogorov’s work on Hilbert’s thirteenthproblem.

Arnold had two younger siblings: a brother,Dmitry, and a sister, Katya, who was the youngest.The family lived in a small apartment in the centerof Moscow. During one of my visits, I was shown atent in the backyard of the building where Arnoldused to spend his nights, even in cold weather. Itseems likely that Arnold’s excellent knowledge ofhistory and geography of Moscow, which many ofhis friends remember with admiration, originatedat that time.

Like me, Arnold loved nature and the outdoors.We did hiking and mountain climbing together.Since I knew Arnold so closely, I often observedthat his ideas both in science and in life came tohim as revelations. I remember one particular oc-casion, when we were climbing in the CaucasusMountains and spent a night with some shepherdsin their tent. In the morning we discovered that theshepherds were gone and had left us alone withtheir dogs. Caucasian dogs are very big, strong,and dangerous, for they are bred and trained tofight wolves. We were surrounded by fiercely bark-ing dogs, and we did not know what to do. Then,all of a sudden, Arnold had an idea. He startedshouting very loudly at the dogs, using all the ob-scenities he could think of. I never heard him usesuch language either before or after this incident,nor did anybody else. It was a brilliant idea, for itworked! The dogs did not touch Arnold and barely


A hiking expedition, 1960s.

touched me. The shepherds returned shortly after-wards, and we were rescued.

On another occasion, roughly at the same time,as Anosov, Arnold, and I walked from the mainMoscow University building to a subway station,which usually took about fifteen minutes, Arnoldtold us that he recently came up with the Galoistheory entirely on his own and explained his ap-proach to us. The next day, Arnold told us thathe found a similar approach in the book by FelixKlein on the mathematics of the nineteenth cen-tury. Arnold was always very fond of this book,and he often recommended it to his students.

Other examples of Arnold’s revelations includehis discovery of the Arnold-Maslov cocycle inthe theory of semi-classical approximations andArnold inequalities for the number of ovals in realalgebraic curves. Many other people who knewArnold personally could provide more examplesof this kind.

Arnold became a graduate student at theMoscow State University in 1959. Naturally itwas A. N. Kolmogorov who became his advisor.In 1957 Kolmogorov gave his famous lecturecourse on dynamical systems, which played apivotal role in the subsequent development ofthe theory. The course was given three years af-ter Kolmogorov’s famous talk at the AmsterdamCongress of Mathematics.

Kolmogorov began his lectures with the expo-sition of the von Neumann theory of dynamicalsystems with pure point spectrum. Everythingwas done in a pure probabilistic way. Later Kol-mogorov found a similar approach in the book byFortet and Blank–Lapierre on random processes,intended for engineers.

This part of Kolmogorov’s lectures had aprofound effect on researchers working onthe measure-theoretic isomorphism in dynam-ical systems, a long-standing problem that goesback to von Neumann. It was shown that whenthe spectrum is a pure point one, it is the only

isomorphism invariant of a dynamical systemand that two systems with the same pure pointspectrum are isomorphic. The excitement aroundthese results was so profound that people beganto believe that the isomorphism theory of sys-tems with continuous spectrum would be just astraightforward generalization of the theory ofsystems with pure point spectrum. However, thiswas refuted by Kolmogorov himself. He proposedthe notion of entropy as a new isomorphism in-variant for systems with continuous spectrum.Since the entropy is zero for systems with purepoint spectrum, it does not distinguish betweensuch systems, but systems with continuous spec-trum might have positive entropy that must bepreserved by isomorphisms. This was a path-breaking discovery, which had a tremendousimpact on the subsequent development of thetheory.

The second part of Kolmogorov’s lectureswas centered around his papers on the preser-vation of invariant tori in small perturbationsof integrable Hamiltonian systems, which werepublished in the Doklady of the Soviet Academyof Sciences. Unfortunately there were no writ-ten notes of these lectures. V. M. Tikhomirov,one of Kolmogorov’s students, hoped for manyyears to locate such notes, but he did not suc-ceed. Arnold used to claim in his correspondencewith many people that good mathematics stu-dents of Moscow University could reconstructKolmogorov’s proof from the text of his pa-pers in the Doklady. However, this was anexaggeration. Recently two Italian mathemati-cians, A. Giorgilli and L. Chierchia, produced aproof of Kolmogorov’s theorem, which was com-plete and close to Kolmogorov’s original proof, asthey claimed.

Apparently Kolmogorov himself never wrotea detailed proof of his result. There might beseveral explanations. At some point, he had plansto work on applications of his technique to thefamous three-body problem. He gave a talk on thistopic at a meeting of the Moscow MathematicalSociety. However, he did not prepare a writtenversion of his talk. Another reason could be thatKolmogorov started to work on a different topicand did not want to be distracted. There mightbe a third reason, although some people woulddisagree with it. It is possible that Kolmogorovunderestimated the significance of his papers.For example, some graduate exams on classicalmechanics included the proof of Kolmogorov’stheorem, so it was easy to assume that the proofwas already known. The theory of entropy, intro-duced by Kolmogorov roughly at the same time,seemed a hotter and more exciting area. He mighthave felt compelled to turn his mind to this newtopic.


A. Kirillov, I. Petrovskii, V. Arnold.

Arnold immediately started to work on allthe problems raised in Kolmogorov’s lectures.In 1963 the Moscow Mathematical Society cele-brated Kolmogorov’s sixtieth birthday. The mainmeeting took place in the Ceremony Hall of theMoscow State University, with about one thou-sand people attending. The opening lecture wasgiven by Arnold on what was later called KAM the-ory, where KAM stands for Kolmogorov, Arnold,Moser. For that occasion, Arnold prepared the firstcomplete exposition of the Kolmogorov theorem.I asked Arnold why he did that, since Kolmogorovpresented his proof in his lectures. Arnold repliedthat the proof of the fact that invariant toriconstitute a set of positive measure was notcomplete. When Arnold asked Kolmogorov aboutsome details of his proof, Kolmogorov replied thathe was too busy at that time with other problemsand that Arnold should provide the details byhimself. This was exactly what Arnold did. I be-lieve that when Kolmogorov prepared his papersfor publication in the Doklady, he did have com-plete proofs, but later he might have forgottensome details. Perhaps it can be expressed betterby saying that it required from him an effort thathe was not prepared to make at that time.

In the following years, Kolmogorov ran a semi-nar on dynamical systems, with the participationof many mathematicians and physicists. At somepoint, two leading physicists, L. A. Arzimovich andM. A. Leontovich, gave a talk at the seminar on theexistence of magnetic surfaces. Subsequently thisproblem was completely solved by Arnold, whosubmitted his paper to the main physics journalin the Soviet Union, called JETP. After some time,the paper was rejected. According to Arnold, thereferee report said that the referee did not under-stand anything in that paper and hence nobodyelse would understand it. M. A. Leontovich helpedArnold to rewrite his paper in the form accessibleto physicists, and it was published eventually. Ac-cording to Arnold, this turned out to be one of hismost quoted papers.

Arnold’s first paper related to the KAM theorywas about smooth diffeomorphisms of the circlethat were close to rotations. Using the methodsof the KAM theory, Arnold proved that such dif-feomorphisms can be reduced to rotations by ap-plying smooth changes of variables. The problemin the general case was called the Arnold prob-lem. It was completely solved by M. Herman andJ.-C. Yoccoz.

A. N. Kolmogorov proved his theorem in theKAM theory for the so-called nondegenerate per-turbations of integrable Hamiltonian systems.Arnold extended this theorem to degenerateperturbations, which arise in many applicationsof the KAM theory.

Arnold proposed an example of the Hamil-tonian system which exhibits a new kind ofinstability and which was later called the Arnolddiffusion. The Arnold diffusion appears in manyphysical problems. New mathematical results onthe Arnold diffusion were recently proved byJ. Mather. V. Kaloshin found many applicationsof the Arnold diffusion to problems of celestialmechanics.

In later years Arnold returned to the theoryof dynamical systems only occasionally. One canmention his results in fluid mechanics (see hisjoint book with B. Khesin [16]) and a series of pa-pers on singularities in the distribution of massesin the universe, motivated by Y. B. Zeldovich. Butall this was done in later years.

Steve Smale

Vladimir I. ArnoldMy first meeting with V. I. Arnold took place inMoscow in September 1961 (certainly I had beenvery aware of him through Moser). After a confer-ence in Kiev, where I had gotten to know Anosov,I visited Moscow, where Anosov introduced me toArnold, Novikov, and Sinai. As I wrote later [35],I was extraordinarily impressed by such a power-ful group of four young mathematicians and thatthere was nothing like that in the West. At mynext visit to Moscow for the world mathematicscongress in 1966 [36], I again saw much of DimaArnold. At that meeting he introduced me to Kol-mogorov.

Perhaps the last time I met Dima was in June2003 at the one-hundred-year memorial confer-ence for Kolmogorov, again in Moscow. In the in-tervening years we saw each other on a numberof occasions in Moscow, in the West, and even inAsia.

Arnold was visiting Hong Kong at the invita-tion of Volodya Vladimirov for the duration of the

Stephen Smale is professor of mathematics at Toyota

Technological Institute at Chicago and City University of

Hong Kong. His email address is [email protected].


fall semester of 1995, while we had just moved toHong Kong. Dima and I often were together on thefantastic day hikes in the Hong Kong countrysideparks. His physical stamina was quite impressive.At that time we two were also the focus of a well-attended panel on contemporary issues of mathe-matics at the Hong Kong University of Science andTechnology. Dima expressed himself in his usualprovocative way! I recall that we found ourselveson the same side in most of the controversies, andcatastrophe theory in particular.

Dima Arnold was a great mathematician, andhere I will just touch on his mathematical contri-butions which affected me the most.

While I never worked directly in the area ofKAM, nevertheless those results had a great im-pact in my scientific work. For one thing theydirected me away from trying to analyze theglobal orbit structures of Hamiltonian ordinarydifferential equations, in contrast to what I wasdoing for (unconstrained) equations. Thus KAMcontributed to my motivation to study mechan-ics in 1970 from the point of view of topology,symmetry, and relative equilibria rather than itsdynamical properties. The work of Arnold hadalready affected those subjects via his big paperon fluid mechanics and symmetry in 1966. SeeJerry Marsden’s account of how our two worksare related [32]. I note that Jerry died even morerecently than Dima.

KAM shattered the chain of hypotheses, er-godic, quasi-ergodic, and metric transitivity goingfrom Boltzmann to Birkhoff. That suggested tome some kind of non-Hamiltonian substitute inthese hypotheses in order to obtain foundationsfor thermodynamics [37].

I read Arnold’s paper on braids and the coho-mology of swallowtails. It was helpful in my workon topology and algorithms, which Victor Vassilievdrastically sharpened.

Dima could express important ideas simply andin such a way that these ideas could transcenda single discipline. His work was instrumental intransforming Kolmogorov’s early sketches into arevolutionary recasting of Hamiltonian dynamicswith sets of invariant curves, tori of positive mea-sure, and Arnold diffusion.

It was my good fortune to have been a part ofDima Arnold’s life and his mathematics.

Mikhail Sevryuk

Some Recollections of Vladimir IgorevichA very large part of my life is connected withVladimir Igorevich Arnold. I became his stu-dent in the beginning of 1980 when I was still a

Mikhail Sevryuk is a senior researcher at the Rus-

sian Academy of Sciences. His email address is

[email protected].

At Arnoldfest, Toronto, 1997.

freshman at the Department of Mechanics andMathematics of Moscow State University. Underhis supervision, I wrote my term papers, master’sthesis, and doctoral thesis. At the end of my firstyear in graduate school, Arnold suggested that Iwrite a monograph on reversible dynamical sys-tems for Springer’s Lecture Notes in Mathematicsseries, and working on this book was one of thecornerstones of my mathematical biography. Forthe last time, I met Vladimir Igorevich (V. I., forshort) on November 3, 2009, at his seminar atMoscow State.

If I had to name one characteristic feature ofArnold as I remember him, I would choose hisagility. He walked fast at walkways of MoscowState (faster than most of the students, not tomention the faculty), his speech was fast andclear, his reaction to one’s remark in a conversa-tion was almost always instantaneous, and oftenutterly unexpected. His fantastic scientific pro-ductivity is well known, and so is his enthusiasmfor sports.

V. I. always devoted a surprising amount oftime and effort to his students. From time totime, he had rather weak students, but I donot recall a single case when he rejected evena struggling student. In the 1980s almost everymeeting of his famous seminar at Moscow Statehe started with “harvesting”: collecting notes ofhis students with sketches of their recent math-ematical achievements or drafts of their papers(and Arnold returned the previously collectedones with his corrections and suggestions). Aftera seminar or a lecture, he often continued talkingwith participants for another 2–3 hours. Arnold’sgenerosity was abundant. Many times, he gavelong written mathematical consultations, even topeople unknown to him, or wrote paper reportssubstantially exceeding the submitted papers.


Dubna, 2006.

In recent years, he used all his energy to stop arapid deterioration of mathematical (and not onlymathematical) education in Russia.

I tried to describe my experience of being V. I.’sstudent at Moscow State in [34]. I would like toemphasize here that Arnold did not follow anypattern in supervising his students. In some caseshe would inform a student that there was a cer-tain “uninhabited” corner in the vast mathematicalland, and if the student decided to “settle” at thatcorner, then it was this student’s task to find themain literature on the subject, to study it, to posenew problems, to find methods of their solution,and to achieve all this practically single-handedly.Of course, V. I. kept the progress under control. (Irecall that, as a senior, I failed to submit my “har-vest” for a long time, but finally made substantialprogress. Arnold exclaimed, “Thank God, I havestarted fearing that I would have to help you!”)But in other situations, Arnold would actively dis-cuss a problem with his student and invite him tocollaborate— this is how our joint paper [13] cameabout. When need be, V. I. could be rather harsh.Once I witnessed him telling a student, “You areworking too slowly. I think it will be good if youstart giving me weekly reports on your progress.”Arnold never tried to spare one’s self-esteem.

V. I. had a surprising feeling of the unity ofmathematics, of natural sciences, and of allnature. He considered mathematics as beingpart of physics, and his “economics” defini-tion of mathematics as a part of physics inwhich experiments are cheap is often quoted.

(Let me add in parentheses that I would preferto characterize mathematics as the natural sci-ence that studies the phenomenon of infinityby analogy with a little-known but remarkabledefinition of topology as the science that studiesthe phenomenon of continuity.) However, Arnoldnoted other specific features of mathematics: “Itis a fair observation that physicists refer to thefirst author, whereas mathematicians to the latestone.” (He considered adequate references to be ofparamount importance and paid much attentionto other priority questions; this was a naturalextension of his generosity, and he encouragedhis students to “over-acknowledge”, rather thanto “under-acknowledge”.)

V. I. was an avid fighter against “Bourbakism”, asuicidal tendency to present mathematics as a for-mal derivation of consequences from unmotivatedaxioms. According to Arnold, one needs mathe-matics to discover new laws of nature as opposedto “rigorously” justify obvious things. V. I. tried toteach his students this perception of mathematicsand natural sciences as a unified tool for under-standing the world. For a number of reasons, afterhaving graduated from university, I had to workpartially as a chemist, and after Arnold’s schoolthis caused me no psychological discomfort.

Fundamental mathematical achievements ofArnold, as well as those of his teacher, A. N. Kol-mogorov, cover almost all mathematics. It wellmay be that V. I. was the last universal mathemati-cian. My mathematical specialization is the KAMtheory. V. I. himself described the contributionsof the three founders; see, e.g., [15], [17]. For thisreason, I shall only briefly recall Arnold’s role inthe development of the KAM theory.

KAM theory is the theory of quasiperiodic mo-tions in nonintegrable dynamical systems. In 1954Kolmogorov made one of the most astonishingdiscoveries in mathematics of the last century.Consider a completely integrable Hamiltoniansystem with n degrees of freedom, and let (I,ϕ)be the corresponding action-angle variables. Thephase space of such a system is smoothly foli-ated into invariant n-tori {I = const} carryingconditionally periodic motions ϕ̇ = ω(I). Kol-mogorov showed that if det(∂ω/∂I) ≠ 0, then(in spite of the general opinion of the physicalcommunity of that time) most of these tori (inthe Lebesgue sense) are not destroyed by a smallHamiltonian perturbation but only slightly de-formed in the phase space. To be more precise,a torus {I = I0} persists under a perturbationwhenever the frequencies ω1(I

0), . . . ,ωn(I0) are

Diophantine (strongly incommensurable). Theperturbed tori (later called Kolmogorov tori) carryquasiperiodic motions with the same frequencies.To prove this fundamental theorem, Kolmogorovproposed a new, powerful method of construct-ing an infinite sequence of canonical coordinate


Vladimir Arnold.

transformations with accelerated (“quadratic”)convergence.

Arnold used Kolmogorov’s techniques toprove analyticity of the Denjoy homeomorphismconjugating an analytic diffeomorphism of a circlewith a rotation (under the condition that this dif-feomorphism is close to a rotation and possessesa Diophantine rotation number). His paper [4]with this result contained also the first detailedexposition of Kolmogorov’s method. Then, in a se-ries of papers, Arnold generalized Kolmogorov’stheorem to various systems with degeneracies.In fact, he considered two types of degeneraciesoften encountered in mechanics and physics: theproper degeneracy, where some frequencies of theperturbed tori tend to zero as the perturbationmagnitude vanishes, and the limit degeneracy,where the unperturbed foliation into invarianttori is singular and includes tori of smaller di-mensions. The latter degeneracy is modeled bya one-degree-of-freedom Hamiltonian systemhaving an equilibrium point surrounded by in-variant circles (the energy levels). These studiesculminated in Arnold’s famous (and technicallyextremely hard) result [5] on stability in planetary-like systems of celestial mechanics where boththe degeneracies combine.

Kolmogorov and Arnold dealt only with an-alytic Hamiltonian systems. On the other hand,J. K. Moser examined the finitely smooth case.The acronym “KAM” was coined by physicistsF. M. Izrailev and B. V. Chirikov in 1968.

Arnold always regarded his discovery of theuniversal mechanism of instability of the actionvariables in nearly integrable Hamiltonian sys-tems with more than two degrees of freedom [6]as his main achievement in the Hamiltonian per-turbation theory. He also constructed an explicitexample where such instability occurs. Chaotic

evolution of the actions along resonances be-tween the Kolmogorov tori was called “Arnold’sdiffusion” by Chirikov in 1969. In the case of twodegrees of freedom, the Kolmogorov 2-tori dividea three-dimensional energy level, which makes anevolution of the action variables impossible.

All these works by Arnold took place in 1958–1965. At the beginning of the eighties, he returnedto the problem of quasiperiodic motions fora short time and examined some interestingproperties of the analogs of Kolmogorov tori inreversible systems. That was just the time whenI started my diploma work. So V. I. forced me togrow fond of reversible systems and KAM theory,for which I’ll be grateful to him forever.

I would like to touch on yet one more side ofArnold’s research. In spite of what is occasion-ally claimed, Arnold did not hate computers: heconsidered them as an absolutely necessaryinstrument of mathematical modeling whenindeed large computations were involved. Heinitiated many computer experiments in dynam-ical systems and number theory and sometimesparticipated in them (see [17]). But of course hestrongly disapproved of the aggressive penetra-tion of computer technologies into all pores ofsociety and the tendency of a man to becomea helpless and mindless attachment to artificialintelligence devices. One should be able to divide111 by 3 without a calculator (and, better still,without scrap paper).

V. I. had a fine sense of humor. It is impossibleto forget his somewhat mischievous smile. In con-clusion, here are a couple of stories which mighthelp to illustrate the unique charm of this person.

I remember how a speaker at Arnold’s seminarkept repeating the words “one can lift” (a struc-ture from the base to the total space of a bundle).Arnold reacted: “Looks like your talk is about re-sults in weight-lifting.”

On another occasion, Arnold was lecturing, andthe proof of a theorem involved tedious computa-tions: “Everyone must make these computationsonce—but only once. I made them in the past, soI won’t repeat them now; they are left to the audi-ence!”

In the fall of 1987 the Gorbachev perestroika

was gaining steam. A speaker at the seminar wasdrawing a series of pictures depicting the pere-stroika (surgery) of a certain geometrical object asdepending on a parameter. Arnold: “Something isnot quite right here. Why is your central stratumalways the same? Perestroika always starts at thecenter and then propagates to the periphery.”


Askold Khovanskii andAlexander Varchenko

Arnold’s Seminar, First YearsIn 1965–66, V. I. Arnold was a postdoc in Paris, lec-turing on hydrodynamics and attending R. Thom’sseminar on singularities. After returning toMoscow, Vladimir Igorevich started his semi-nar, meeting on Tuesdays from 4 to 6 p.m. Itcontinued until his death on June 3 of 2010. Webecame Arnold’s students in 1966 and 1968, re-spectively. The seminar was an essential part ofour life. Among the first participants were R. Bog-danov, N. Brushlinskaya, I. Dolgachev, D. Fuchs,A. Gabrielov, S. Gusein-Zade, A. Kushnirenko,A. Leontovich, O. Lyashko, N. Nekhoroshev,V. Palamodov, A. Tyurin, G. Tyurina, V. Zakalyukin,and S. Zdravkovska.

V. I. Arnold had numerous interesting ideas,and to realize his plans he needed enthusiasticcolleagues and collaborators. Every semester hestarted the seminar with a new list of problemsand comments. Everyone wanted to be involved inthis lively creative process. Many problems weresolved, new theories were developed, and newmathematicians were emerging.

Here we will briefly describe some of the topicsof the seminar in its first years, as well as the skioutings which were an integral part of the seminar.

Hilbert’s 13th Problem and Arrangements ofHyperplanes

An algebraic function x = x(a1, . . . , ak) is a mul-tivalued function defined by an equation of theform

xn + P1(a1, . . . , ak)xn−1 + · · · + Pn(a1, . . . , ak) = 0

where Pi ’s are rational functions.Hilbert’s 13th Problem: Show that the function

x(a, b, c), defined by the equation

x7 + x3 + ax2 + bx+ c = 0,

cannot be represented by superpositions of contin-uous functions in two variables.

A. N. Kolmogorov and V. I. Arnold proved that infact such a representation does exist [3], thus solv-ing the problem negatively. Despite this result it isstill believed that the representation is impossibleif one considers the superpositions of (branchesof) algebraic functions only.

Can an algebraic function be represented asa composition of radicals and arithmetic opera-tions? Such a representation does exist if and only

Askold Khovanskii is professor of mathematics

at the University of Toronto. His email address is

[email protected].

Alexander Varchenko is professor of mathematics at the

University of North Carolina at Chapel Hill. His email

address is [email protected].

V. Arnold, Yu. Chekanov, V. Zakalyukin, andA. Khovanskii at Arnoldfest, Toronto, 1997.

if the Galois group of the equation over the fieldof its coefficients is solvable. Hence, the generalalgebraic function of degree k ≥ 5, defined by theequation a0x

k + a1xk−1 + · · · + ak = 0, cannot be

represented by radicals.In 1963, while teaching gifted high school stu-

dents at Moscow boarding school No. 18, foundedby Kolmogorov, V. I. Arnold discovered a topolog-ical proof of the insolvability by radicals of thegeneral algebraic equation of degree ≥ 5, a proofwhich does not rely on Galois theory. Arnold’s lec-tures at the school were written down and pub-lished by V. B. Alekseev in [1].

V. I. Arnold often stressed that when establish-ing the insolvability of a mathematical problem,topological methods are the most powerful andthose best suited to the task. Using such topolog-ical methods, V. I. Arnold proved the insolvabilityof a number of classical problems; see [18], [14]. In-spired by that approach, a topological Galois the-ory was developed later; see [28]. The topologicalGalois theory studies topological obstructions tothe solvability of equations in finite terms. For ex-ample, it describes obstructions to the solvabilityof differential equations by quadratures.

The classical formula for the solution by radi-cals of the degree four equation does not definethe roots of the equation only. It defines a 72-valued algebraic function. V. I. Arnold introducedthe notion of an exact representation of an al-gebraic function by superpositions of algebraicfunctions in which all branches of algebraic func-tions are taken into account. He proved that thealgebraic function of degree k = 2n, defined bythe equation xk + a1x

k−1 + ·· · + ak = 0, does nothave an exact representation by superpositions ofalgebraic functions in< k−1 variables; see [8] andthe references therein. The proof is again topo-logical and based on the characteristic classesof algebraic functions, introduced for that pur-pose. The characteristic classes are elements of


the cohomology ring of the complement to the dis-criminant of an algebraic function. To prove thattheorem V. I. Arnold calculated the cohomologyring of the pure braid group.

Consider the complement in Ck to the union ofthe diagonal hyperplanes,

U = { y ∈ Ck | yi ≠ yj for all i ≠ j}.

The cohomology ring H∗(U,Z) is the cohomologyring of the pure braid group on k strings. The co-homology ringH∗(U,Z)was described in [7]. Con-sider the ring A of differential forms on U gener-ated by the 1-forms wij =

12πid log(yi − yj), 1 ≤

i, j ≤ k, i ≠ j. Then the relations wij = wji and

wij ∧wjk +wjk ∧wki +wki ∧wij = 0

are the defining relations ofA. Moreover, the mapA→ H∗(U,Z), α ֏ [α], is an isomorphism.

This statement says that each cohomologyclass in H∗(U,Z) can be represented as an exte-rior polynomial inwij with integer coefficients andthe class is zero if and only if the polynomial iszero. As an application, V. I. Arnold calculated thePoincaré polynomial PD(t) =

∑ki=0 rankH i(D) t i ,

P∆(t) = (1+ t)(1+ 2t) · · · (1+ (n− 1)t).

Arnold’s paper [7] was the beginning of the mod-ern theory of arrangements of hyperplanes; see,for example, the book by P. Orlik and H. Terao.

Real Algebraic Geometry

By Harnack’s theorem, a real algebraic curve of de-gree n in the real projective plane can consist of atmost g+1 ovals, where g = (n−1)(n−2)/2 is thegenus of the curve. The M-curves are the curves forwhich this maximum is attained. For example, anM-curve of degree 6 has 11 ovals. Harnack provedthat the M-curves exist.

If the curve is of even degree n = 2k, then eachof its ovals has an interior (a disc) and an exterior(a Möbius strip). An oval is said to be positive if itlies inside an even number of other ovals and issaid to be negative if it lies inside an odd numberof other ovals. The ordinary circle, x2 + y2 = 1, isan example of a positive oval.

In his 16th problem, Hilbert asked how to de-scribe the relative positions of the ovals in theplane. In particular, Hilbert conjectured that 11ovals on an M-curve of degree 6 cannot lie externalto one another. This fact was proved by Petrovskyin 1938 [33].

The first M-curve of degree 6 was constructedby Harnack, the second by Hilbert. It was believedfor a long time that there were no other M-curvesof degree 6. Only in the 1960s did Gudkov con-struct a third example and prove that there areonly three types of M-curves of degree 6; see [26].

Experimental data led Gudkov to the followingconjecture: If p andm are the numbers of positive

Harnack’s, Hilbert’s, and Gudkov’s M-curves.

and negative ovals of an M-curve of degree 2k, thenp −m = k2 mod 8.

V. I. Arnold was a member of Gudkov’s Doctorof Science thesis defense committee and becameinterested in these problems. V. I. Arnold relatedGudkov’s conjecture and theorems of divisibilityby 16 in the topology of oriented closed four-dimensional manifolds developed by V. Rokhlinand others. Starting with an M-curve, V. I. Arnoldconstructed a four-dimensional manifold withan involution and using the divisibility theoremsproved that p −m = k2 mod 4; see [9]. Soon afterthat, V. A. Rokhlin, using Arnold’s construction,proved Gudkov’s conjecture in full generality.

This paper by V. I. Arnold began a revitalizationof real algebraic geometry.

Petrovsky–Oleinik Inequalities

Petrovsky’s paper [33] led to the discovery ofremarkable estimates for the Euler characteristicsof real algebraic sets, called Petrovsky-Oleinikinequalities. V. I. Arnold found in [10] unexpectedgeneralizations of these inequalities and newproofs of the inequalities based on singularitytheory.

Consider in Rn+1 the differential one-form α =P0dx0 + P1dx1 + · · · + Pndxn, whose componentsare homogeneous polynomials of degree m. Whatare possible values of the index ind of the form αat the point 0 ∈ Rn+1?

Let us introduce Petrovsky’s numberΠ(n,m) asthe number of integral points in the intersection ofthe cube 0 ≤ x0, . . . , xn ≤m−1 and the hyperplanex0+· · ·+xn = (n+1)(m−1)/2. V. I. Arnold provedin [10] that

|ind| ≤ Π(n,m) and ind ≡ Π(n,m) mod 2.

His elegant proof of these relations is based onthe Levin-Eisenbud-Khimshiashvili formula for theindex of a singular point of a vector field.

Let P be a homogeneous polynomial of de-gree m + 1 in homogeneous coordinates on RPn.Petrovsky-Oleinik inequalities give upper boundsfor the following quantities:

a) |χ(P = 0) − 1| for odd n, where χ(P = 0)is the Euler characteristic of the hypersur-face P = 0 in RPn, and


b) |2χ(P ≤ 0)−1| for even n andm+1, whereχ(P ≤ 0) is the Euler characteristic of thesubset P ≤ 0 in RPn.

V. I. Arnold noticed in [10] that in both casesa) and b) the estimated quantity equals the abso-lute value of the index at 0 ∈ Rn+1 of the gradientof P . Thus, the Petrovsky-Oleinik inequalities areparticular cases of Arnold’s inequalities for α =

dP .Furthermore, Arnold’s inequalities are ex-

act (unlike the Petrovsky-Oleinik ones): forany integral value of ind with the properties|ind| ≤ Π(n,m) and ind ≡ Π(n,m) mod 2 thereexists a homogeneous 1-form α (not necessarilyexact) with this index (proved by Khovasnkii).

Critical Points of Functions

Critical points of functions was one of the maintopics of the seminar in its first years. V. I. Arnoldclassified simple singularities of critical pointsin 1972, unimodal ones in 1973, and bimodalones in 1975. Simple critical points form seriesAn,Dn, E6, E7, E8 in Arnold’s classification. Alreadyin his first papers V. I. Arnold indicated (some-times without proofs) the connections of simplecritical points with simple Lie algebras of thecorresponding series. For example, the Dynkindiagram of the intersection form on vanishingcohomology at a simple singularity of an oddnumber of variables equals the Dynkin diagramof the corresponding Lie algebra, the monodromygroup of the simple singularity equals the Weylgroup of the Lie algebra, and the singularity indexof the simple singularity equals 1/N, where N isthe Coxeter number of the Lie algebra.

One of the main problems of that time was tostudy characteristics of critical points. The meth-ods were developed to calculate the intersectionform on vanishing cohomology at a critical point(Gabrielov, Gusein-Zade), monodromy groups(Gabrielov, Gusein-Zade, Varchenko, Chmu-tov), and asymptotics of oscillatory integrals(Varchenko). The mixed Hodge structure on van-ishing cohomology was introduced (Steenbrink,Varchenko), and the Hodge numbers of the mixedHodge structure were calculated in terms of New-ton polygons (Danilov, Khovanskii); see [12], [22]and the references therein.

The emergence of extensive new experimentaldata led to new discoveries. For example, ac-cording to Arnold’s classification, the unimodalsingularities form one infinite series Tp,q,r and 14exceptional families. Dolgachev discovered thatthe 14 exceptional unimodal singularities canbe obtained from automorphic forms associatedwith the discrete groups of isometries of theLobachevsky plane generated by reflections at thesides of some 14 triangles [23]. For the anglesπ/p,π/q,π/r of such a triangle, the numbers

p, q, r are integers, called Dolgachev’s triple. Ac-cording to Gabrielov [25], the intersection formon vanishing cohomology at an exceptional uni-modal singularity is described by another tripleof integers, called Gabrielov’s triple. V. I. Arnoldnoticed that Gabrielov’s triple of an exceptionalunimodal singularity equals Dolgachev’s tripleof (in general) another exceptional unimodal sin-gularity, while Gabrielov’s triple of that othersingularity equals Dolgachev’s triple of the initialsingularity. Thus, there is an involution on the setof 14 exceptional unimodal singularities, calledArnold’s strange duality. Much later, after discov-ery of the mirror symmetry phenomenon, it wasrealized that Arnold’s strange duality is one of itsfirst examples.

Newton Polygons

While classifying critical point of functions,Arnold noticed that, for all critical points ofhis classification, the Milnor number of the criti-cal point can be expressed in terms of the Newtonpolygon of the Taylor series of that critical point.Moreover, an essential part of Arnold’s classifi-cation was based on the choice of the coordinatesystem simplifying the Newton polygon of the cor-responding Taylor series. (According to Arnold,he used “Newton’s method of a moving ruler(line, plane)”.) V. I. Arnold formulated a generalprinciple: in the family of all critical points withthe same Newton polygon, discrete characteris-tics of a typical critical point (the Milnor number,singularity index, Hodge numbers of vanishingcohomology, and so on) can be described in termsof the Newton polygon.

That statement was the beginning of the the-ory of Newton polygons. Newton polygons wereone of the permanent topics of the seminar. Thefirst result, the formula for the Milnor numberin terms of the Newton polygon, was obtainedby Kouchnirenko in [30]. After Kouchnirenko’sreport at Arnold’s seminar, Lyashko formulateda conjecture that a similar statement must holdin the global situation: the number of solutionsof a generic system of polynomial equations in nvariables with a given Newton polygon must beequal to the volume of the Newton polygon mul-tiplied by n!. Kouchnirenko himself proved thisconjecture. David Bernstein [21] generalized thestatement of Kouchnirenko’s theorem to the caseof polynomial equations with different Newtonpolygons and found a simple proof of his general-ization. Khovanskii discovered the connection ofNewton polygons with the theory of toric varietiesand using this connection calculated numerouscharacteristics of local and global complete in-tersections in terms of Newton polygons; see[11] and the references therein. Varchenko cal-culated the zeta-function of the monodromy and


asymptotics of oscillatory integrals in terms ofNewton polygons; see [12].

Nowadays Newton polygons are a working toolin many fields. Newton polygons appear in realand complex analysis, representation theory, andreal algebraic geometry; and the Newton polygonsprovide examples of mirror symmetry and so on.

Skiing and Swimming

Every year at the end of the winter Arnold’s semi-nar went to ski on the outskirts of Moscow. Thistradition started in 1973. While the number ofseminar participants was between twenty andthirty people, no more than ten of the bravestparticipants came out to ski. People preparedfor this event the whole winter. The meeting wasat 8 a.m. at the railway station in Kuntsevo, thewestern part of Moscow, and skiing went on untilafter sunset, around 6 p.m. The daily distance wasabout 50 km.

Usually Arnold ran in front of the chain ofskiers, dressed only in swimming trunks. He ranat a speed a bit above the maximal possible speedof the slowest of the participants. As a result,the slowest participant became exhausted afteran hour of such an outing and was sent back toMoscow on a bus at one of the crossroads. Thenthe entire process was repeated again and anotherparticipant was sent back to Moscow after anotherhour. Those who were able to finish the skiingwere very proud of themselves.

Only one time was the skiing pattern different.In that year we were joined by Dmitri Boriso-vich Fuchs, a tall, unflappable man, who was atone time a serious mountain hiker. Early in themorning when Arnold started running away fromthe station with us, Dmitri Borisovich unhur-riedly began to walk in the same direction. Soonhe completely disappeared from our view, andArnold stopped and began waiting impatiently forFuchs to arrive. Arnold again rushed to run andFuchs, again unperturbed, unhurriedly followedthe group. So proceeded the entire day. That daynone of the participants of the run were senthome in the middle of the day.

Several times we were joined by Olya Krav-chenko and Nadya Shirokova, and every time theykept up the run as well as the best.

All participants of the ski-walk brought sand-wiches, which they ate at a stop in the middleof the day. Before sandwiches there was bathing.In Moscow suburbs you will come across smallrivers which are not frozen even in winter. Wewould meet at such a stream and bathe, lyingon the bottom of the streambed as the waterwas usually only knee deep. We certainly did notuse bathing suits, and there were no towels. Thetradition of bathing in any open water at any timeof the year Arnold had adopted from his teacher,

Summer expedition, 1960s.

Kolmogorov. This tradition was taken up by manyparticipants of the seminar.

Arnold thought that vigorous occupation withmathematics should be accompanied by vigorousphysical exercise. He skied regularly in the winter(about 100 km per week), and in summer rode abicycle and took long walks.

There is a funny story connected to the tra-dition of bathing in any available open water.In 1983 the Moscow mathematicians were takenout to the Mathematical Congress in Warsaw.This congress had been boycotted by Westernmathematicians. The large Soviet delegation wassupposed to compensate for the small number ofWestern participants. A special Moscow-Warsaw-Moscow train had been arranged, which deliveredus to Poland with Arnold. Once, walking acrossWarsaw in the evening with Arnold, we arrived ata bridge across the Vistula. While on the bridgewe decided to bathe, as required by tradition. Wereached the water in total darkness and swam fora few minutes. In the morning we found, to ouramazement, that we were floating more in mudthan water.

Michael Berry

Memories of Vladimir ArnoldMy first interaction with Vladimir Arnold was re-ceiving one of his notoriously caustic letters. In1976 I had sent him my paper (about caustics, in-deed) applying the classification of singularities ofgradient maps to a variety of phenomena in opticsand quantum mechanics. In my innocence, I hadcalled the paper “Waves and Thom’s theorem”. Hisreply began bluntly:

Thank you for your paper. Refer-ences:…

Michael Berry is professor of physics at the

University of Bristol. His email address is

[email protected].


There followed a long list of his papers hethought I should have referred to. After declaringthat in his view René Thom (whom he admired)never proved or even announced the theoremsunderlying his catastrophe theory, he continued:

I can’t approve your system ofreferring to English translationswhere Russian papers exist. Thishas led to wrong attributions ofresults, the difference of 1 yearbeing important—a translationdelay is sometimes of 7 years…

and

…theorems and publications arevery important in our science (…atpresent one considers as a publi-cation rather 2-3 words at Buresor Fine Hall tea, than a paper withproofs in a Russian periodical)

and (in 1981)

I hope you’ll not attribute these re-sult [sic] to epigons.

He liked to quote Isaac Newton, often in scribbledmarginal afterthoughts in his letters:

A man must either resolve to putout nothing new, or to become aslave to defend it

and (probably referring to Hooke)

Mathematicians that find out, set-tle and do all the business mustcontent themselves with beingnothing but dry calculators anddrudges and another that doesnothing but pretend and graspat all things must carry away allthe invention as well of those thatwere to follow him as of those thatwent before.

(I would not accuse Vladimir Arnold of compar-ing himself with Newton, but was flattered to beassociated with Hooke, even by implication.)

I was not his only target. To my colleague JohnNye, who had politely written “I have much ad-mired your work…,” he responded:

I understand well your letter, youradmiration have not led neither toread the [reference to a paper] norto send reprints….

This abrasive tone obviously reflected a toughand uncompromising character, but I was never of-fended by it. From the beginning, I recognized anunderlying warm and generous personality, andthis was confirmed when I finally met him in thelate 1980s. His robust correspondence arose fromwhat he regarded as systematic neglect by Westernscientists of Russian papers in which their resultshad been anticipated. In this he was sometimes

right and sometimes not. And he was unconvincedby my response that scientific papers can legiti-mately be cited to direct readers to the most ac-cessible and readable source of a result rather thanto recognize priority with the hard-to-find originalpublication.

He never lost his ironic edge. In Bristol, whenasked his opinion of perestroika, he declared:“Maybe the fourth derivative is positive.” And ata meeting in Paris in 1992, when I found, in myconference mailbox, a reprint on which he hadwritten: “to Michael Berry, admiringly,” I swelledwith pride—until I noticed, a moment later, thatevery other participant’s mailbox contained thesame reprint, with its analogous dedication!

In 1999, when I wrote to him after his accident,he replied (I preserve his inimitable style):

…from the POINCARÉ hospi-tal…the French doctors insistedthat I shall recover for the follow-ing arguments: 1) Russians are2 times stronger and any Frenchwould already die. 2) This particu-lar person has a special optimismand 3) his humour sense is spe-cially a positive thing: even unableto recognize you, he is laughing….I do not believe this story, becauseit would imply a slaughtering ofher husband for Elia, while I amstill alive.

(Elia is Arnold’s widow.)There are mathematicians whose work has

greatly influenced physics but whose writings arehard to understand; for example, I find Hamilton’spapers unreadable. Not so with Arnold’s: throughhis pellucid expositions, several generations ofphysicists came to appreciate the significance ofpure mathematical notions that we previouslyregarded as irrelevant. “Arnold’s cat” made usaware of the importance of mappings as modelsfor dynamical chaos. And the exceptional tori thatdo not persist under perturbation (as Kolmogorov,Arnold, and Moser showed that most do) made usaware of Diophantine approximation in numbertheory: “resonant torus” to a physicist = “rationalnumber” to a mathematician.

Most importantly, Arnold’s writings were oneof the two routes by which, in the 1970s, thenotion of genericity slipped quietly into physics(the other route was critical phenomena in statis-tical mechanics, where it was called universality).Genericity emphasizes phenomena that are typi-cal rather than the special cases (often with highsymmetry) corresponding to exact solutions ofthe governing equations in terms of special func-tions. (And I distinguish genericity from abstractgenerality, which can often degenerate into whatMichael Atiyah has called “general nonsense”.)


This resulted in a shift in our thinking whosesignificance cannot be overemphasized.

It suddenly occurs to me that in at least four re-spects Arnold was the mathematical counterpartof Richard Feynman. Like Feynman, Arnold mademassive original contributions in his field, withenormous influence outside it; he was a masterexpositor, an inspiring teacher bringing new ideasto new and wide audiences; he was uncompromis-ingly direct and utterly honest; and he was a col-orful character, bubbling with mischief, endlesslysurprising.

Acknowledgments

The photographs are courtesy of the Arnold familyarchive, F. Aicardi, Ya. Eliashberg, E. Ferrand, B.and M. Khesin, J. Pöschel, M. Ratner, S. Tretyakova,and I. Zakharevich.

References[1] V. B. Alekseev, Abel’s Theorem in Problems and

Solutions, based on the lectures of ProfessorV. I. Arnold, with a preface and an appendix byArnold and an appendix by A. Khovanskii, KluwerAcademic Publishers, Dordrecht, 2004.

[2] D. Anosov, Geodesic flows on closed Riemannianmanifolds of negative curvature, Trudy Mat. Inst.

Steklov 90 (1967).[3] V. I. Arnold, On functions of three variables, Dokl.

Akad. Nauk SSSR 114 (1957), 679–681.[4] V. Arnold, Small denominators. I. Mappings of a

circle onto itself, Izvestiya AN SSSR, Ser. Mat. 25

(1961), 21–86.[5] , Small denominators and problems of stabil-

ity of motion in classical and celestial mechanics,Uspekhi Mat. Nauk 18 (1963), no. 6, 91–192.

[6] , Instability of dynamical systems with manydegrees of freedom, Dokl. Akad. Nauk SSSR 156

(1964), 9–12.[7] , The cohomology ring of the group of dyed

braids, Mat. Zametki 5 (1969), 227–231.[8] V. I. Arnold, The cohomology classes of algebraic

functions that are preserved under Tschirnhausentransformations, Funkt. Anal. Prilozhen 4 (1970),no. 1, 84–85.

[9] V. Arnold, The situation of ovals of real algebraicplane curves, the involutions of four-dimensionalsmooth manifolds, and the arithmetic of integralquadratic forms, Funkt. Anal. Prilozhen 5 (1971), no.3, 1–9.

[10] , The index of a singular point of a vectorfield, the Petrovsky–Oleinik inequalities, and mixedHodge structures, Funct. Anal. Appl. 12 (1978),no. 1, 1–14.

[11] V. I. Arnold, A. B. Givental, A. G. Khovan-

skii, A. N. Varchenko, Singularities of functions,

wave fronts, caustics and multidimensional inte-

grals, Mathematical Physics Reviews, Vol. 4, 1–92,Harwood Acad. Publ., Chur, 1984.

[12] V. Arnold, S. Gusein-Zade, A. Varchenko,Singularities of Differentiable Maps, Vol. I. The

Classification of Critical Points, Caustics and Wave

Fronts, Birkhäuser, Boston, MA, 1985.

[13] V. Arnold, M. Sevryuk, Oscillations and bi-furcations in reversible systems, in Nonlinear

Phenomena in Plasma Physics and Hydrodynamics,Mir, Moscow, 1986, 31–64.

[14] V. I. Arnold, V. A. Vassiliev, Newton’s “Principia”read 300 years later, Notices Amer. Math. Soc. 36

(1989), no. 9, 1148–1154; 37 (1990), no. 2, 144.[15] V. Arnold, From superpositions to KAM theory,

in Vladimir Igorevich Arnold, Selected–60, PHASIS,Moscow, 1997, 727–740 (in Russian).

[16] V. Arnold, B. Khesin, Topological Methods in

Hydrodynamics, Springer-Verlag, New York, 1998.[17] V. Arnold, From Hilbert’s superposition problem

to dynamical systems, in The Arnoldfest, Amer.Math. Soc., Providence, RI, 1999, 1–18.

[18] V. I. Arnold, I. G. Petrovskii, Hilbert’s topologicalproblems, and modern mathematics, Russian Math.

Surveys 57 (2002), no. 4, 833–845.[19] M. Atiyah, J. Berndt, Projective planes, Severi

varieties and spheres, Surveys in Differential Geom-

etry, Vol. VIII (Boston, MA, 2002), 1–27, Int. Press,Somerville, MA, 2003.

[20] M. Audin, Cobordismes d’immersions lagrangiennes

et legendriennes, Travaux en Cours, 20, Hermann,Paris, 1987.

[21] D. Bernstein, On the number of roots of a systemof equations, Funkt. Anal. i Prilozhen. 9 (1975), no. 3,1–4.

[22] V. I. Danilov, A. G. Khovanskii, Newton polyhe-dra and an algorithm for calculating Hodge–Delignenumbers, Math. USSR—Izv. 29 (1987), 279–298.

[23] I. Dolgachev, Conic quotient singularities of com-plex surfaces, Funkt. Anal. i Prilozhen. 8 (1974),no. 2, 75–76.

[24] A. Eskin, A. Okounkov, Asymptotics of numbersof branched coverings of a torus and volumes ofmoduli spaces of holomorphic differentials, Invent.

Math. 145 (2001), 59–103.[25] A. Gabrielov, Dynkin diagrams of unimodal sin-

gularities, Funkt. Anal. i Prilozhen. 8 (1974), no. 3,1–6.

[26] D. Gudkov, Topology of real projective algebraicvarieties, Uspekhi Mat. Nauk 29 (1974), no. 4, 3–79.

[27] M. Khovanov, P. Seidel, Quivers, Floer cohomol-ogy, and braid group actions, J. Amer. Math. Soc.

15 (2002), 203–271.[28] A. G. Khovanskii, Topological Galois Theory,

MTSNMO, Moscow, 2008.[29] M. Kontsevich, A. Zorich, Connected components

of the moduli spaces of Abelian differentials withprescribed singularities, Invent. Math. 153 (2003),631–678.

[30] A. Kouchnirenko, Polyèdres de Newton etnombres de Milnor, Invent. Math. 32 (1976), 1–31.

[31] N. Kuiper, The quotient space of CP2 by complexconjugation is the 4-sphere, Math. Ann. 208 (1974),175–177.

[32] J. Marsden, Steve Smale and Geometric Mechanics,The Collected Papers of Stephen Smale, vol. 2, 871–888, World Scientific Publ., River Edge, NJ, 2000.

[33] I. G. Petrovskii, On the topology of real planealgebraic curves, Ann. of Math. (2) 39 (1938),187–209.

[34] M. Sevryuk, My scientific advisor V. I. Arnold,Matem. Prosveshchenie, Ser. 3 2 (1998), 13–18 (inRussian).


[35] S. Smale, On how I got started in dynamical sys-tems, 1959–1962, From Topology to Computation:

Proceedings of the Smalefest, 22–26, Springer, NewYork, 1993.

[36] , On the steps of Moscow University,From Topology to Computation: Proceedings of the

Smalefest, 41–52, Springer, New York, 1993.[37] , On the problem of revising the ergodic hy-

pothesis of Boltzmann and Birkhoff, The Collected

Papers of Stephen Smale, vol. 2, 823–830, WorldScientific Publ., River Edge, NJ, 2000.

[38] V. Vassiliev, Lagrange and Legendre Characteristic

Classes, Gordon and Breach Science Publ., New York,1988.

[39] M. V. Yakobson, The number of periodic trajecto-ries for analytic diffeomorphisms of a circle, Funct.

Anal. Appl. 19 (1985), no. 1, 91–92.

About the Cover

A mathematician’s dinner table

conversation

It is often said that mathematicians are eager to draw on napkins, but in our experience it is actually rather rare. The doodles on the cover are by the late mathematician Vladimir Arnold, the subject of articles in this and the next issue of the Notices. Emmanuel Ferrand saved the napkin and took the photograph. He writes,

“Arnold wrote on this napkin at the oc-casion of a private meal at the Institut des Hautes Études Scientifiques near Paris. As far as I remember, it was early in 2006, and this is consistent with the problems discussed on this napkin. Most of what is written here is related to the question of the enumeration of the topological types of Morse functions on surfaces. The two drawings with the let-ters A, B, C correspond to two of his favorite examples, the height functions of Stromboli and Etna, two famous volcanos in Italy. These are specifically referred to in Arnold’s papers on the classification of Morse functions, and some of the problems he posed have been solved—some, for example, by Liviu Nico-laescu.

“It is important to note that Arnold’s prob-lems are one of his major contributions to mathematics. He wrote a list of about twenty open problems every year. He was not in-terested in leaving problems for the next generation like the Poincaré conjecture or the Riemann Hypothesis. On the contrary, the ‘half life’ of his problems was about five years. He also insisted not to take his ques-tions too literally, but to consider them as a loose direction of research instead. He was not that much interested in questions whose answer is only ‘yes’ or ‘no’. His seminars in Moscow and in Paris were structured around this problem list. Most of his problems were collected in the book Arnold’s Problems from Springer-Verlag. There are still many interest-ing open questions waiting for the curious mathematician here!”

As anyone familiar with Arnold’s writings knows, his line drawings, although not exactly requiring high technology to produce, were a charming and important part of his exposi-tion. As the article in the next issue of the Notices points out, he insisted that figures play an important role in all of his students’ work, as well.

—Bill Casselman

Graphics editor

([email protected])


tribute to vladimir arnold - national chiao tung universitysmchang/news/2012_v_arnold.pdf ·...

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