Tribute to
Vladimir Arnold
Boris Khesin and Serge Tabachnikov, Coordinating Editors
Vladimir Arnold, an eminent mathematician of our time, passed away on June 3, 2010, nine days before his seventy-third birthday. This article, along with one in the next issue of the Notices, touches on his outstanding personality and his great contribution to mathematics.
A word about spelling: we use “Arnold”, as opposed to “Arnol’d”; the latter is closer to the Rus- sian pronunciation, but Vladimir Arnold preferred the former (it is used in numerous translations of his books into English), and we use it throughout.
Arnold in His Own Words
In 1990 the second author interviewed V. Arnold for a Russian magazine Kvant (Quantum). The readership of this monthly magazine for physics and mathematics consisted mostly of high school students, high school teachers, and undergraduate students; the magazine had a circulation of about 200,000. As far as we know, the interview was never translated into English. We translate excerpts from this interview;1 the footnotes are ours.
Q: How did you become a mathematician? What was the role played by your family, school, math- ematical circles, Olympiads? Please tell us about your teachers.
A: I always hated learning by rote. For that reason, my elementary school teacher told my parents that a moron, like myself, would never manage 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 Kvant magazine (July 1990), http://kvant.mirror1.mccme.
ru/.
DOI: http://dx.doi.org/10.1090/noti810
My first mathematical revelation was when I met my first real teacher of mathematics, Ivan Vassilievich Morozkin. I remember the problem about two old ladies who started simultaneously
Vladimir Igorevich Arnold
from two towns to- ward each other, met at noon, and who reached the oppo- site towns at 4 p.m.
and 9 p.m., respec- tively. The question was when they started their trip.
We didn’t have al- gebra yet. I invented an “arithmetic” solu- tion (based on a scal- ing—or similarity—
argument) and expe- rienced a joy of dis- covery; the desire to experience this joy again was what made me a mathematician.
A. A. Lyapunov organized at his home “Chil- dren Learned Society”. The curriculum included mathematics and physics, along with chemistry and biology, including genetics that was just re- cently banned2(a son of one of our best geneticists was my classmate; in a questionnaire, he wrote:
“my mother is a stay-at-home mom; my father is a stay-at-home dad”).
Q: You have been actively working in mathe- matics for over thirty years. Has the attitude of society towards mathematics and mathematicians changed?
A: The attitude of society (not only in the USSR) to fundamental science in general, and to mathe- matics in particular, is well described by I. A. Krylov
2In 1948 genetics was officially declared “a bourgeois pseudoscience” in the former Soviet Union.
Vladimir Arnold, circa 1985.
in the fable “The hog under the oak”.3In the 1930s and 1940s, mathematics suffered in this country less than other sciences. It is well known that Viète was a cryptographer in the service of Henry IV of France. Since then, certain areas of mathematics have been supported by all governments, and even Beria4 cared about preservation of mathematical culture in this country.
In the last thirty years the prestige of mathe- matics has declined in all countries. I think that mathematicians are partially to be blamed as well (foremost, Hilbert and Bourbaki), particularly the ones who proclaimed that the goal of their science was investigation of all corollaries of arbitrary systems of axioms.
Q: Does the concept of fashion apply to mathe- matics?
A: Development of mathematics resembles a fast revolution of a wheel: sprinkles of water are flying in all directions. Fashion—it is the stream that leaves the main trajectory in the tangential direction. These streams of epigone works attract the most attention, and they constitute the main mass, but they inevitably disappear after a while because they parted with the wheel. To remain on the wheel, one must apply the effort in the direction perpendicular to the main stream.
A mathematician finds it hard to agree that the 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 “strange attractors” illustrates the fruitfulness of word creation 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 science about truth. It seems to me that modern science (i.e., theoretical physics along with mathematics) is a new religion, a cult of truth, founded by Newton three hundred years ago.
Q: When you prove a theorem, do you “create”
or “discover” it?
A: I certainly have a feeling that I am discovering something 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 which really teach will never be popular; lectures which are popular will never teach.” This Faraday effect is easy to explain: according to N. Bohr, clearness and truth 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 telling A. N. Kolmogorov about his participation in Concours Général (roughly corresponding to our Olympiads), he was still very excited: Hadamard won only the second prize, while the student who had won the first prize also became a mathematician, but a much weaker one!
Some Olympiad winners later achieve nothing, and many outstanding mathematicians had no success in Olympiads at all.
Mathematicians differ dramatically by their time scale: some are very good tackling 15-minute problems, some are good with the problems that require an hour, a day, a week, the problems that take a month, a year, decades of thinking.
A. N. Kolmogorov considered his “ceiling” to be two weeks of concentrated thinking.
Success in an Olympiad largely depends on one’s sprinter qualities, whereas serious mathe- matical research requires long distance endurance (B. N. Delaunay used to say, “A good theorem takes not 5 hours, as in an Olympiad, but 5,000 hours”).
There are contraindications to becoming a re- search mathematician. The main one is lack of love of mathematics.
Teaching at Moscow State University, 1983.
But mathematical talents can be very diverse:
geometrical and intuitive, algebraic and compu- tational, logical and deductive, natural scientific and inductive. And all kinds are useful. It seems to me that one’s difficulties with the multipli- cation table or a formal definition of half-plane should not obstruct one’s way to mathematics.
An extremely important condition for serious mathematical 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 a solution (or at least an idea for a solution) always comes. Under scrutiny, an error is often found.
But this is a new difficulty that is overcome in the same way.
The Hog Under the Oak A Hog under a mighty Oak
Had 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 dead
Two 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 around You’d have a chance to figure out Where your beloved fruit is born.”
Likewise, an ignoramus in defiance Is scolding scientists and science, And all preprints at lanl_dot_gov, Oblivious of his partaking fruit thereof.
Arnold’s Doctoral Students
The list below includes those who defended their Ph.D. theses under Arnold’s guidance. We have to admit that it was difficult to compile. Along with
straightforward cases when Arnold supervised the thesis and was listed as the person’s Ph.D. advisor, there were many other situations. For example, in Moscow State University before perestroika, a Ph.D. advisor for a foreigner had to be a member of the Communist Party, so in such cases there was a different nominal Ph.D. advisor while Arnold was supervising the student’s work. In other cases there were two co-advisors or there was a differ- ent advisor of the Ph.D. thesis, while the person defended the Doctor of Science degree (the second scientific degree in Russia) under Arnold’s super- vision. In these “difficult cases” the inclusion in the list below is based on “self-definition” as an Arnold student rather than on a formality. We tried to make the list as complete and precise as possible, but we apologize in advance for possible omissions: there were many more people whose work Arnold influenced greatly and who might feel they belong to Arnold’s school.
Names are listed chronologically according to the 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 first degree 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 Concern
No estь i Boжii sud ...
M. . Lermontov, “Smertь Poзta”6 Posthumous memoirs seem to have the unintended effect of reducing the person’s life to a collection of stories. For most of us it would probably be a just and welcome outcome, but for Vladimir Arnold, I think, it would not. He tried and managed to tell us many different things about mathematics, education, and beyond, and in many cases we’ve been rather slow listening or thinking, so I believe we will be returning again and again not only to our memories of him but to his own words as well. What is found below is not a memoir, but a recommendation letter, albeit a weak one, for he did not get the prize, and yet hopefully useful as an interim attempt to overview his mathematical heritage.
January 25, 2005 Dear Members of [the name of the committee], You have requested my commentaries on the work of Vladimir Arnold. Writing them is an honorable and 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].
6Yet, there is God’s Court, too. . ., M. Yu. Lermontov,
“Death of Poet”.
At Dubna, 2006.
these areas) it introduces powerful new techniques too.
On the other hand, writing this letter is not easy, mainly because the ways Arnold’s work con- tributes to our knowledge are numerous and go far beyond my personal comprehension. As Arnold’s student, I am quite familiar with those aspects of his work which inspired my own research. Outside these areas, hopefully, I will be able to convey the conventional wisdom about Arnold’s most fa- mous achievements. Yet this leaves out the ocean of numerous, possibly less famous but extremely influential, contributions, of which I have only partial knowledge and understanding. So, I will have to be selective here and will mention just a handful of examples which I am better familiar with and which for this reason may look chosen randomly.
Perhaps the most legendary, so to speak, of Arnold’s contributions is his work on small denominators,7 followed by the discovery of Arnold’s diffusion,8and known now as part of the Kolmogorov–Arnold–Moser theory. Among other things, this work contains an explanation (or, de- pending on the attitude, a proof, and a highly technical one) of stability of the solar planetary system. Even more importantly, the KAM theory provides 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 motion in classical and celestial mechanics, Uspekhi Mat. Nauk 18(1963), no. 6, 91–192, following Small denominators I. Mappings of a circle onto itself, Izvestia AN SSSR, Ser.
Mat. 25 (1961), 21–86, Small denominators II. Proof of a theorem of A. N. Kolmogorov on the preservation of con- ditionally periodic motions under a small perturbation of 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 of freedom, DAN SSSR 156 (1964), 9–12.
is widely regarded as one of the major discoveries of twentieth-century mathematical physics.
Symplectic geometry has established itself as a universal geometric language of Hamiltonian mechanics, calculus of variations, quantization, representation theory and microlocal analysis of differential equations. One of the first mathe- maticians who understood the unifying nature of symplectic geometry was Vladimir Arnold, and his work played a key role in establishing this status of symplectic geometry. In particular, his monograph Mathematical Methods of Classical Mechanics9has become a standard textbook, but thirty years ago it indicated a paradigm shift in a favorite subject of physicists and engineers. The traditional “an- alytical” or “theoretical” mechanics got suddenly transformed into an active region of modern math- ematics populated with Riemannian metrics, Lie algebras, differential forms, fundamental groups, and symplectic manifolds.
Just as much as symplectic geometry is merely a language, symplectic topology is a profound problem. 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 others belong to this area. It would not be too much of an overstatement to say that symplectic topology has developed from attempts to solve a single problem: to prove the Arnold conjecture about Hamiltonian fixed points and Lagrangian intersec- tions.10While the conjecture has been essentially proved11 and many new problems and ramifica- tions discovered, the theory in a sense continues to explore various facets of that same topological rigidity property of phase spaces of Hamiltonian mechanics that goes back to Poincaré and Birkhoff and whose symplectic nature was first recognized by Arnold in his 1965 notes in Comptes Rendus.
Arnold’s work in Riemannian geometry of infinite-dimensional Lie groups had almost as much of a revolutionizing effect on hydrodynam- ics as his work in small denominators produced in classical mechanics. In particular, Arnold’s seminal
9Nauka, Moscow, 1974.
10First stated in Sur une propriété topologique des ap- plications globalement canoniques de la mécanique classique, 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 Fourier12draws on his observation that flows of incompressible fluids can be interpreted as geodesics of right-invariant metrics on the groups of volume-preserving dif- feomorphisms. Technically speaking, the aim of the paper is to show that most of the sectional curvatures of the area-preserving diffeomorphism group of the standard 2-torus are negative and thus the geodesics on the group typically diverge exponentially. From time to time this result makes the news as a “mathematical proof of impossibility of long-term weather forecasts”. More importantly, the work had set Euler’s equations on coadjoint orbits as a blueprint and redirected the attention in many models of continuum mechanics toward symmetries, conservation laws, relative equilib- ria, symplectic reduction, topological methods (in works of Marsden, Ratiu, Weinstein, Moffat, and Freedman among many others).13
Due to the ideas of Thom and Pham and fundamental results of Mather and Malgrange, singularity theory became one of the most active fields of the seventies and eighties, apparently with two leading centers: Brieskorn’s seminar in Bonn and Arnold’s seminar in Moscow. The theory of critical points of functions and its applications to classification of singularities of caustics, wave fronts and short-wave asymptotics in geometrical optics as well as their relationship with the ADE-classification are perhaps the most famous (among uncountably many other) results
12Sur la géométrie différentielle des groupes de Lie de di- mension infinie et ses applications à l’hydrodynamique des 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 role in this area went, however, far beyond his own papers.
Imagine a seminar of about thirty partici- pants: undergraduates writing their first research papers, graduate students working on their dis- sertation problems, postgraduates employed else- where as software engineers but unwilling to give up their dream of pursuing mathematics even if only as a hobby, several experts—Fuchs, Dolgachev, Gabrielov, Gusein-Zade, Khovansky, Kushnirenko, Tyurin, Varchenko, Vassiliev—and the leader, Arnold—beginning each semester by formulating a bunch of new problems, giving talks or listening to talks, generating and generously sharing new ideas and conjectures, editing his stu- dents’ papers, and ultimately remaining the only person in his seminar who would keep in mind everyone else’s works-in-progress and understand their relationships. Obviously, a lion’s share of his students’ achievements (and among the quite famous ones are the theory of Newton polyhedra by Khovansky and Kushnirenko or Varchenko’s re- sults on asymptotical mixed Hodge structures and semicontinuity of Steenbrink spectra) is due to his help, 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—we will learn later why).
Moreover, under Arnold’s influence, the elite branch of topology and algebraic geometry study- ing singular real and complex hypersurfaces was transformed into a powerful tool of applied
14Normal forms of functions near degenerate criti- cal points, the Weyl groups Ak, Dk, Ek and Lagrangian singularities, 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 all kinds of mathematical objects (metamorphoses of wave fronts and caustics, evolutes, evolvents and envelopes of plane curves, phase diagrams in thermodynamics and convex hulls, accessibility regions in control theory, differential forms and Pfaff equations, symplectic and contact struc- tures, solutions of Hamilton-Jacobi equations, the Hamilton-Jacobi equations themselves, the boundaries between various domains in func- tional spaces of all such equations, etc.) and merging with the theory of bifurcations (of equi- libria, limit cycles, or more complicated attractors in ODEs and dynamical systems). Arnold had de- veloped a unique intuition and expertise in the subject, so that when physicists and engineers would 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 small dimensions right on the spot. In this regard, the situation would resemble experimental physics or chemistry, where personal expertise is often more important than formally registered knowledge.
Having described several (frankly, quite obvi- ous) broad areas of mathematics reshaped by Arnold’s seminal contributions, I would like to turn now to some more specific classical problems which attracted his attention over a long time span.
The affirmative solution of the 13th Hilbert problem (understood as a question about super- positions of continuous functions) given by Arnold in his early (essentially undergraduate) work15was the beginning of his interest in the “genuine” (and still open) Hilbert’s problem: Can the root of the general degree 7 polynomial considered as an algebraic function of its coefficients be writ- ten as a superposition of algebraic functions of 2 variables? The negative16solution to the more gen- eral question about polynomials of degree n was given by Arnold in 1970 for n = 2r.17 The result was generalized by V. Lin. Furthermore, Arnold’s approach, based on his previous study of coho- mology of braid groups, later gave rise to Smale’s concept of topological complexity of algorithms and Vassiliev’s results on this subject. Even more importantly, Arnold’s study of braid groups via topology of configuration spaces18was generalized by Brieskorn to E. Artin’s braid groups associated with 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 smaller number 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’s approach to hypergeometric functions, Aomoto’s work on Yang-Baxter equations, and Varchenko–
Schekhtman’s hypergeometric “Bethe ansatz” for solutions of Knizhnik–Zamolodchikov equations in conformal field theory.
Arnold’s result19 on the 16th Hilbert prob- lem, Part I, about disposition of ovals of real plane algebraic curves, was immediately improved by Rokhlin (who applied Arnold’s method but used more powerful tools from the topology of 4-manifolds). This led Rokhlin to his proof of a famous conjecture of Gudkov (who cor- rected Hilbert’s expectations in the problem), inspired many new developments (due to Viro and Kharlamov among others), and is consid- ered a crucial breakthrough in the history of real algebraic geometry.
Among other things, the paper of Arnold out- lines an explicit diffeomorphism between S4 and the quotient of CP2by complex conjugation.20The fact was rediscovered by Kuiper in 1974 and is known as Kuiper’s theorem [31]. Arnold’s argu- ment, based on hyperbolicity of the discriminant in the space of Hermitian forms, was recently revived in a far-reaching paper by M. Atiyah and J. Berndt [19].
Another work of Arnold in the same field21uni- fied the Petrovsky-Oleinik inequalities concerning topology of real hypersurfaces (or their comple- ments) and brought mixed Hodge structures (just introduced by Steenbrink into complex singularity theory) into real algebraic geometry.
Arnold’s interest in the 16th Hilbert problem, Part II, on the number of limit cycles of polynomial ODE systems on the plane has been an open-ended search for simplifying formulations. One such for- mulation22 (about the maximal number of limit cycles born under a nonconservative perturbation of a Hamiltonian system and equivalent to the problem about the number of zeroes of Abelian integrals over a family of real algebraic ovals) gen- erated extensive research. The results here include the general deep finiteness theorems of Khovan- sky and Varchenko, Arnold’s conjecture about nonoscillatory 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 Hodge structures, Funct. Anal. Appl. 12 (1978), no. 1, 1–14.
22V. I. Arnold, O. A. Oleinik, Topology of real algebraic varieties, 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 Hamiltonian systems,23various attempts to prove this conjec- ture (including a series of papers by Petrov-Tan’kin on Abelian integrals over elliptic curves, my own application of Sturm’s theory to nonoscillation of hyperelliptic integrals, and more recent estimates of Grigoriev, Novikov–Yakovenko), and further work by Horozov, Khovansky, Ilyashenko and oth- ers. Yet another modification of the problem (a discrete one-dimensional analogue) suggested by Arnold led to a beautiful and nontrivial theorem of 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 has been approached by many authors, with a paradox- ical outcome: there are many relatively straightfor- ward and relatively successful generalizations, but none as unique and satisfactory as the elementary continued fraction theory. Arnold’s approach to this problem24is based on his discovery of a rela- tionship between graded algebras and Klein’s sails (i.e., convex hulls of integer points inside simplicial convex cones in Euclidean spaces). Arnold’s prob- lems and conjectures on the subject have led to the results of E. Korkina and G. Lauchaud generaliz- ing Lagrange’s theorem (which identifies quadratic irrationalities with eventually periodic continued fractions) and to the work of Kontsevich–Sukhov generalizing Gauss’s dynamical system and its ergodic 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 integral convex 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 classical prototype.
The above examples show how Arnold’s interest in specific problems helped to transform them into central areas of modern research. There are other classical results which, according to Arnold’s in- tuition, are scheduled to generate such new areas, but to my understanding have not yet achieved the status of important mathematical theories in spite of interesting work done by Arnold himself and some others. But who knows? To mention one: the Four-Vertex Theorem, according to Arnold, is the seed of a new (yet unknown) branch of topology (in the same sense as the Last Poincaré Theorem was the seed of symplectic topology). Another ex- ample: a field-theoretic analogue of Sturm theory, broadly understood as a study of topology of zero levels (and their complements) of eigenfunctions of selfadjoint linear partial differential operators.
Perhaps with the notable exceptions of KAM- theory and singularity theory, where Arnold’s contributions are marked not only by fresh ideas but also by technical breakthroughs (e.g., a heavy- duty tool in singularity theory—his spectral se- quence),25a more typical path for Arnold would be to invent a bold new problem, attack its first nontrivial cases with his bare hands, and then leave developing an advanced machinery to his followers. I have already mentioned how the the- ory of hyperplane arrangements emerged in this fashion. Here are some other examples of this sort where 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 bifurcations of wave fronts and caustics.26 The general ho- motopy theory formulation was then given by Ya. Eliashberg, and the corresponding “Thom rings” computed in an award-winning treatise by M. Audin [20]. A geometric realization of La- grange and Legendre characteristic numbers as the enumerative theory of singularities of global caustics and wave fronts was given by V. Vassiliev [38]. The method developed for this task, namely associating a spectral sequence to a stratification of functional spaces of maps according to types of singularities, was later applied by Vassiliev sev- eral more times, of which his work on Vassiliev invariants of knots is the most famous one.
Arnold’s definition27 of the asymptotic Hopf invariant as the average self-linking number of
25A spectral sequence for the reduction of functions to normal 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 a simply connected 3-fold and his “ergodic” theorem about coincidence of the invariant with Moffatt’s helicity gave the start to many improvements, generalizations, and applications of topological methods in hydro- and magneto-dynamics due to M. H. Freedman et al., É. Ghys, B. Khesin, K. Moffatt, and many others.28
As one can find out, say, on MathSciNet, Arnold is one of the most prolific mathematicians of our time. His high productivity is partly due to his fearless curiosity and enormous appetite for new problems.29 Paired with his taste and intuition, these qualities often bring unexpected fruit, sometimes in the areas quite remote from the domain of his direct expertise. Here are some examples.
Arnold’s observation30 on the pairs of triples of numbers computed by I. Dolgachev and A. Gabrielov and characterizing respectively uniformization 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 and Dolgachev–Nikulin, the phenomenon received a beautiful explanation in terms of geometry of K3-surfaces. As became clear in the early nineties, Arnold’s Strange Duality was the first, and highly nontrivial, manifestation of Mirror Symmetry: a profound conjecture discovered by string theo- rists and claiming a sort of equivalence between symplectic topology and complex geometry (or singularity 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 study of dynamics on Riemann surfaces defined by levels of closed 1-forms, which led to a number of remarkable results of Kontsevich–Zorich [29] and others related to ergodic theory on Teichmüller spaces and conformal field theory, and of Eskin–
Okounkov [24] in the Hurwitz problem of counting ramified covers over elliptic curves.
Arnold seems to be the first to suggest32that monodromy (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 on the Fukaya category (consisting of all Lagrangian submanifolds in the fibers and of their Floer com- plexes). This construction is now an important ingredient of the Mirror Symmetry philosophy and gave rise to the remarkable results of M.
Khovanov and P. Seidel about faithfulness of such Hamiltonian representations of braid groups [27].
The celebrated Witten’s conjecture proved by M. Kontsevich in 1991 characterizes intersection theory on Deligne–Mumford moduli spaces of Riemann surfaces in terms of KdV-hierarchy of integrable systems. A refreshingly new proof of this result was recently given by Okounkov–
Pandharipande. A key ingredient in their argument is an elementary construction of Arnold from his work on enumerative geometry of trigonometric polynomials.33
Among many concepts owing Arnold their exis- tence, let me mention two of general mathematical stature which do not carry his name.
One is the Maslov index, which proved to be important in geometry, calculus of variations, numbers theory, representation theory, quanti- zation, index theory of differential operators, and whose topological origin was explained by Arnold.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-forms with incommensurable periods, Funct. Anal. Appl. 25 (1991), no. 2, 1–12.
32See Some remarks on symplectic monodromy of Milnor fibrations, The Floer memorial volume, 99–103, Progr.
Math., 133, Birkhäuser, Basel, 1995.
33Topological classification of complex trigonometric polynomials and the combinatorics of graphs with an identical number of vertices and edges, Funct. Anal.
Appl. 30 (1996), no. 1, 1–17.
34In his paper On a characteristic class entering into conditions of quantization, Funct. Anal. Appl. 1 (1967), 1–14.
controversy over which of the known integrability mechanisms is most fundamental, but there is a consensus that integrability means a complete set of Poisson-commuting first integrals.
This definition and “Liouville’s Theorem” on geometric consequences of the integrability prop- erty (namely, foliation of the phase space by Lagrangian tori) are in fact Arnold’s original inventions.
Similar to the case with integrable systems, there are other examples of important develop- ments which have become so common knowledge that Arnold’s seminal role eventually became in- visible. Let me round up these comments with a peculiar 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, picked up 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 each author’s contribution was correct, so that neither one could alone be held responsible). By the way, Arnold cites this episode as the reason why he refrained 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 and its applications, including those I described above and those I missed: classical and celestial mechan- ics, cosmology and hydrodynamics, dynamical systems and bifurcation theory, ordinary and par- tial differential equations, algebraic and geometric topology, number theory and combinatorics, real and complex algebraic geometry, symplectic and contact geometry and topology, and perhaps some others. I can think of few mathematicians whose work and personality would influence the scien- tific community at a comparable scale. And beyond this community, Arnold is a highly visible (and possibly controversial) figure, the subject of sev- eral interviews, of a recent documentary movie, and even of the night sky show, where one can watch an asteroid, Vladarnolda, named after him.
I am sure there are other mathematicians who also deserve [the name of the prize], but awarding it to Vladimir Arnold will hardly be perceived by anyone as a mistake.
35Arnold, V. I., Sinai, Ya. G., On small perturbations of the automorphisms 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 were very close friends since the beginning of the twen- tieth century. Both families lived in Odessa, which was a big city in the southern part of Russia and now is a part of Ukraine. At that time, Odessa was a center of Jewish intellectual life, which produced many scientists, musicians, writers, and other sig- nificant figures.
My maternal grandfather, V. F. Kagan, was a well-known geometer who worked on the founda- tions of geometry. During World War I, he gave the very first lecture course in Russia on the special relativity theory. At various times his lec- tures were attended by future famous physicists L. 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’s maternal grandmother. He was the founder and the leader of a major school of theoretical physics that included A. A. Andronov, G. S. Landsberg, and M. A. Leontovich, among others. A. A. Andronov is known to the mathematical community for his famous paper “Robust systems”, coauthored with L. S. Pontryagin, which laid the foundations of the theory of structural stability of dynami- cal systems. A. A. Andronov was the leader of a group of physicists and mathematicians working in Nizhny Novgorod, formerly Gorky, on nonlinear oscillations. 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. Tamm was a Nobel Prize winner in physics in the fifties.
N. D. Papaleksi was a great expert on nonlinear optics.
V. I. Arnold was born in Odessa, where his mother 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 his family had a young prodigy soon became widely known. In those days, when we were both in high school, we did not really know each other. On one occasion, Arnold visited my grandfather to borrow a mathematics book, but I was not there at the time. We met for the first time when we were both students at the mathematics department of the Moscow State University; he was walking by with Professor A. G. Vitushkin, who ran a fresh- man seminar on real analysis, and Arnold was one of the most active participants. When Arnold was a third-year undergraduate student, he was inspired by A. N. Kolmogorov to work on super- position of functions of several variables and the related Hilbert’s thirteenth problem. Eventually this work became Arnold’s Ph.D. thesis. When I visited the University of Cambridge recently, I was very pleased to learn that one of the main lecture courses there was dedicated to Arnold’s and Kolmogorov’s work on Hilbert’s thirteenth problem.
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 center of Moscow. During one of my visits, I was shown a tent in the backyard of the building where Arnold used to spend his nights, even in cold weather. It seems likely that Arnold’s excellent knowledge of history and geography of Moscow, which many of his friends remember with admiration, originated at 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 observed that his ideas both in science and in life came to him as revelations. I remember one particular oc- casion, when we were climbing in the Caucasus Mountains and spent a night with some shepherds in their tent. In the morning we discovered that the shepherds were gone and had left us alone with their dogs. Caucasian dogs are very big, strong, and dangerous, for they are bred and trained to fight 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 started shouting very loudly at the dogs, using all the ob- scenities he could think of. I never heard him use such language either before or after this incident, nor did anybody else. It was a brilliant idea, for it worked! 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 main Moscow University building to a subway station, which usually took about fifteen minutes, Arnold told us that he recently came up with the Galois theory entirely on his own and explained his ap- proach to us. The next day, Arnold told us that he found a similar approach in the book by Felix Klein 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 include his discovery of the Arnold-Maslov cocycle in the theory of semi-classical approximations and Arnold inequalities for the number of ovals in real algebraic curves. Many other people who knew Arnold personally could provide more examples of this kind.
Arnold became a graduate student at the Moscow State University in 1959. Naturally it was A. N. Kolmogorov who became his advisor.
In 1957 Kolmogorov gave his famous lecture course on dynamical systems, which played a pivotal role in the subsequent development of the theory. The course was given three years af- ter Kolmogorov’s famous talk at the Amsterdam Congress of Mathematics.
Kolmogorov began his lectures with the expo- sition of the von Neumann theory of dynamical systems with pure point spectrum. Everything was done in a pure probabilistic way. Later Kol- mogorov found a similar approach in the book by Fortet and Blank–Lapierre on random processes, intended for engineers.
This part of Kolmogorov’s lectures had a profound effect on researchers working on the measure-theoretic isomorphism in dynam- ical systems, a long-standing problem that goes back to von Neumann. It was shown that when the spectrum is a pure point one, it is the only
isomorphism invariant of a dynamical system and that two systems with the same pure point spectrum are isomorphic. The excitement around these results was so profound that people began to believe that the isomorphism theory of sys- tems with continuous spectrum would be just a straightforward generalization of the theory of systems with pure point spectrum. However, this was refuted by Kolmogorov himself. He proposed the notion of entropy as a new isomorphism in- variant for systems with continuous spectrum.
Since the entropy is zero for systems with pure point spectrum, it does not distinguish between such systems, but systems with continuous spec- trum might have positive entropy that must be preserved by isomorphisms. This was a path- breaking discovery, which had a tremendous impact on the subsequent development of the theory.
The second part of Kolmogorov’s lectures was centered around his papers on the preser- vation of invariant tori in small perturbations of integrable Hamiltonian systems, which were published in the Doklady of the Soviet Academy of Sciences. Unfortunately there were no writ- ten notes of these lectures. V. M. Tikhomirov, one of Kolmogorov’s students, hoped for many years to locate such notes, but he did not suc- ceed. Arnold used to claim in his correspondence with many people that good mathematics stu- dents of Moscow University could reconstruct Kolmogorov’s proof from the text of his pa- pers in the Doklady. However, this was an exaggeration. Recently two Italian mathemati- cians, A. Giorgilli and L. Chierchia, produced a proof of Kolmogorov’s theorem, which was com- plete and close to Kolmogorov’s original proof, as they claimed.
Apparently Kolmogorov himself never wrote a detailed proof of his result. There might be several explanations. At some point, he had plans to work on applications of his technique to the famous three-body problem. He gave a talk on this topic at a meeting of the Moscow Mathematical Society. However, he did not prepare a written version of his talk. Another reason could be that Kolmogorov started to work on a different topic and did not want to be distracted. There might be a third reason, although some people would disagree with it. It is possible that Kolmogorov underestimated the significance of his papers.
For example, some graduate exams on classical mechanics included the proof of Kolmogorov’s theorem, so it was easy to assume that the proof was already known. The theory of entropy, intro- duced by Kolmogorov roughly at the same time, seemed a hotter and more exciting area. He might have felt compelled to turn his mind to this new topic.
A. Kirillov, I. Petrovskii, V. Arnold.
Arnold immediately started to work on all the problems raised in Kolmogorov’s lectures.
In 1963 the Moscow Mathematical Society cele- brated Kolmogorov’s sixtieth birthday. The main meeting took place in the Ceremony Hall of the Moscow State University, with about one thou- sand people attending. The opening lecture was given by Arnold on what was later called KAM the- ory, where KAM stands for Kolmogorov, Arnold, Moser. For that occasion, Arnold prepared the first complete exposition of the Kolmogorov theorem.
I asked Arnold why he did that, since Kolmogorov presented his proof in his lectures. Arnold replied that the proof of the fact that invariant tori constitute a set of positive measure was not complete. When Arnold asked Kolmogorov about some details of his proof, Kolmogorov replied that he was too busy at that time with other problems and that Arnold should provide the details by himself. This was exactly what Arnold did. I be- lieve that when Kolmogorov prepared his papers for publication in the Doklady, he did have com- plete proofs, but later he might have forgotten some details. Perhaps it can be expressed better by saying that it required from him an effort that he was not prepared to make at that time.
In the following years, Kolmogorov ran a semi- nar on dynamical systems, with the participation of many mathematicians and physicists. At some point, two leading physicists, L. A. Arzimovich and M. A. Leontovich, gave a talk at the seminar on the existence of magnetic surfaces. Subsequently this problem was completely solved by Arnold, who submitted his paper to the main physics journal in the Soviet Union, called JETP. After some time, the paper was rejected. According to Arnold, the referee report said that the referee did not under- stand anything in that paper and hence nobody else would understand it. M. A. Leontovich helped Arnold to rewrite his paper in the form accessible to physicists, and it was published eventually. Ac- cording to Arnold, this turned out to be one of his most quoted papers.
Arnold’s first paper related to the KAM theory was about smooth diffeomorphisms of the circle that were close to rotations. Using the methods of the KAM theory, Arnold proved that such dif- feomorphisms can be reduced to rotations by ap- plying smooth changes of variables. The problem in the general case was called the Arnold prob- lem. It was completely solved by M. Herman and J.-C. Yoccoz.
A. N. Kolmogorov proved his theorem in the KAM theory for the so-called nondegenerate per- turbations of integrable Hamiltonian systems.
Arnold extended this theorem to degenerate perturbations, which arise in many applications of the KAM theory.
Arnold proposed an example of the Hamil- tonian system which exhibits a new kind of instability and which was later called the Arnold diffusion. The Arnold diffusion appears in many physical problems. New mathematical results on the Arnold diffusion were recently proved by J. Mather. V. Kaloshin found many applications of the Arnold diffusion to problems of celestial mechanics.
In later years Arnold returned to the theory of dynamical systems only occasionally. One can mention his results in fluid mechanics (see his joint book with B. Khesin [16]) and a series of pa- pers on singularities in the distribution of masses in the universe, motivated by Y. B. Zeldovich. But all this was done in later years.
Steve Smale
Vladimir I. Arnold
My first meeting with V. I. Arnold took place in Moscow in September 1961 (certainly I had been very 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 to Arnold, Novikov, and Sinai. As I wrote later [35], I was extraordinarily impressed by such a power- ful group of four young mathematicians and that there was nothing like that in the West. At my next visit to Moscow for the world mathematics congress in 1966 [36], I again saw much of Dima Arnold. At that meeting he introduced me to Kol- mogorov.
Perhaps the last time I met Dima was in June 2003 at the one-hundred-year memorial confer- ence for Kolmogorov, again in Moscow. In the in- tervening years we saw each other on a number of occasions in Moscow, in the West, and even in Asia.
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 to Hong Kong. Dima and I often were together on the fantastic day hikes in the Hong Kong countryside parks. 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 and Technology. Dima expressed himself in his usual provocative way! I recall that we found ourselves on the same side in most of the controversies, and catastrophe theory in particular.
Dima Arnold was a great mathematician, and here I will just touch on his mathematical contri- butions which affected me the most.
While I never worked directly in the area of KAM, nevertheless those results had a great im- pact in my scientific work. For one thing they directed me away from trying to analyze the global orbit structures of Hamiltonian ordinary differential equations, in contrast to what I was doing for (unconstrained) equations. Thus KAM contributed to my motivation to study mechan- ics in 1970 from the point of view of topology, symmetry, and relative equilibria rather than its dynamical properties. The work of Arnold had already affected those subjects via his big paper on fluid mechanics and symmetry in 1966. See Jerry Marsden’s account of how our two works are related [32]. I note that Jerry died even more recently than Dima.
KAM shattered the chain of hypotheses, er- godic, quasi-ergodic, and metric transitivity going from Boltzmann to Birkhoff. That suggested to me some kind of non-Hamiltonian substitute in these hypotheses in order to obtain foundations for thermodynamics [37].
I read Arnold’s paper on braids and the coho- mology of swallowtails. It was helpful in my work on topology and algorithms, which Victor Vassiliev drastically sharpened.
Dima could express important ideas simply and in such a way that these ideas could transcend a single discipline. His work was instrumental in transforming Kolmogorov’s early sketches into a revolutionary recasting of Hamiltonian dynamics with sets of invariant curves, tori of positive mea- sure, and Arnold diffusion.
It was my good fortune to have been a part of Dima Arnold’s life and his mathematics.
Mikhail Sevryuk
Some Recollections of Vladimir Igorevich A very large part of my life is connected with Vladimir 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 and Mathematics of Moscow State University. Under his supervision, I wrote my term papers, master’s thesis, and doctoral thesis. At the end of my first year in graduate school, Arnold suggested that I write a monograph on reversible dynamical sys- tems for Springer’s Lecture Notes in Mathematics series, and working on this book was one of the cornerstones of my mathematical biography. For the last time, I met Vladimir Igorevich (V. I., for short) on November 3, 2009, at his seminar at Moscow State.
If I had to name one characteristic feature of Arnold as I remember him, I would choose his agility. He walked fast at walkways of Moscow State (faster than most of the students, not to mention the faculty), his speech was fast and clear, his reaction to one’s remark in a conversa- tion was almost always instantaneous, and often utterly unexpected. His fantastic scientific pro- ductivity is well known, and so is his enthusiasm for sports.
V. I. always devoted a surprising amount of time and effort to his students. From time to time, he had rather weak students, but I do not recall a single case when he rejected even a struggling student. In the 1980s almost every meeting of his famous seminar at Moscow State he started with “harvesting”: collecting notes of his students with sketches of their recent math- ematical achievements or drafts of their papers (and Arnold returned the previously collected ones with his corrections and suggestions). After a seminar or a lecture, he often continued talking with participants for another 2–3 hours. Arnold’s generosity was abundant. Many times, he gave long written mathematical consultations, even to people unknown to him, or wrote paper reports substantially exceeding the submitted papers.
Dubna, 2006.
In recent years, he used all his energy to stop a rapid deterioration of mathematical (and not only mathematical) education in Russia.
I tried to describe my experience of being V. I.’s student at Moscow State in [34]. I would like to emphasize here that Arnold did not follow any pattern in supervising his students. In some cases he would inform a student that there was a cer- tain “uninhabited” corner in the vast mathematical land, and if the student decided to “settle” at that corner, then it was this student’s task to find the main literature on the subject, to study it, to pose new problems, to find methods of their solution, and to achieve all this practically single-handedly.
Of course, V. I. kept the progress under control. (I recall that, as a senior, I failed to submit my “har- vest” for a long time, but finally made substantial progress. Arnold exclaimed, “Thank God, I have started 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 to collaborate— this is how our joint paper [13] came about. When need be, V. I. could be rather harsh.
Once I witnessed him telling a student, “You are working too slowly. I think it will be good if you start 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 of mathematics, of natural sciences, and of all nature. He considered mathematics as being part of physics, and his “economics” defini- tion of mathematics as a part of physics in which experiments are cheap is often quoted.
(Let me add in parentheses that I would prefer to characterize mathematics as the natural sci- ence that studies the phenomenon of infinity by analogy with a little-known but remarkable definition of topology as the science that studies the phenomenon of continuity.) However, Arnold noted other specific features of mathematics: “It is a fair observation that physicists refer to the first author, whereas mathematicians to the latest one.” (He considered adequate references to be of paramount importance and paid much attention to other priority questions; this was a natural extension of his generosity, and he encouraged his students to “over-acknowledge”, rather than to “under-acknowledge”.)
V. I. was an avid fighter against “Bourbakism”, a suicidal tendency to present mathematics as a for- mal derivation of consequences from unmotivated axioms. According to Arnold, one needs mathe- matics to discover new laws of nature as opposed to “rigorously” justify obvious things. V. I. tried to teach his students this perception of mathematics and natural sciences as a unified tool for under- standing the world. For a number of reasons, after having graduated from university, I had to work partially as a chemist, and after Arnold’s school this caused me no psychological discomfort.
Fundamental mathematical achievements of Arnold, as well as those of his teacher, A. N. Kol- mogorov, cover almost all mathematics. It well may be that V. I. was the last universal mathemati- cian. My mathematical specialization is the KAM theory. V. I. himself described the contributions of the three founders; see, e.g., [15], [17]. For this reason, I shall only briefly recall Arnold’s role in the development of the KAM theory.
KAM theory is the theory of quasiperiodic mo- tions in nonintegrable dynamical systems. In 1954 Kolmogorov made one of the most astonishing discoveries in mathematics of the last century.
Consider a completely integrable Hamiltonian system with n degrees of freedom, and let (I, ϕ) be the corresponding action-angle variables. The phase space of such a system is smoothly foli- ated into invariant n-tori {I = const} carrying conditionally periodic motions ˙ϕ = ω(I). Kol- mogorov showed that if det(∂ω/∂I) ≠ 0, then (in spite of the general opinion of the physical community of that time) most of these tori (in the Lebesgue sense) are not destroyed by a small Hamiltonian perturbation but only slightly de- formed in the phase space. To be more precise, a torus {I = I0} persists under a perturbation whenever the frequencies ω1(I0), . . . , ωn(I0) are Diophantine (strongly incommensurable). The perturbed tori (later called Kolmogorov tori) carry quasiperiodic motions with the same frequencies.
To prove this fundamental theorem, Kolmogorov proposed 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 to prove analyticity of the Denjoy homeomorphism conjugating an analytic diffeomorphism of a circle with a rotation (under the condition that this dif- feomorphism is close to a rotation and possesses a Diophantine rotation number). His paper [4]
with this result contained also the first detailed exposition of Kolmogorov’s method. Then, in a se- ries of papers, Arnold generalized Kolmogorov’s theorem to various systems with degeneracies.
In fact, he considered two types of degeneracies often encountered in mechanics and physics: the proper degeneracy, where some frequencies of the perturbed tori tend to zero as the perturbation magnitude vanishes, and the limit degeneracy, where the unperturbed foliation into invariant tori is singular and includes tori of smaller di- mensions. The latter degeneracy is modeled by a one-degree-of-freedom Hamiltonian system having an equilibrium point surrounded by in- variant circles (the energy levels). These studies culminated in Arnold’s famous (and technically extremely hard) result [5] on stability in planetary- like systems of celestial mechanics where both the 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 physicists F. M. Izrailev and B. V. Chirikov in 1968.
Arnold always regarded his discovery of the universal mechanism of instability of the action variables 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 explicit example where such instability occurs. Chaotic
evolution of the actions along resonances be- tween the Kolmogorov tori was called “Arnold’s diffusion” by Chirikov in 1969. In the case of two degrees of freedom, the Kolmogorov 2-tori divide a three-dimensional energy level, which makes an evolution of the action variables impossible.
All these works by Arnold took place in 1958–
1965. At the beginning of the eighties, he returned to the problem of quasiperiodic motions for a short time and examined some interesting properties of the analogs of Kolmogorov tori in reversible systems. That was just the time when I started my diploma work. So V. I. forced me to grow 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 of Arnold’s research. In spite of what is occasion- ally claimed, Arnold did not hate computers: he considered them as an absolutely necessary instrument of mathematical modeling when indeed large computations were involved. He initiated many computer experiments in dynam- ical systems and number theory and sometimes participated in them (see [17]). But of course he strongly disapproved of the aggressive penetra- tion of computer technologies into all pores of society and the tendency of a man to become a helpless and mindless attachment to artificial intelligence devices. One should be able to divide 111 by 3 without a calculator (and, better still, without scrap paper).
V. I. had a fine sense of humor. It is impossible to forget his somewhat mischievous smile. In con- clusion, here are a couple of stories which might help to illustrate the unique charm of this person.
I remember how a speaker at Arnold’s seminar kept 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, and the proof of a theorem involved tedious computa- tions: “Everyone must make these computations once—but only once. I made them in the past, so I 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 was drawing a series of pictures depicting the pere- stroika (surgery) of a certain geometrical object as depending on a parameter. Arnold: “Something is not quite right here. Why is your central stratum always the same? Perestroika always starts at the center and then propagates to the periphery.”
