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An Interview with Vladimir Arnold arnold.qxp 3/24/98 11:38 AM Page 432 An Interview with Vladimir Arnol′d by S. H. Lui Utilius scandalum nasci permittur quam veritas relinquatur. (One should speak the truth even at risk of provoking a scandal.) —Decretalium V of Pope Gregory IX, 1227–1241 ladimir Arnol′d is currently professor young children start thinking about such prob- of mathematics at both the Steklov lems even before they have any knowledge of Mathematical Institute, Moscow, and numbers. Children five to six years old like them Ceremade, Université de Paris– very much and are able to solve them, but they Dauphine. Professor Arnol′d obtained may be too difficult for university graduates, Vhis Ph.D. from the Moscow State University in who are spoiled by formal mathematical train- 1961. He has made fundamental contributions ing. A typical example is: in dynamical systems, singularity theory, sta- You take a spoon of wine from a barrel of bility theory, topology, algebraic geometry, mag- wine, and you put it into your cup of tea. Then neto-hydrodynamics, partial differential equa- you return a spoon of the (nonuniform!) mixture tions, and other areas. Professor Arnol′d has of tea from your cup to the barrel. Now you won numerous honors and awards, including have some foreign substance (wine) in the cup the Lenin Prize, the Crafoord Prize, and the Har- and some foreign substance (tea) in the barrel. vey Prize. Which is larger: the quantity of wine in the cup This interview took place on November 11, or the quantity of tea in the barrel at the end of 1995. The following articles may be of interest your manipulations? to the reader: Slightly older children, knowing the first few 1) Conversation with Vladimir Igorevich numbers, like the following problem. Jane and Arnol′d, by S. Zdravkovska, Mathematical Intel- John wish to buy a children’s book. However, Jane ligencer 9:4 (1987). needs seven more cents to buy the book, while 2) A mathematical trivium, by V. I. Arnol′d, John needs one more cent. They decide to buy Russian Math. Surveys 46:1 (1991). only one book together but discover that they 3) Will Russian mathematics survive?, by V. I. Arnol′d, Notices of the AMS 40:2 (1993). S. H. Lui is an assistant professor at the Hong Kong Uni- 4) Why Mathematics?, by V. I. Arnol′d, Quan- versity of Science and Technology. His e-mail address tum, 1994. is [email protected]. 5) Will mathematics survive? Report on the This article previously appeared in the February Zurich Congress, by V. I. Arnol′d, Mathematical 1996 issue of the Hong Kong Mathematics Society Intelligencer 17:3 (1995). Newsletter. Lui: Please tell us a little bit about your early ed- Editor’s Note: As this article went to press, ucation. Were you already interested in math- V. I. Arnol′d submitted an update on the in- ematics as a child? terview, based on subsequent correspondence and events. It was received too late to be in- Arnol′d: The Russian mathematical tradition cluded in the article. goes back to the old merchant problems. Very 432 NOTICES OF THE AMS VOLUME 44, NUMBER 4 arnold.qxp 3/24/98 11:38 AM Page 433 do not have enough money. What is the price of nected with at most K others by wires of fixed the book? (One should know that books in Rus- thickness. He explained that when N is very sia are very cheap!) large (while K is fixed), the diameter of the cube Many Russian families have the tradition of grows like √N by the following argument: the giving hundreds of such problems to their chil- grey matter (the body of the neurons) is on the dren, and mine was no exception. The first real surface of the human brain, while the white mat- mathematical experience I had was when our ter (the connections) fills the interior part. Since schoolteacher I. V. Morozkin gave us the fol- the brain is embedded into the head as eco- lowing problem: Two old women started at sun- nomically as possible, a sufficiently complicated rise and each walked at a constant velocity. One brain of N neurons can only be embedded in a went from A to B and the other from B to A. They cube of size √N (while a trivial brain, like that met at noon and, continuing with no stop, arrived of a worm, needs only the size √3 N). respectively at B at 4 p.m. and at A at 9 p.m. At Kolmogorov’s work on what is now called what time was the sunrise on this day? KAM theory of Hamiltonian systems was a by- I spent a whole day thinking on this oldie, and product of compulsory exercises that he gave to the solution (based on what is now called scal- all second-year undergraduate students. One of ing arguments, dimensional analysis, or toric the problems was the study of some nontrivial variety theory, depending on your taste) came completely integrable systems (like the motion as a revelation. The feeling of discovery that I had of a heavy particle along the surface of a hori- then (1949) was exactly the same as in all the sub- zontal torus of revolution). No computers were sequent much more serious problems—be it the available then! He observed that the motion in discovery of the relation between algebraic geom- all such classical examples was quasiperiodic etry of real plane curves and four-dimensional and tried to find examples of more complicated topology (1970) or between singularities of caus- motion (“mixing”, or in today’s language, “chaos”) tics and of wave fronts and simple Lie algebra in the case of nonintegrable perturbed systems. and Coxeter groups (1972). It is the greed to ex- His attempts were unsuccessful. The problem perience such a wonderful feeling more and which motivated his study is still open—no one more times that was, and still is, my main mo- has been able to find an invariant torus carry- tivation in mathematics. ing mixing flows in generically perturbed sys- tems. However, the by-products of this investi- Lui: What was it like studying at Moscow State gation are far more important than the initial University? Can you tell us something about the technical problem on mixing. They include the professors (Petrovskii, Kolmogorov, Pontriagin, discovery of the persistent nonresonant tori, Rokhlin,…)? the “accelerated convergence” method and the Arnol′d: The atmosphere of the Mechmat related implicit function theorems in function (Moscow State University Mechanics and Math- spaces, the proof of stability of motion in many ematics Faculty) in the fifties when I was a stu- Hamiltonian systems (e.g., gyroscopes and plan- dent is described in detail in the book Golden etary orbits), and the proof of the existence of Years of Moscow Mathematics, edited by magnetic surfaces in the Tokamak geometry, S. Zdravkovska and P. L. Duren and published which is used in the study of plasma containment jointly by the AMS and LMS in 1993. It contains for controlled thermonuclear fusion. reminiscences of many people. In particular, my That consequences of an investigation are article was on A. N. Kolmogorov, who was my su- more important than the original question is a pervisor. general phenomenon. The initial goal of Colum- The constellation of great mathematicians in bus was to find a new way to India. The discov- the same department when I was studying at the ery of the New World was just a by-product. Mechmat was really exceptional, and I have never Pontriagin was already very weak when I was seen anything like it at any other place. Kol- a student at Mechmat, but he was perhaps the mogorov, Gelfand, Petrovskii, Pontriagin, best of the lecturers. He had just turned from P. Novikov, Markov, Gelfond, Lusternik, Khinchin, topology to control theory, and his personality and P. S. Alexandrov were teaching students like had also changed a lot. He later explained his rea- Manin, Sinai, S. Novikov, V. M. Alexeev, Anosov, sons for switching to applied mathematics and A. A. Kirillov, and me. his antisemitic ideas in his autobiography pub- All these mathematicians were so different! lished in the Russian Mathematical Surveys. It was almost impossible to understand Kol- When he submitted this paper to the Editorial mogorov’s lectures, but they were full of ideas Board, the KGB representative suggested that the and were really rewarding! I recall his explana- article should not be published as it was be- tion of his theory of the size of the minimal cause of its extreme openness. I would prefer to cube into which you can embed every graph hav- see the original text published—what you now ing N vertices (balls of fixed size), each con- find is rather softened. Some people claim that APRIL 1997 NOTICES OF THE AMS 433 arnold.qxp 3/24/98 11:38 AM Page 434 his antisemitism might be simply a manifesta- became an active antisemite. My personal rela- tion of his fear that some part of his blood might tion with Pontriagin was rather good. He invited be Jewish and that this might be discovered. me to his house and to his seminar and showed However, Pontriagin was not always like this! genuine interest in my work, especially on sin- During the war his best student, V. A. Rokhlin, gularity theory. This was partially due to our was wounded and imprisoned by the Germans. common interests in differential topology and Later, Rokhlin was liberated by the Americans, control and game theory.
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