Tribute to Vladimir Arnold Boris Khesin and Serge Tabachnikov, Coordinating Editors Vladimir Arnold, an eminent mathematician of My first mathematical revelation was when I our time, passed away on June 3, 2010, nine days met my first real teacher of mathematics, Ivan before his seventy-third birthday. This article, Vassilievich Morozkin. I remember the problem along with one in the next issue of the Notices, about two old ladies who started simultaneously touches on his outstanding personality and his from two towns to- great contribution to mathematics. ward each other, met A word about spelling: we use “Arnold”, as at noon, and who opposed to “Arnol’d”; the latter is closer to the Rus- reached the oppo- sian pronunciation, but Vladimir Arnold preferred site towns at 4 p.m. the former (it is used in numerous translations of and 9 p.m., respec- his books into English), and we use it throughout. tively. The question was when they started Arnold in His Own Words their trip. In 1990 the second author interviewed V. Arnold We didn’t have al- for a Russian magazine Kvant (Quantum). The gebra yet. I invented readership of this monthly magazine for physics an “arithmetic” solu- and mathematics consisted mostly of high school tion (based on a scal- students, high school teachers, and undergraduate ing—or similarity— students; the magazine had a circulation of about argument) and expe- 200,000. As far as we know, the interview was never rienced a joy of dis- translated into English. We translate excerpts from covery; the desire to Vladimir Igorevich Arnold this interview;1 the footnotes are ours. experience this joy Q: How did you become a mathematician? What again was what made me a mathematician. was the role played by your family, school, math- A. A. Lyapunov organized at his home “Chil- ematical circles, Olympiads? Please tell us about dren Learned Society”. The curriculum included your teachers. mathematics and physics, along with chemistry A: I always hated learning by rote. For that and biology, including genetics that was just re- 2 reason, my elementary school teacher told my cently banned (a son of one of our best geneticists parents that a moron, like myself, would never was my classmate; in a questionnaire, he wrote: manage to master the multiplication table. “my mother is a stay-at-home mom; my father is a stay-at-home dad”). Boris Khesin is professor of mathematics at the University Q: You have been actively working in mathe- of Toronto. His email address is [email protected]. matics for over thirty years. Has the attitude of edu. society towards mathematics and mathematicians Serge Tabachnikov is professor of mathematics at Penn- changed? sylvania State University. His email address is tabachni@ A: The attitude of society (not only in the USSR) math.psu.edu. to fundamental science in general, and to mathe- 1Full text is available in Russian on the website of Kvant maticsin particular, iswell describedby I. A.Krylov magazine (July 1990), http://kvant.mirror1.mccme. ru/. 2In 1948 genetics was officially declared “a bourgeois DOI: http://dx.doi.org/10.1090/noti810 pseudoscience” in the former Soviet Union. 378 Notices of the AMS Volume 59, Number 3 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 veryfirst popularizers, M. Faraday, Vladimir Arnold, circa 1985. arrived at the conclusion that “Lectures which really teach will never be popular; lectures which are popular will never teach.” This Faraday effect is in the fable “The hog under the oak”.3 In the 1930s easy to explain: according to N. Bohr, clearness and and 1940s, mathematics suffered in this country truth are in a quantum complementarity relation. less than other sciences. It is well known that Viète Q: Many readers of Kvant aspire to become was a cryptographer in the service of Henry IV of mathematicians. Are there “indications” and “con- France. Since then, certain areas of mathematics traindications” to becoming a mathematician, or have been supported by all governments, and even can anyone interested in the subject become one? Beria4 cared about preservation of mathematical Is it necessary for a mathematician-to-be to success- culture in this country. fully participate in mathematical Olympiads? In the last thirty years the prestige of mathe- A: When 90-year-old Hadamard was telling matics has declined in all countries. I think that A. N. Kolmogorov about his participation in mathematicians are partially to be blamed as well Concours Général (roughly corresponding to our (foremost, Hilbert and Bourbaki), particularly the Olympiads), he was still very excited: Hadamard ones who proclaimed that the goal of their science won only the second prize, while the student was investigation of all corollaries of arbitrary who had won the first prize also became a systems of axioms. mathematician, but a much weaker one! Q: Does the concept of fashion apply to mathe- Some Olympiad winners later achieve nothing, matics? and many outstanding mathematicians had no A: Development of mathematics resembles a success in Olympiads at all. fast revolution of a wheel: sprinkles of water are Mathematicians differ dramatically by their time flying in all directions. Fashion—it is the stream scale: some are very good tackling 15-minute that leaves the main trajectory in the tangential problems, some are good with the problems that direction. These streams of epigone works attract require an hour, a day, a week, the problems the most attention, and they constitute the main that take a month, a year, decades of thinking. mass, but they inevitably disappear after a while A. N. Kolmogorov considered his “ceiling” to be because they parted with the wheel. To remain two weeks of concentrated thinking. on the wheel, one must apply the effort in the Success in an Olympiad largely depends on direction perpendicular to the main stream. A mathematician finds it hard to agree that one’s sprinter qualities, whereas serious mathe- the introduction of a new term not supported by matical research requires long distance endurance (B. N. Delaunay used to say, “A good theorem takes 3See A. Givental and E. Wilson-Egolf’s (slightly modern- not 5 hours, as in an Olympiad, but 5,000 hours”). ized) translation of this early nineteenth-century Russian There are contraindications to becoming a re- fable at the end of this interview. search mathematician. The main one is lack of 4The monstrous chief of Stalin’s secret police. love of mathematics. March 2012 Notices of the AMS 379 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- Teaching at Moscow State University, 1983. 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 But mathematical talents can be very diverse: possible, but we apologize in advance for possible geometrical and intuitive, algebraic and compu- omissions: there were many more people whose tational, logical and deductive, natural scientific work Arnold influenced greatly and who might and inductive. And all kinds are useful. It seems feel they belong to Arnold’s school. to me that one’s difficulties with the multipli- Names are listed chronologically according to cation table or a formal definition of half-plane the defense years, which are given in parentheses. should not obstruct one’s way to mathematics. Many former Arnold’s students defended the sec- An extremely important condition for serious ond degree, the Doctor of Science or Habilitation, mathematical research is good health. but we marked it only in the cases where the first Q: Tell us about the role of sport in your life. degree was not under Arnold’s supervision. A: When a problem resists a solution, I jump on my cross-country skis. After forty kilometers a Edward G. Belaga (1965) solution (or at least an idea for a solution) always Andrei M. Leontovich (1967) comes. Under scrutiny, an error is often found. Yulij S. Ilyashenko (1969) (1994, DSci) But this is a new difficulty that is overcome in the Anatoly G. Kushnirenko (1970) same way. Askold G. Khovanskii (1973) Nikolai N. Nekhoroshev (1973) The Hog Under the Oak Alexander S. Pyartli (1974) A Hog under a mighty Oak Alexander N.