# Vladimir Igorevich Arnold

### Quick Info

Odessa, USSR (now Ukraine)

Paris, France

**Vladimir Arnold**won a Wolf prize for his work on dynamical systems, differential equations, and singularity theory.

### Biography

**Vladimir Arnold**'s parents were Igor Vladimorovich Arnold and Nina Alexandrova Isakovich. Several generations of Arnold's family had been scientists. His interest in mathematics began when he was as young as five years old. He explained that this was a consequence of the Russian mathematical tradition [4]:-

Very young children start thinking about [old merchant] problems even before they have any knowledge of numbers. Children five to six years old like them very much and are able to solve them, but they may be too difficult for university graduates, who are spoiled by formal mathematical training. ... Many Russian families have the tradition of giving hundreds of [mathematical] problems to their children, and mine were no exception.When he was twelve years old he was given challenging problems by his schoolteacher. He quoted one such problem in [4]:-

Two old women started at sunrise and each walked as a constant velocity. One went from A to B and the other from B to A. They met at noon and, continuing with no stop, arrived respectively at B at 4 p.m. and at A at 9 p.m. At what time was sunrise on this day?Arnold said:-

I spent a whole day thinking on this oldies, and the solution ... came as a revelation. The feeling of discovery I had then was exactly the same as in all the subsequent much more serious problems ...He entered Moscow State University in 1954 as an undergraduate student in the Faculty of Mechanics and Mathematics. He was awarded his first degree in 1959 with a dissertation

*On mappings of a circle to itself*written with Kolmogorov as advisor. Speaking of his undergraduate years he said [4]:-

The constellation of great mathematicians in the same department when I was studying at the Faculty of Mechanics and Mathematics was really exceptional, and I have never seen anything like it at any other place. Kolmogorov, Gelfand, Petrovsky, Pontryagin, P Novikov, Markov, Gelfond, Lusternik, Khinchin and P S Aleksandrov were teaching students like Manin, Sinai, Sergi Novikov, V M Alexeev, Anosov, A A Kirillov, and me. All these mathematicians were so different! It was almost impossible to understand Kolmogorov's lectures, but they were full of ideas and were really rewarding! ... Pontryagin was already very weak when I was a student at the Faculty of Mechanics and Mathematics, but he was perhaps the best of the lecturers.Arnold continued to study as a postgraduate student at Moscow State University for his Candidate's Degree (equivalent to a Ph.D.) still with Kolmogorov as advisor. He was awarded the degree in 1961 by the Institute of Applied Mathematics in Moscow for his thesis

*On the representation of continuous functions of 3 variables by the superpositions of continuous functions of 2 variables*. The examining committee for the theis, which contained a solution to Hilbert's 13th problem, consisted of A G Vitushkin and L V Keldysh. Following this he was appointed as an assistant in the Faculty of Mechanics and Mathematics at Moscow State University. He continued to work towards his doctorate (equivalent to the habilitation) and this was awarded by the Institute of Applied Mathematics in Moscow in 1963 for the thesis

*Small denominators and stability problems in classical and celestial mechanics*. He was examined by N N Bogolyubov, V M Volosov, and G N Duboshin. Following the award, Arnold was promoted.

In 1965 Arnold became a Professor in the Faculty of Mechanics and Mathematics at Moscow State University, a position he held until 1986 when he took up the position of Principal Researcher at the Steklov Institute of Mathematics in Moscow. In addition to his Russian positions, in 1993 he was appointed Professor at the University Paris-Dauphine in France. He held this position until 2005. Arnold married Voronina Elionora Aleksandrova in 1976; they had one son.

An excellent overview of Arnold's contributions is given in the citation for the Wolf Prize awarded to him in 2001:-

Vladimir I Arnold has made significant contributions to an astounding number of different mathematical disciplines. His many research papers, books, and lectures, plus his enormous erudition and enthusiasm, have had a profound influence on an entire generation of mathematicians. Arnold's Ph.D. thesis contained a solution to Hilbert's 13th problem. His work on Hamiltonian dynamics, which includes cocreation of KAM (Kolmogorov- Arnold- Moser) theory and the discovery of "Arnold diffusion", made him world famous at an early age. Arnold's contributions to the theory of singularities complement Thom's catastrophe theory and have transformed this field. Arnold has also made innumerable and fundamental contributions to the theory of differential equations, symplectic geometry, real algebraic geometry, the calculus of variations, hydrodynamics, and magneto- hydrodynamics. He has often discovered links between problems in diverse areas.Indeed the number of different disciplines in which Arnold has worked is truly astounding. The areas are Dynamical Systems, Differential Equations, Hydrodynamics, Magnetohydrodynamics, Classical and Celestial Mechanics, Geometry, Topology, Algebraic Geometry, Symplectic Geometry, and Singularity Theory.

Let us also indicate the range of the books referred to in this citation. He published

*Problèmes ergodiques de la mécanique classique*Ⓣ (with A Avez) (1967),

*Ordinary differential equations*(Russian) (1971),

*Mathematical methods of classical mechanics*(Russian) (1974),

*Supplementary chapters to the theory of ordinary differential equations*(Russian) (1978),

*Singularity theory*(1981),

*Singularities of differentiable mappings*(Russian) (with A N Varchenko and S M Gusein-Zade) (1982),

*Catastrophe theory*(1984),

*Huygens and Barrow, Newton and Hooke*(Russian) (1989),

*Contact geometry and wave propagation*(1989),

*Singularities of caustics and wave fronts*(1990),

*The theory of singularities and its applications*(1991),

*Topological invariants of plane curves and caustics*(1994),

*Lectures on partial differential equations*(Russian) (1997),

*Topological methods in hydrodynamics*(with B A Khesin) (1998), and

*Arnold problems*(Russian) (2000).

Serge L Tabachnikov reviewing the last mentioned book gives details of Arnold's Seminar:-

Mathematical life in the Soviet Union, in particular in Moscow, was famous for its seminars: the seminars of Gelfand, Sinai, Kirillov, Manin, Novikov, to mention just a few. Most of these seminars met weekly for two hours, in late afternoon. One of the most celebrated ones is Arnold's seminar, existing for more than 30 years. For a number of very well-known mathematicians this seminar was a formative experience. Every semester, the opening meeting of the seminar was devoted to open problems. Arnold discussed about a dozen research problems with detailed comments. Many of these problems were later solved (or partially solved) by participants of the seminar. According to Arnold, the half-life of a problem is seven years. Many seminar participants are Arnold's graduate students. His philosophy is that a student should learn from his teacher that a certain problem is open; the choice of a particular research problem is then up to the student (to quote from Arnold's Preface: "To choose a problem for him is like choosing a bride for one's son").Arnold has been honoured throughout the world. He has been elected to membership of the London Mathematical Society (1976), the National Academy of Sciences of the United States (1983), the Academy of Sciences of Paris (1984), the Academy of Arts and Sciences of the United States (1987), the Royal Soceity of London (1988), Accademia Nazionale dei Lincei in Rome (1988), the Russian Academy of Sciences (1990), the American Philosophical Society (1990), the Academy of Natural Sciences of Russia (1991), and the Academia Europaea (1991). He has received many prizes, for example the Young Mathematicians Prize of the Moscow Mathematical Society (1958), the Lenin Prize (with Andrei Kolmogorov) (1965), the Crafoord Prize of the Swedish Academy of Sciences (with Louis Nirenberg) (1982), the Lobachevsky Prize of Russian Academy of Sciences (1992), the Harvey Prize, Technion, Haifa, Israel (1994), the Petr L Kapitsa Medal for Scientific Discoveries, Russian Academy of Natural Sciences (1997), Dannie Heineman Prize for Mathematical Physics (2001), Prize of the American Institute of Physics (2001), and the Wolf Prize in Mathematics (2001).

The Harvey Prize was awarded:-

... In recognition of his basic contribution to the stability theory of Dynamical Systems, his pioneering work on singularity theory and seminal contributions to analysis and geometry.The Wolf Prize was awarded:-

... for his deep and influential work in a multitude of areas of mathematics, including dynamical systems, differential equations, and singularity theory.In addition to these honours, Arnold has been awarded honorary degrees from the University P and M Curie, Paris (1979), Warwick University, Coventry (1988), Utrecht University, Netherlands (1991), University of Bologna, Italy (1991), University Complutense, Madrid (1994), and the University of Toronto, Canada (1997).

Arnold is openly critical of the education system of many countries. For example in [4] he pokes fun at education in the United States:-

Recently, even the National Academy of Sciences decided that scientific education in America should be enhanced. What they propose is to eliminate from the curriculum unnecessary scientific facts too difficult for American children and replace them by really fundamental basic knowledge, such as all objects have properties and all organisms have nature! Undoubtedly they will go far with this! Two years ago, I read in USA Today that American parents have formed a list of really necessary knowledge for children of each age category. At ten they have to know that water has two phases, and at fifteen that the moon has phases and rotates round the earth. In Russia we still teach children in primary school that water has three phases, but the new American culture will undoubtedly win in the near future. There are, however, some remarkable advantages in the free American system, where a high school student may take, say, a course on the history of jazz instead of algebra.Arnold has also been critical of the French education system. In an address on teaching of mathematics, given in the Palais de Découverte in Paris on 7 March 1997, he said:-

To the question "what is 2 + 3" a French primary school pupil replied: "3 + 2, since addition is commutative". He did not know what the sum was equal to and could not even understand what he was asked about! Another French pupil (quite rational, in my opinion) defined mathematics as follows: "There is a square, but that still has to be proved". Judging by my teaching experience in France, the university students' idea of mathematics (even of those taught mathematics at the École Normale Supérieure - I feel sorry most of all for these obviously intelligent but deformed kids) is as poor as that of this pupil. Mentally challenged zealots of "abstract mathematics" threw all the geometry (through which connection with physics and reality most often takes place in mathematics) out of teaching. Calculus textbooks by Goursat, Hermite, Picard were recently dumped by the student library of the Universitiés Paris 6 and 7 (Jussieu) as obsolete and, therefore, harmful (they were only rescued by my intervention).Arnold reached 65 years of age in 2002 and the Moscow Mathematical Journal devoted two issues to papers dedicated to celebrate the occasion. The editors of the Journal wrote an introduction and the quote we give from that forms a fitting end to this biography:-

Arnold is one of the very best mathematicians of the world, one of the founders of the Independent University of Moscow, president of its Board of Trustees and its Scientific Committee, member of our Editorial Board. The face of modern mathematics would be unrecognisable without his work in dynamical systems, classical and celestial mechanics, singularity theory, topology, real and complex algebraic geometry, symplectic and contact geometry, hydrodynamics, variation calculus, differential geometry, potential theory, mathematical physics, superposition theory, etc.Arnold was awarded the State Prize of the Russian Federation (2007), and in the following year he received the prestigious Shaw Prize in mathematical sciences. The Shaw Prize was awarded in equal shares to Vladimir Arnold and Ludwig Faddeev:-

Arnold is a rare teacher, his school is famous and numerous, he has a special gift for finding new beautiful problems to interest and involve young researchers. He is an extraordinary lecturer at all levels of mathematical education and research. Difficult modern theories become quite clear and simple in his exposition. One could hardly imagine modern mathematical education without his brilliant textbooks. The Moscow mathematical school owes a lot to his seminar.

Coming from a family of scientists in several generations, he brings together their scientific approach and deep interest to all sides of the life, his knowledge being extremely vast and his curiosity towards everything around him quite amazing.

... for their widespread and influential contributions to Mathematical Physics.The Press Release for the award of the Shaw Prize begins:-

Vladimir Arnold, together with Andrei Kolmogorov and Jurgen Möser, made fundamental contributions to the study of stability in dynamical systems, exemplified by the motion of the planets round the sun. This work laid the foundation for all subsequent developments right up to the present time. Arnold also produced extremely fruitful ideas, relating classical mechanics to questions of topology. This includes the famous "Arnold Conjecture" which has only recently seen important progress. In classical hydrodynamics the basic equations of an ideal fluid were derived by Euler in 1757 and major steps towards understanding them were taken by Helmholtz in 1858, and Kelvin in 1869. The next significant breakthrough was made by Arnold a century later and this has provided the basis for more recent work. ...

### References (show)

- D V Anosov, A A Bolibrukh, V A Vasil'ev, et al., Vladimir Igorevich Arnol'd (on the occasion of his sixtieth birthday) (Russian)
*Uspekhi Mat. Nauk***52**(5) (317) (1997), 235-255. - D V Anosov, A A Bolibrukh, V A Vasil'ev, et al., Vladimir Igorevich Arnol'd (on the occasion of his sixtieth birthday),
*Russian Math. Surveys***52**(5) (1997), 1117-1139. - M Audin and P Iglésias, Questions à V I Arnol'd,
*Gaz. Math. No.***52**(1992), 5-12. - S H Lui, An interview with Vladimir Arnol'd,
*Notices Amer. Math. Soc.***44**(4) (1997), 432-438. - C Musès, The living legend of Vladimir Arnol'd, master system theorist and philosopher of mathematics,
*Kybernetes***22**(7) (1993), 50-52. - Vladimir Igorevich Arnol'd (on the occasion of his sixtieth birthday) (Russian),
*Regul. Khaoticheskaya Din.***2**(3-4) (1997), 3-8. - Vladimir Igorevich Arnol'd (on the occasion of his fiftieth birthday) (Russian),
*Uspekhi Mat. Nauk***42**(4) (256) (1987), 197. - Vladimir Igorevich Arnol'd (French),
*C. R. Acad. Sci. Sér. Gén. Vie Sci.***1**(6) (1984), 511. - V M Zakalyukin, V I Arnol'd (on the occasion of his sixtieth birthday) (Russian),
*Tr. Mat. Inst. Steklova***221**(1998), 7-8. - V M Zakalyukin, V I Arnol'd (on the occasion of his sixtieth birthday),
*Proc. Steklov Inst. Math.*(2) (221) (1998), 1-2. - S Zdravkovska, Conversation with Vladimir Igorevich Arnol'd,
*Math. Intelligencer***9**(4) (1987), 28-32.

### Additional Resources (show)

Other pages about Vladimir Arnold:

Other websites about Vladimir Arnold:

### Honours (show)

### Cross-references (show)

Written by J J O'Connor and E F Robertson

Last Update September 2009

Last Update September 2009