Vladimir Igorevich Arnold

V. I. Arnold

Vladimir Igorevich Arnold, a great mathematician of our time, passed away in Paris on June 3, 2010.

The mathematical creative work of Arnold began more than fifty years ago, and influenced the entire face of modern mathematics. He was a member of all major academies of the world, and the winner of many national and international prizes. He created a brilliant mathematical school. His influence on mathematical research and education is enormous and will last for many years.

It is difficult to list even the main mathematical discoveries of Arnold. At the age of 19, Arnold, together with his teacher Andrey Kolmogorov, solved the 13th Hilbert problem. Hilbert conjectured the impossibility of presenting the solution of a generic equation of degree 7 as a composition of continuous functions depending only on two variables. Arnold and Kolmogorov disproved this conjecture.

Another work of Arnold done in his student years was his first contribution to the famous KAM (Kolmogorov–Arnold–Moser) theory. This theory, initiated by Kolmogorov, was aimed at the solution of classical problems of perturbations of completely integrable Hamiltonian systems. There was an additional difficulty in cases of celestial mechanics, for example, similar to the Solar System. (This problem was finally solved by M. Hermann and his school 40 years later.) Within just four years, Arnold developed this theory, which led to the solution of several two hundred year old problems. The KAM theory yields a new kind of stability for Hamiltonian systems with two degrees of freedom. For systems with three or more degrees of freedom, Arnold discovered a new kind of instability, called Arnold's diffusion. His conjecture about the generality of this effect remains one of the central problems in classical mechanics.

Arnold was one of the founders of singularity theory. With his students he initiated the modern period in the development of bifurcation theory.

Another Hilbert problem, the first part of the 16th, is associated with the topology of level curves of polynomials in the plane. For Arnold, it served as the starting point of a new mathematical theory, real algebraic geometry.

Another branch of mathematics started by Arnold is symplectic topology. This unexpected fusion of geometry, algebraic geometry, and classical mechanics arose from the famous Arnold Conjecture concerning the number of fixed points of an area-preserving map of the two-dimensional torus.

Arnold always had a feeling for Mathematics as a whole. For him, it was not a tower built of abstract notions, but rather a natural part of the beautiful world surrounding us. This feeling of harmony—of the beauty and unity of the world—is characteristic of all Arnold's works. He can be called Pushkin of mathematics.

Arnold had an unusual creative method he inherited from his teacher A. N. Kolmogorov. When a problem resisted, he would grab his cross country skis and run forty kilometers or more, wearing his swimming suit only. His colleagues sometimes met him dressed like that in a piercing wind. According to him, this practice would always lead him to a new idea. Another trait inherited from Kolmogorov was his habit of going for a swim whenever he encountered any open water. Not only did he swim in winter himself, but also taught many of his students to do this.

Since the early sixties, Arnold's seminar met every Tuesday at the faculty of Mathematics and Mechanics of Moscow State University. The seminar was not limited to any narrow topic and investigated a wide range of mathematics. Every semester, Arnold opened the seminar with a new list of problems. He had so many ideas that even he was not able to bring them to an end. So he formulated these ideas as problems, and offered them to seminar participants. In many cases, solving these problems gave rise to new domains and new big names in mathematics. Problems formulated by Arnold for his seminar constitute a thick book.

Arnold not only discovered the diffusion named after him, but also suggested that it cannot be fast. The verification of this conjecture resulted in creation of Nekhoroshev theory that was then developed all over the world. Arnold's problems lead to the creation of the modern theory of local bifurcations. His topological proof of Abel's theorem was the starting point of a new topological Galois theory. His problems on discriminant surfaces in functional spaces led to the invention of completely new and unexpected knot invariants. His problems influenced the creation of Newton polyhedra theory. His former students explained mirror symmetry, solved Dulac's problem, found a mixed Hodge structure in vanishing cohomologies, created the theory of fewnomials, and made a lot of other discoveries. The “School of Arnold”, which unites not only his direct students, but also a number of other mathematicians highly influenced by Arnold, is one of the most important scientific schools in Russia, and in the whole world.

Arnold was always involved in various sorts of activity, and whatever he did, he did it with talent and passion. People who participated in his walking tours in the city of Paris, which he loved so much, sometimes fainted of exhaustion. All day long, Arnold would move from block to block, form century to century, from language to language. People would say of him: a man of the Renaissance.

During the last 14 year Arnold was the President of the Moscow Mathematical Society. In 1995–1998 he was a vice-president of the International Mathematical Union.

Arnold was one of the founders of the Independent University of Moscow, and gave the first lecture courses there. He supported the creation of the Moscow Center for Continuous Mathematical Education, and headed its Board of Trustees. Both organizations became crystallization centers in Russian mathematical life and education.

Arnold was a member of Editorial Boards of many important Russian and international mathematical journals. Since foundation of the Moscow Mathematical Journal, he was a member of its Editorial Board. As the editor-in-chief of the remarkable journal “Functional Analysis and its Applications”, Arnold had his own definite opinion on so many articles, had such a clear vision of mathematics as a whole, that it looked unbelievable.

Arnold was an ardent fighter against the disastrous reforms of education in Russia aimed at its emasculation. In his speech at the parliamentary hearings at the State Duma, Arnold lashed out against the reform plan which “gives the overall impression of a training plan for slaves serving as a raw material appendage of the ruling masters.” He himself was a personification of freedom in everything: in his creative work, in work with his students, in life. He taught this feeling of freedom to everybody who had the good fortune of being influenced by him.

D. Anosov, V. Buchstaber, S. Gusein-Zade, Yu. Ilyashenko,
A. Khovanskii, S. Lando, S. Novikov, A. Sossinsky,
M. Tsfasman, V. Vassiliev

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