I was born on 12 June 1937 in Odessa and studied at the Moscow University from 1954 to 1959.

I was a Candidate of physical-mathematical sciences, for the Thesis, resolving the Hilbert's 13-th problem, Applied Mathematics (Keldysh) Institute in 1961 and attained the physical-mathematical sciences doctor in 1963, for the Thesis on the stability of the Hamiltonian systems, at the same Institute. The graduated studies were supervised by A.N. Kolmogorov.

Since 1965 I have worked as a professor at the chair of differential equations of the mathematical-mechanical faculty of the Moscow State University and since 1986 also at the Steklov Mathematical Institute, Moscow. I was elected a member of the Russian Academy of Sciences in 1990.

I served as the vice-president of the International Union of Mathematicians (1999-2003), being also the President of the Moscow Mathematical Society.

The list of scientifical journals, on whose Editorial Boards I participated, includes, for instance:

Doklady RAN, Izvestia RAN, Russian Mathematical Surveys, Functional Analysis and its Applications, Functional Analysis and Other Mathematics, Proceedings of Petrovski Seminar, Inventiones Mathematicae, Physica D-Nonlinear Phenomena, Quantum, Bulletin des Sciences Mathematiques, Selecta, Journal of Geometry and Physics, Topological Methods in Nonlinear Analysis.

Being Moscow University's professor for 30 years, I worked also as the professor at the University Paris-Dauphine from 1993 to 2005 (remaining now its honorary professor).

I have published several dozens of books. Examples are:

Ergodic Problems of Classical Mechanics (with A. Avez);

The above list contains 10 university textbooks.

Most known mathematical papers of mine deal with Hamiltonian systems (including the discovery of the “Arnold diffusion” and the creation of the symplectic topology).

My articles on the “quantum catastrophies theory” include the studies of the bifurcations of the caustics, based on my discovery of unexpected interrelations between the simple critical points of functions and simple Lie algebras (and also to Coxeter reflections' groups).

The real algebraic geometry of plane curves was related by me to the four-dimensional topology (and to quantum fields theory) – this discovery generated many studies by many mathematicians of the algebraic geometry part of the 16th problem of Hilbert.

My recent works on arithmetical turbulence provide unexpected statistical properties of the Young diagrams of the cycles of random permutations of N→∞ points.

Many domains of modern mathematics, generated by my articles, include, for instance:

Lagrange and Legendre cobordism theories (in symplectic and contact topologies);

geometryof icosahedron);

To understand the natural interrelations between such different subjects as mentioned above, I recommend reading my articles (approximately 700) explaining these interrelations.

9 September 2008, Hong Kong