1. Biographical Note for Vladimir Arnold
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 scientific 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);
- Ordinary Differential Equations;
- Mathematical Methods of Classical Mechanics;
- Geometrical Methods of theory of Ordinary Differential Equations;
- Catastrophes Theory;
- Singularities of Caustics and of Wave Fronts;
- Problems for Children from 5 to 15 years old;
- Huygens and Barrow, Newton and Hooke - first steps of calculus and of catastrophes theory;
- Yesterday and Long Ago;
- Contact Geometry and Wave Propagation;
- Lectures of Partial Derivatives equations;
- Pseudoperiodic Topology (with M Kontsevitch and A Zoritch);
- Mild and Soft Mathematical Models;
- Continued Fractions;
- Euler Groups and Geometric Progressions Arithmetics;
- Dynamics, Statistics and Projective Geometry of Galois Fields;
- New Obscurantism and Russia's Educational System;
- Is Mathematics Needed at High Schools?
- Geometry of Complex Numbers, Quaternions and Spins;
- Experimental Mathematics;
- What is Mathematics;
- Experimental Discoveries of Mathematical Facts;
- Science of Mathematics and Arts of Mathematicians;
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 catastrophe 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);
- Statistics of the most frequent representations of finite groups;
- Ergodic theory of the segments' permutations;
- Planetary dynamo theory (in magnetohydrodynamics);
- Statistics of the higher-dimensional continued fractions;
- Theory of singularities of the distribution of galaxies;
- Arnold's discovery of the "strange duality" of Lobachevsky triangles (leading to the mirror symmetry theory of the quantum fields physics);
- Asymptotical statistics of the Fermat-Euler geometrical progressions of residues;
- Theory of the weak asymptotics (for the distributions of the solutions of Diophantine problems);
- Description of the boundary singularities of the optimal control problems (in terms of the geometry of icosahedron);
- Topological Galois theory (of radical insolvability for the algebraic equations of degrees ≥ 5);
- Creation of the characteristic classes theories for the Braids and for the algebraic functions;
- Arnold's discovery of the topological reasons of the divergences of the permutation theory's series (including the classification of the neigbourhoods and in the orbits spaces of dynamical systems);
- Asymptotical study of irreducible representations frequencies (in the eigenspaces of the Laplacian on a symmetrical Riemannian manifold);
- Topological classification of the immersed smooth plane curves;
- Ergodic theory and projective geometry of Galois fields;
- Statistics of the convex polygons, whose vertices are integer points on the plane;
- Topological interpretation of the Maxwell's multipole formula for the spherical harmonics;
- Palindromicity theory for the periodic continued fractions of the quadric irrationalities (x2 + px + q = 0);
- Arnold's discovery of the validity of the Gauss-Kuz'min statistics for the random periodic continued fractions;
- Arnold's discovery of the violation of the Gauss-Kuz'min statistics for the periodic continued fractions of eigenvalues of the random matrices (in SL(2, Z));
- Arnold's invention of the characteristic class, involved in the quantization conditions;
- Arnold's symplectic geometry theory of the Lagrange tore in completely integrable Hamilton systems;
- The ergodic and number-theoretical "Arnold's cats" of physicists (F Dyson, I Persival, ...).
9 September 2008, Hong Kong
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.
Both of the 2008 Shaw Laureates in the Mathematical Sciences, Vladimir Arnold and Ludwig Faddeev, are part of this great tradition. Arnold's contributions are mainly in classical mechanics, emphasizing the geometrical aspects as developed over the centuries by Newton, Riemann and Poincare. Faddeev has been attracted more by the challenges of quantum theory and the algebraic formalism that is related to it.
Arnold has made many important contributions to a wide variety of problems on the Analysis/geometry frontier, but his most famous is the Kolmogorov-Arnold-Möser (KAM) theory. This theory shows the persistence of quasi-periodic orbits of dynamical systems under suitable perturbations. Originating in Newton's work on a single planetary orbit, it deals with the more general N-body problem, has been enormously influential and has important applications from the solar system to particle accelerators. Arnold also pointed out (in 1964) a subtle instability, now called Arnold diffusion, which has been much studied by mathematicians and physicists.
The general theory of Hamiltonian mechanics (in which energy is conserved) has an elegant formulation in geometrical terms - symplectic geometry. Arnold made a deep study of this subject and formulated some profound conjectures relating Hamiltonian flows to topology. These were very influential, leading to a fruitful development over the subsequent decades, culminating in the proof of some versions of the Arnold conjectures by Andreas Floer and others.
Geometrical structures frequently exhibit singularities, a subject of study by algebraists and geometers for a long time, but Arnold's interest in them centres around their appearance as caustics in wave-propagation. He emphasized the geometrical approach in this field but incorporated new results from algebra and topology.
Arnold was also a pioneer of the geometric approach to the study of the Euler equations for the dynamics of ideal fluids, an approach that has had great influence over the last thirty years. In 1966 he obtained general criteria for the stability, both linear and non-linear, of the Euler equations, while in 1974 he provided an interpretation of the helicity invariant of the Euler equations as an asymptotic Hopf invariant of linked vortex lines.
Another connection between geometry and analysis emerges from real algebraic geometry associated with the characteristics of partial differential equations. Whereas complex algebraic geometry has a rich and beautiful theory, most of this disappears over the real numbers. For plane curves one of Hilbert's famous problems asked questions about the disposition of real ovals. Arnold attacked this problem by a highly ingenious and effective topological approach giving the best results at the time.