V.I. Arnold

Vladimir I. Arnold

This issue of the Journal is dedicated to the memory of Vladimir Igorevich Arnold, mathematicus incomparabilis of the modern times. To our great sorrow he passed away on 3 June 2010, leaving a feeling of painful emptiness in all those who had touched the huge and brilliant universe he carried inside and a glimpse of which was shining in the depth of his eyes.

However, his most precious heritage stays with us: his articles, his books, his lectures and memories of the people who had the luck of knowing him in person.

He always worked hard—not for himself, but for the science and mathematical community.

He was so enthusiastic giving seminars on his new achievements that nobody could stay away.

He possessed the gift to express his thoughts and theorems in a perfect form, geometrically clear, starting with the simplest illustrations and then going up to hardly visible peaks.

Like a child exploring the world, he posed more questions than anybody, even the whole science, can hardly answer over decades. However, these questions were stated by a genius with a deep intuition of the nature and for future of the science. He always verified all new results against works of classics, his favourites being Huygens, Newton, Poincaré. He used to comment that this or that was already known in certain sense to the classics—and this was his regular proof of the necessity and importance of a new result.

This collection of papers by Arnold's former students and others influenced by his genius is a far from complete reflection of the extraordinary breadth of his mathematical interests. This breadth was not a disjoint union of particular subjects—Arnold's interests even in seemingly very distant areas were deeply inter-related and based on similar general principles.

Over many years, Arnold was one of the most cited scientists. His works in dynamical systems, classical and celestial mechanics, mathematical physics, topology, real and complex algebraic geometry, singularity theory and in many other fields he had touched were always pioneering and rich of new ideas, conjectures and open problems.

Such pattern started from his student papers which were written under the supervision of Andrei Nikolaevich Kolmogorov in the mid-50ies and completed the solution of Hilbert's thirteenth problem on representation of an algebraic function as a superposition of functions in fewer variables. Later the area was related by Arnold to the topology of discriminants, which are on the one hand the main objects in singularity theory while on the other they have proven to be a very fruitful field for topology itself, and the foundation for the connection was laid by Arnold's early work on the cohomology of braid groups.

The next seminal step was the work on what is now known as Kolmogorov–Arnold–Mozer theory. The area emerged from the fundamental problem of mechanics—going back to Poincaré—on perturbations of quasi-periodic motions in dynamical systems and is based on the theory of small denominators in systems close to resonance. Today KAM theory has become a major field in theory of dynamical systems, plasma and nuclear physics, celestial mechanics. The mechanism of Arnold's diffusion in Hamiltonian systems close to integrable is now understood as one of the general laws of behaviour in physical systems.

Arnold brought a range of new concepts into hydrodynamics, showing deep relations between equations of fluid motion and geodesics on infinite-dimensional groups of diffeomorphisms. He found principal phenomena of A-stability in planar flows, defined famous ABC-flows in the theory of magnetic dynamo, introduced topological invariants of magnetic fields.

Creation of real algebraic geometry was initiated by Arnold's proof of Gudkov's conjecture on describing all possible placement types for ovals of projective algebraic curves of degree 6. Arnold's approach related the conjecture to 4-dimen­sional complex topology.

Arnold is also a creator of modern theory of bifurcations in dynamical systems, which was logically related to KAM theory and various other geometrical problems in differential equations on the one hand and to non-linear functional analysis on the other. This was a straight bridge to a broad range of problems involving infinite-dimensional group action on functional spaces and metamorphoses in processes depending on parameters.

In the late 60ies Arnold's interests were concentrated on rapidly growing singularity theory of which he was one of the creators. He introduced many basic concepts there, such as notions of simple singularity and versal deformation, and found striking relations of crystallographic symmetry groups to simple singularities of functions, of caustics and wave fronts in optics, to oscillating integrals etc.

Arnold's conjecture which generalises the geometric Poincaré–Birkhoff theorem on a fixed point of an annulus area preserving mapping was a challenging first step in creation of symplectic topology. The Arnold–Maslov characteristic class of a Lagrangian submanifold opened a new field of interconnections between topology, physics and variational calculus.

Mathematics students all round the world have read his textbooks on ordinary differential equations and mechanics, many became his adepts for the entire life.

We know that, for many decades to come, future achievements in mathematics and beyond will be very much influenced by Arnold's heritage.

V. Goryunov and V. Zakalyukin

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Moscow Mathematical Journal
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Independent University of Moscow

Online ISSN 1609-4514
© 2011, Independent University of Moscow

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