Vladimir Arnold

Vladimir Arnold Igorewitsch (Russian Владимир Игоревич Арнольд, scientific transliteration Vladimir Arnol'd Igorevič; born June 12, 1937 in Odessa, USSR, † June 3, 2010 in Paris, France) was a Russian mathematician of international reputation.

Life and work

Arnold was the son of Russian mathematician Igor Vladimirovich Arnold ( 1900-1948 ). He studied from 1954 at Andrei Nikolaevich Kolmogorov in Moscow with the completion in 1959 and his doctorate in 1961 (Russian candidate titles) and was from 1965 to 1986 professor at the Moscow State University, since 1986 at the Steklov Institute of Mathematics in Moscow and at the same time since 1993 the University of Paris 9

As ( undergraduate ) Student Kolmogorov he dissolved in 1956 the 13th Hilbert problem: Is every continuous function of three variables by continuous functions of two variables to represent it? For four or more variable Kolmogorov had already shown the reducibility on two variables. Arnold proved this for the case of three variables, also with Kolmogorov tree construction ( it was founded in 1961 his doctoral dissertation ). In his lectures in Toronto in 1997, he referred to the basic idea of its solution as an almost trivial to show, then, that many important subsequent work by him had its roots in extensions of this idea. The correct formulation of Hilbert's problem is open for Arnold the question of reducibility for such algebraic functions and still.

After its initial release him Kolmogorov set free his choice of dissertation topic, and he examined diffeomorphisms oval curves ( in the manner of the later studied by Sinai billiard ). Henri Poincaré had already been studied in such a circle and ellipse, where this figure ( chaotic ) is by Poincaré in general ( depending on the choice of the rotation angle ) ergodic periodically at rational angles. At Arnold's disappointment, the area of his thesis presented but as an active work area Kolmogorov out, and from their collaboration, the KAM theorem ( Kolmogorov, Arnold, Jürgen Moser ) was born about dynamic systems, especially the celestial mechanics. The qualitative theory of dynamical systems ( differential equations) remained continue to be a focus of Arnold's work. He wrote about well-known textbooks, his Mathematical methods of classical mechanics, which are known by their informal contexts and applications seeking style and avoid unnecessary abstractions. 1961, came to Moscow to initial discussions with Stephen Smale, whose theory of structurally stable systems had just emerged.

In the 1950s, Arnold investigated his own words, applications that were later known in chaos theory, in a paper on heart rhythms, inspired by the mathematician Israel Gelfand, who was interested in applications of mathematics in biology. In 1964 he discovered his namesake Arnold diffusion. This is according to Arnold his most important contribution to the " KAM theory " and describes the general cause of instability in ( deterministic ) dynamical systems with several degrees of freedom.

Arnold dealt in 1963 with the much more complicated dynamic systems of hydrodynamics, also a work area Kolmogorov. Arnold formulated his study of the Navier -Stokes and Euler equations as " differential geometry infinite dimensional Lie groups ", which he defined curvature. A byproduct was after Arnold evidence that weather forecasts are impossible to last more than two weeks. At the same time he tried the existence of a - later called - " strange attractors " to prove. The former studies were very hampered by the lack of sufficient computer capacity.

The mid-1960s he began to be interested in singularity theory, later one of his main areas of work. According to the company, this work also had time to obstructions against the resolution of singularities of equations to examine their roots in the "correct" formulation of a Hilbert problem in algebraic geometry nth degree. The topology of the plane minus singularities is the braid group (English: braid group) writable. Arnold examined its cohomology ring.

In various essays he has against the Bourbaki tradition of teaching specifically, pronounced in France, where he taught from the 1990s. He also lamented the neglect of Russian works in the "Western" literature, which often led to " new discoveries " and incomplete or incorrect attributions, partly because of the language barrier, but partly also by Arnold from ignorance. Arnold was very interested in the history of mathematics. In an interview he said he had learned through the study of Felix Klein's history of mathematics in the 19th century a large part of his knowledge. The "Russian method" of literature begins because even in the collected works of Felix Klein ( Arnold supplemented Poincaré ) and published in the early 20th century volumes, edited by Felix Klein and other " Encyclopedia of mathematical sciences " to. To specifically to highlight the contributions of the Russian mathematician in the right light, have their leaders, among them Arnold, with the publication of a new, modern encyclopedia ( a series of review articles and books, as formerly in Russia, especially for the "Russian Mathematical Surveys were " written ) started.

Arnold is also known for the various demands placed by him problems, eg about the existence of fixed points in compact symplectic manifolds symplectic pictures ( such as occur in classical mechanics ) - partially solved by Andreas Floer.

He was awarded the prize of the Moscow Mathematical Society, and in 1965, together with Andrei Kolmogorov the Lenin Prize, among others, in 1958. In 1962 he was invited speaker at the International Congress of Mathematicians in Stockholm ( perturbation theory and the problem- of stability for planetary systems) and in 1966 in Moscow ( The problem of the stability and ergodic properties of classical dynamical systems). In 1974 he gave a plenary lecture at the International Congress of Mathematicians (ICM ) in Vancouver (Critical Points of Smooth Functions) and 1983 a plenary lecture at the ICM in Warsaw ( Singularities of Ray Systems ). In 1992 he gave a plenary lecture at the first European Congress of Mathematicians in Paris ( Vasiliev 's Theory of Discriminants and Knots).

In 1982 he was awarded, together with Louis Nirenberg of Courant Institute of Mathematical Sciences at New York University, endowed with 400,000 Swedish crowns Crafoord Prize " for outstanding achievements in the theory of nonlinear partial differential equations ", awarded by the Swedish Academy of Sciences. With another 400,000 skr the research has been sponsored in this area in Sweden.

In 2001 he received the Dannie Heineman Prize, also the 2001 Wolf Prize in Mathematics. In 2008 he was awarded the Shaw Prize (together with Faddejew ).

In 1991 he was one of the founders of the Moscow Independent University and was long before its Executive Board.

Among his doctoral students include Alexander Givental, Sabir Gussein -Zadeh, Askold Chowanski, Boris chessin, Viktor Vasilyev, Alexander Wartschenko.

Works

  • Collected Works, Vol 1 ( Representations of functions, celestial mechanics, KAM Theory 1957-1965 ), Springer 2009
  • Yesterday and long ago, Springer 2007 ( memories )
  • Lectures on partial differential equations, Springer 2004, ISBN 3-540-43578-6
  • Ordinary Differential Equations, 1980, 2nd ed. , Berlin, Springer 2001, ISBN 3-540-66890- X (English as early as 1973, MIT press )
  • Mathematical methods of classical mechanics, Birkhäuser, 1988, ISBN 3-7643-1878-3 ( engl. 2nd edition, 1989, Springer, Graduate texts in mathematics )
  • With Avez Ergodic problems of classical mechanics, New York, Benjamin 1968
  • Topological methods in hydrodynamics, Springer 1998
  • Geometric methods in the theory of ordinary differential equations, ISBN 3-7643-1879-1
  • Arnold's problems, 2nd edition, Springer 2004 ( a problem list from 2002 is on its home page )
  • Mathematics - frontiers and perspectives, American Mathematical Society 2000
  • Catastrophe theory, 3rd edition, Springer 1993
  • Bifurcation theory and catastrophe theory, 2nd edition Springer 1999
  • Singularities of caustics and wave fronts, Kluwer 1990
  • With Varchenko, Gusein - Zade: Singularities of Differentiable Maps, 2 volumes, Birkhauser, 1985, 1988
  • Huygens and Barrow, Newton and Hooke, Birkhauser 1990
  • From Hilbert's Superposition Problem to Dynamical systems, American Mathematical Monthly, August / September 2006 ( overview of his mathematical career, Lecture Toronto 1997 online here, in Bolibruch, Osipov, Sinai (Editor) Mathematical Events of the Twentieth Century, Springer, 2006 p 19)
  • Arnold was the editor and co-author of the series " Encyclopedia of mathematical sciences" by Springer Verlag (among others in the series " Dynamical Systems ").
  • Dynamical systems, in Jean -Paul Pier ( Editor) Development of mathematics 1950-2000, Birkhäuser 2000
  • Singularity theory, in Jean -Paul Pier ( Editor) Development of mathematics 1950-2000, Birkhäuser 2000
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