Vladimir Igorevich Arnold

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Vladimir Igorevich Arnold

Wladimir Igorewitsch Arnold ( Russian Влади́мир И́горевич Арно́льд , scientific transliteration Vladimir Igorevič Arnolʹd ; born  June 12, 1937 in Odessa , USSR ; † June 3, 2010 in Paris , France ) was a Russian mathematician with an international reputation.

life and work

Arnold was the son of the Russian mathematician Igor Arnold (1900-1948). He studied with Andrei Kolmogorow in Moscow from 1954, graduating in 1959 and gaining his doctorate in 1961 (Russian candidate title) and was professor at Moscow State University from 1965 to 1986, at the Steklow Institute for Mathematics in Moscow since 1986 and at the same time since 1993 University of Paris 9 .

As a (pre-graduate) student of Kolmogorov, he solved Hilbert's 13th problem in 1956 : Can every continuous function of three variables be represented by continuous functions of two variables? For four or more variables Kolmogorow had already shown that they can be reduced to two variables. Arnold proved this for the case of three variables, also with Kolmogorov's tree construction (this became his dissertation in 1961). In his lectures in Toronto in 1997 he described the basic idea of ​​his solution as almost trivial, only to show that many of his later important works had their roots in extensions of this idea. For Arnold, the correct formulation of Hilbert's problem is the question of such reducibility for algebraic functions and is still open.

After his first publication, Kolmogorow gave him the choice of his dissertation topic, and he investigated diffeomorphisms of oval curves (in the manner of the billiards later examined by Sinai ). Henri Poincaré had already investigated such for circles and ellipses, where this mapping according to Poincaré is generally (depending on the choice of the angle of rotation) ergodic (chaotic), and periodic for rational angles. To Arnold's disappointment, however, the area of ​​his diploma thesis turned out to be Kolmogorow's active field of work, and from their collaboration the KAM theorem (Kolmogorow, Arnold, Jürgen Moser ) about dynamic systems, especially celestial mechanics , arose . The qualitative theory of dynamic systems (differential equations) remained a focus of Arnold's work. He wrote well-known textbooks about it, such as his Mathematical Methods of Classical Mechanics , which are known for their informal, context and application-seeking style and avoid unnecessary abstractions. In 1961 the first discussions arose in Moscow with Stephen Smale , whose theory of structurally stable systems was just emerging.

In the 1950s, Arnold, in his own words, also investigated applications that later became known in chaos theory , for example in a work on heart rhythms, inspired by the mathematician Israel Gelfand , who was interested in applications of mathematics in biology. In 1964 he discovered the Arnold Diffusion named after him . According to Arnold, this is his most important contribution to the “KAM theory” and describes the general cause of instability in (deterministic) dynamic systems with several degrees of freedom.

From 1963 Arnold also dealt with the much more complicated dynamic systems of hydrodynamics, also one of Kolmogorov's areas of work. Arnold formulated his investigation of the Navier-Stokes and Euler equations as “differential geometry of infinitely dimensional Lie groups”, the curvature of which he determined. According to Arnold, a by-product was the proof that weather forecasts for more than two weeks are impossible. At the same time he tried to prove the existence of what was later called a " strange attractor ". The examinations at that time were, however, very hindered by the lack of sufficient computer capacity.

In the mid-1960s he began to be interested in singularity theory, which later became one of his main areas of work. According to his own information, this work also had its roots in the “correct” formulation of a Hilbert problem in algebraic geometry (the 16th Hilbert problem , where he made significant progress in 1972), this time about obstructions to the resolution of singularities of equations n -th degree to investigate. The topology of the plane minus singularities with the braid group (English: braid group) writable. Arnold examined her cohomology ring.

In various essays he spoke out against the Bourbaki tradition of teaching, especially in France, where he taught from the 1990s. He also complained about the neglect of Russian works in “Western” literature, which often led to “new discoveries” and incomplete or incorrect attributions, partly because of the language barrier, but partly also from ignorance, according to Arnold. Arnold was very interested in the history of mathematics. In an interview, he said he learned much of his knowledge by studying Felix Klein's history of mathematics in the 19th century . The “Russian method” of literature research begins in the collected works of Felix Klein (Arnold adds Poincaré) and in the early 20th century volumes of the “Encyclopedia of Mathematical Sciences” published by Felix Klein and others. In order to put the contributions of the Russian mathematicians in the right light, their leading representatives, among them Arnold, started to publish a new, modern encyclopedia (a series of overview articles and books, as they were earlier in Russia especially for the “Russian Mathematical Surveys “were written).

Arnold is also known for various problems he posed, such as: B. on the existence of fixed points in symplectic mappings of compact symplectic manifolds (as they occur in classical mechanics) - partially solved by Andreas Floer .

He received the Moscow Mathematical Society Prize in 1958 and the Lenin Prize in 1965 together with Andrei Kolmogorov . In 1962 he gave a lecture at the International Congress of Mathematicians in Stockholm ( Perturbation theory and the problem of stability for planetary systems ), 1966 Invited Speaker at the ICM in Moscow ( The problem of stability and the ergodic properties of classical dynamic systems ) and in 1958 at the in Edinburgh ( Some questions about approximation and representation of functions (Russian)). In 1974 he gave a plenary lecture at the International Congress of Mathematicians (ICM) in Vancouver (Critical Points of Smooth Functions) and in 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 and Louis Nirenberg received from the Courant Institute of Mathematical Sciences at New York University the Crafoord Prize, endowed with 400,000 Swedish kronor, “for exceptional achievements in the theory of nonlinear partial differential equations”, awarded by the Swedish Academy of Sciences. Research in this area in Sweden was funded with a further 400,000 Swedish kronor . In 1983 he was elected to the National Academy of Sciences , 1987 to the American Academy of Arts and Sciences, and 1990 to the American Philosophical Society . Since 1984 he was a member ("associé étranger") of the Académie des sciences .

In 1992 he received the Lobachevsky Medal from the Kazan State University, the Harvey Prize of the Technion Institute in Haifa in 1994 , the Dannie Heineman Prize in 2001 and the 2001 Wolf Prize for Mathematics. In 2008 he was awarded the Shaw Prize (together with Faddejew ). In 2000 the asteroid (10031) Vladarnolda was named after him.

In 1991 he was one of the founders of the Independent University of Moscow and chaired its board for a long time.

His doctoral students include Alexander Givental , Sabir Gussein-Sade , Askold Chowanski , Boris Chessin , Wiktor Wassiljew , Alexander Wartschenko .

Works

  • Collected Works , Vol. 1 (Representations of functions, celestial mechanics, KAM-Theory 1957–1965), Springer 2009, Vol. 2 (Hydrodynamics, Bifurcation theory and algebraic geometry 1965–1972), Springer 2014
  • Yesterday and long ago , Springer 2007 (memories)
  • Lectures on partial differential equations , Springer 2004, ISBN 3-540-43578-6
  • Ordinary differential equations , VEB Dt. Verl. D. Knowledge 1979, 2nd edition, 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 (English 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 (= university books for mathematics , vol. 90). ISBN 3-7643-1879-1
  • Arnold's problems , 2nd edition, Springer 2004 (a list of problems from 2002 is on his homepage)
  • 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, Birkhäuser 1985, 1988
  • Topological invariants of plane curves and caustics , American Mathematical Society 1994
  • Huygens and Barrow, Newton and Hooke , Birkhäuser 1990
  • From Hilberts Superposition problem to Dynamical systems , American Mathematical Monthly, August / September 2006 (overview of his mathematical career, lecture Toronto 1997, online here , also in Bolibruch, Osipov, Sinai (editor) Mathematical Events of the Twentieth Century , Springer 2006, P. 19)
  • Arnold was editor and co-author of the series Encyclopedia of mathematical sciences at Springer Verlag (including the series Dynamische Systeme ).
  • with Valeri Wassiljewitsch Koslow , Anatoli Isserowitsch Neischtadt : Dynamical Systems III: Mathematical Aspects of Classical and Celestial Mechanics, Encyclopedia of Mathematical Sciences, Springer 2006 (first 1987)
  • 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
  • Real algebraic geometry , Unitext, Springer Verlag 2013
  • Mathematical understanding of nature , American Mathematical Society 2014

literature

  • Bierstone Ed. The Arnoldfest , American Mathematical Society 1999 (Conference on Arnold's 60th birthday in Toronto 1997)
  • Smilka Zdravkovska: Conversation with Vladimir Igorevich Arnold , Mathematical Intelligencer, Vol. 9, 1987, No. 4, p. 28 (interview)
  • Leonid Polterovich , Inna Scherbak VI Arnold (1937-2010) , Annual Report DMV, Volume 113, 2011, Issue 4, 185-219

Web links

Commons : Wladimir Arnold  - Collection of pictures, videos and audio files

Individual evidence

  1. a b c From Hilbert's Superposition problem to Dynamical systems , American Mathematical Monthly, August / September 2006
  2. Zhihong Xia from the Georgia Institute of Technology was able to prove in 1994 that even a three-body system can behave accordingly
  3. ^ Lobachevsky Medal
  4. Minor Planet Circ. 39653