Stephen Smale

from Wikipedia, the free encyclopedia
Stephen Smale (2008)

Stephen Smale (born July 15, 1930 in Flint , Michigan , USA ) is an American mathematician , who is mainly known for his work on dynamical systems and for his proof of the Poincaré conjecture for the case . He is a recipient of the Fields Medal and was a professor at the University of California, Berkeley .


Smale grew up on a farm while his father worked at General Motors. He attended a one-class school a mile from his farm for eight years. In high school, his main interest was chemistry. Smale began his studies at the University of Michigan in 1948, with rather mediocre grades initially - he was more interested in travel and political activities on campus. He joined the communist party. At first he studied physics. Because of his declining grades, he even received a warning from his faculty director Joel Henry Hildebrand and turned to mathematics. In 1952 he graduated (bachelor's degree), received his master's degree in 1953 and in 1957 he made his doctoral thesis ( Regular Curves on Riemannian Manifolds ) at the University of Michigan under Raoul Bott , whose first doctoral student he was. With this work he generalized older results from Hassler Whitney , who classified regular closed curves in the plane by their tangent rotation number ( Whitney-Graustein theorem) in 1937 . In 1956 he attended the Topology Conference in Mexico City , which was attended by the world's leading topologists.

In 1959 he caused a sensation at the University of Chicago with the proof of the possibility of turning a sphere in three-dimensional space inside out without creating "cracks" ( Sphere Eversion ). A clear procedure later showed z. B. the blind French mathematician Bernard Morin . More precisely, Smale showed that all immersions from into the regular were homotopic , including the standard embedding for embedding the inverted sphere (“turned inside out”).

With this work he won a grant from the National Science Foundation and was invited to the Institute for Advanced Study , but in 1960 he went to IMPA in Rio de Janeiro to study with Mauricio Peixoto , who specialized in dynamic systems and whom he had met in 1958 . It was here “on the beach of Rio” that he got the ideas for his horseshoe figure and for the proof of the generalized Poincaré conjecture for dimensions larger than 4. He used ideas from the Morse theory . He later generalized ideas from his proof and derived the h-cobordism theorem from them. Today, the Poincaré conjecture in d> 4 is usually proven in reverse as a consequence of this h-cobordism theorem. A roughly simultaneous proof of a version of the Poincaré conjecture in d> 4 by John Stallings led to a priority dispute.

As early as the early 1960s, he began to deal with dynamic systems such as his famous horseshoe figure, which is chaotic but " structurally stable ". With this he generalized studies of the disturbances of stable movements by the Russian mathematicians Andronov and Pontryagin and began his own qualitative, topological studies of dynamic systems. Smale summarized chaotic systems such as the horseshoe or geodetic flows on manifolds of negative curvature as "hyperbolic" systems, characterized by local compression and stretching. At first he believed that these systems are "typical" (their orbits are "close"), but this turned out to be wrong. Smale also made - unusual at the time - contacts with the Soviet mathematicians who were traditionally strong in the theory of dynamic systems, such as Vladimir Arnold , e. B. 1961 in Moscow and at the 1966 International Congress of Mathematicians in Moscow, where he received the Fields Medal .

In the 1970s he began to investigate applications of dynamic systems, e.g. B. the n-body problem , electrical oscillating circuits or the equilibrium of systems from economics. This gave rise to the question of the convergence of the approach to equilibrium points, which led Smale to algorithmic investigations that he also addressed globally.

From the 1990s onwards he tried to combine numerical analysis and the Turing machine- based calculation model of theoretical computer science (working with Lenore Blum , Mike Shub ).

When he once said that his best work was done "on the beach in Rio", the National Science Foundation took this as an opportunity in the 1960s to want to cut his funds, but later refrained from doing so. President Johnson's scientific adviser Donald Hornig took Smale's remarks in Science in 1968 as an example of a reckless attitude by mathematicians to assume that taxpayers' money could be used for math research on the beaches of Rio. He also caused a stir with his left-wing political activities, especially in the 1960s. 1960 and then again from 1964 to 1995 he was a professor in Berkeley , so the center of American student movement, and in May 1965 he was instrumental in organizing the anti- Vietnam war involved -day. In 1966 he aroused displeasure in the US administration when he publicly spoke out against the Vietnam War in Moscow, where he received the Fields Medal. At the same time, the House Committee Against Un-American Activities (HUAC) tried to summon him. During his student days, Smale was a member of the Communist Party's Labor Youth League (and later a secret member of the Communist Party).

After his retirement he was at the University of Hong Kong and is currently at the Toyota Institute for Technology in Chicago.

In 1998 he compiled a list of 18 unsolved problems for the 21st century ( Smale problems , Mathematical Intelligencer 1998 No. 2). This is inspired by Hilbert's 23 problems that Hilbert posed in 1900. Two of them come up again in Smale, on the one hand the Riemann Hypothesis , on the other hand a modern version of part of Hilbert's 16th problem. Some of Smale's problems are also among the Millennium problems (Riemann conjecture, Navier-Stokes equation , P-NP problem , Poincaré conjecture ). Many of his problems are from the theory of dynamic systems or have an algorithm background, his last problem generally asks about the limits of artificial and human intelligence.

Smale has received several awards for his work, in particular the Fields Medal and the Oswald Veblen Prize (both 1966). In 2007 he received the Wolf Prize . He was invited speaker (plenary lecture) at the International Congress of Mathematicians (ICM) in Berkeley in 1986 ( Complexity aspects of numerical analysis ), in Stockholm in 1962 ( Dynamical systems and the topological conjugacy problem for diffeomorphisms ) and in Moscow in 1966 ( Differentiable dynamical systems ). In 1968 he was admitted to the American Academy of Arts and Sciences and in 1970 to the National Academy of Sciences .

Smale has a large collection of minerals and gemstones and has also emerged as a mineral photographer.

His PhD students include Michael Shub , Robert Devaney , Morris Hirsch , John Guckenheimer , Nancy Kopell , Zbigniew Nitecki , Jacob Palis , Sheldon Newhouse .

See also


from Stephen Smale:

  • The Story of the Higher Dimensional Poincaré Conjecture (what actually happened on the beaches of Rio). Mathematical Intelligencer 1990
  • A classification of immersions of the 2-sphere. Bulletin AMS 1958 (how to turn a sphere inside out without cracks), and Transactions AMS 1958
  • Generalized Poincaré's conjecture in dimensions greater than four. Annals of Math., Vol. 74, 1961, p. 391
  • On the structure of manifolds , Amer. J. Math., Vol. 84, 1962, pp. 387-399
  • A survey of some recent results in differential topology. Bulletin AMS 1963
  • Finding a horseshoe on the beaches of Rio. Mathematical Intelligencer 1998
  • Differentiable dynamical systems. Bulletin AMS Vol. 73, 1967, pp. 747-817.
  • On the problem of reviving the ergodic hypothesis of Boltzmann and Birkhoff. In: Helleman (ed.): Nonlinear dynamics. Annales NY Academy of Sciences 1979
  • with Morris Hirsch: Differential equations, dynamical systems and linear algebra. Academic Press 1974

to him and his work:

  • Steve Batterson: The mathematician who broke the dimension barrier. American Mathematical Society 2000
  • Phillips: Turning a sphere inside out. Scientific American, May 1966
  • Hirsch: The work of Stephen Smale in differential topology. In: Hirsch (Ed.): From topology to computation: Proceedings of the Smalefest, Berkeley 1990 . Springer 1993
  • Shub: What is a Horseshoe? Notices AMS, May 2005, online here: [1]
  • Donald J. Albers, GL Alexanderson, Constance Reid : More Mathematical People - Contemporary Conversations , Academic Press 1994

Web links

Individual evidence

  1. ^ Smale Finding a Horseshoe on the Beaches of Rio
  2. ^ Batterson, Stephen Smale , AMS 2000
  3. ^ Mathematics Genealogy Project