Stephen Smale

Stephen Smale ( born July 15, 1930 in Flint, Michigan, United States) is an American mathematician who is mainly known for the case of his work on dynamical systems and for his proof of the Poincaré conjecture.

Life

Smale began his studies at the University of Michigan in 1948, initially with rather mediocre grades - he was interested in more travel and political activities on campus. He joined the Communist Party. Due to his failing grades, he even received a reminder of his faculty conductor Hildebrandt. In 1952, he graduated in 1957 he made his doctoral thesis (Regular Curves on Riemannian Manifolds ) under Raoul Bott, whose first graduate student he was. With this work he generalized earlier results of Hassler Whitney, the classified 1937 regular closed curves in the plane by their turns. In 1956 he attended the topology conference in Mexico City, which was attended by the world's leading topologists.

In 1959, he attended the University of Chicago with the proof of the possibility of a sphere in three-dimensional space from the inside to put on the outside, without creating " cracks " ( Sphere eversion ), stir. A descriptive approach was later shown, for example, the blind French mathematician Bernard Morin. More Smale showed that all continuous embeddings ( immersions ) of were regularly homotopic in, so even the standard embedding to embed the inverted sphere ( " slipped from the outside to the inside" ).

With these works, he won a grant from the National Science Foundation and was invited to the Institute for Advanced Study, but he went in 1960 to Rio de Janeiro to the IMPA to Mauricio Peixoto, who specialized in dynamic systems, and whom he had met in 1958. Here " on the beach in Rio " his ideas for his horseshoe illustration and for the proof of the generalized Poincaré conjecture for dimensions were larger than 4 He used ideas from Morse theory. Ideas of his proof he later generalized and led them from the h- cobordism theorem ago. Today mostly the Poincaré conjecture is proved in d> 4 as a result of this h- cobordism theorem vice versa. An approximately simultaneous proof of a version of the Poincaré conjecture in d> 4 by John Stallings resulted in a priority dispute.

Back in the early 1960s he began to deal with dynamic systems, such as his famous horseshoe map that is chaotic, but " structurally stable". He generalized studies of disturbances stable movements of Russian mathematicians Andronov and Pontryagin and started his own qualitative, topological studies of dynamic systems. Smale grabbed chaotic systems like the horseshoe or geodesic flows on manifolds of negative curvature together as " hyperbolic " systems, characterized by local compression and extension. At first he believed that these systems " typical" ( their orbits " tight" are ), but this turned out to be wrong. Smale also tied - then uncommon - contacts with the traditional in the theory of dynamical systems strong Soviet mathematicians, such as Vladimir Arnold, for example, in Moscow in 1961 and at the International Congress of Mathematicians in 1966 in Moscow, where he received the Fields Medal.

In the 1970s he began to examine applications of dynamic systems, such as the n- body problem, electrical resonant circuits or the equilibria of systems from economics. This led to the question of the convergence of approach to equilibrium points, which Smale led to algorithmic investigations that he was concerned also globally.

From the 1990s, he tried to Numerical Analysis and based on Turing machines computation model of theoretical computer science to unite (working with Lenore Blum, Mike Shub ).

As he once commented to the effect that his best works were " on the beach in Rio ," the National Science Foundation in the 1960s, took this as a reason to want to cut him money, but they took back later refrained. The scientific advisor to President Johnson Donald Hornig took Smale 's remarks in 1968 in Science as an example of a light-hearted attitude of mathematicians to suppose that they could use the taxpayers' money for mathematical research on the beaches of Rio. Even with its left-wing political activities, especially in the 1960s, he created a stir. 1960 and then again from 1964 to 1995 he was a professor at Berkeley, ie in the center of the American student movement, and in May 1965 he was instrumental in the organization of the anti- Vietnam War days. In 1966, he caught at the U.S. Administration indignation when he publicly expressed itself in Moscow, where he took the Fields Medal in reception against the Vietnam War. At the same time, the House Committee tried to summon him against House Un-American machinations ( HUAC ). Smale was a member of the youth organization ( Labor Youth League) of the Communist Party (and later secretly joined the Communist Party ) in his student days.

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

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

Smale has won several awards for his work, especially with the Fields Medal and the Oswald Veblen - Prize (both in 1966). In 2007 he was awarded the Wolf Prize. He was invited speaker ( plenary lecture ) on the ICM 1986 in Berkeley ( Complexity aspects of numerical analysis ), in Stockholm 1962 ( Dynamical systems and the topological conjugacy problem- for diffeomorphisms ) and in Moscow 1966 ( Differentiable dynamical systems ).

Smale has a large mineral and gem collection and has also appeared as a photographer of minerals.

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