Paul A. Smith

from Wikipedia, the free encyclopedia

Paul Althaus Smith (born May 18, 1900 , † June 13, 1980 ) was an American mathematician who dealt with geometric topology .

Smith studied at the University of Kansas and received his doctorate in 1926 under Solomon Lefschetz at Princeton University (where he had gone with Lefschetz from Kansas) ( Approximation of curves and surfaces by algebraic curves and surfaces ). He was a professor at Columbia University . In 1947 he was elected to the National Academy of Sciences .

He is known for two guesses. The Hilbert-Smith conjecture says that topological groups that are locally compact and have a faithful group action as a transformation group on a manifold are Lie groups . The assumption is open. It is also named after David Hilbert , as it is sometimes seen as the correct formulation of Hilbert's 5th problem .

The Smith conjecture, however, has been proven. It says that fixed points of diffeomorphisms of finite order of the 3-sphere cannot be nontrivial nodes. It was proved for straight order by Friedhelm Waldhausen in 1969 and the general case around 1978 by a number of topologists such as William Thurston , William Meeks , Shing-Tung Yau , Hyman Bass , Cameron Gordon , Peter Shalen . In higher dimensions (four and more) the guess is wrong. It is also wrong if one considers more general continuous transformations as diffeomorphisms ( Deane Montgomery , Leo Zippin 1954).

The study of the cohomology of groups of finite order homeomorphisms of manifolds is called Smith's theory. Smith himself justified the theory with research in the late 1930s. He calculated the cohomology of fixed point sets of involutions on spheres and projective spaces.

He was responsible for Samuel Eilenberg's appointment to Columbia University. He was of a calm and reserved nature and was completely absorbed in his work as a topologist - the saying was repeated after him (which shows his roots in the topology): Whenever i see a derivative it gives me nausea .

Individual evidence

  1. ^ National Academy of Sciences (US): Members' Directory. The Academy, Washington DC 1992, p. 230.
  2. ^ Mathematics Genealogy Project . Published in Annals of Mathematics . Series 2, Volume 27, No. 3, 1926, pp. 224-244, doi : 10.2307 / 1967843 .
  3. ^ Smith: Periodic and nearly periodic transformations. In: Raymond L. Wilder , William L. Ayres (Eds.): Lectures in Topology. The University of Michigan Conference of 1940. University of Michigan Press et al. Ann Arbor MI et al. 1941, pp. 159-190.
  4. ^ Smith: Transformations of finite period. II. In: Annals of Mathematics. Series 2, Volume 40, No. 3, 1939, pp. 690-711, doi : 10.2307 / 1968950 .
  5. ↑ Friedhelm Waldhausen : About the Involutions of the 3-Sphere. In: Topology. Volume 8, No. 1, 1969, pp. 81-91, doi : 10.1016 / 0040-9383 (69) 90033-0 .
  6. ↑ John W. Morgan , Hyman Bass (Ed.): The Smith Conjecture (= Pure and Applied Mathematics. 112). Academic Press, Orlando FL et al. 1984, ISBN 0-12-506980-4 .
  7. ^ Smith Conjecture, Mathworld
  8. ^ Charles H. Giffen: The generalized Smith conjecture. In: American Journal of Mathematics . Volume 88, No. 1, 1966, pp. 187-198, doi : 10.2307 / 2373054 .
  9. ^ Smith: Transformations of finite period. In: Annals of Mathematics. Series 2, Volume 39, No. 1, 1938, pp. 127-164, doi : 10.2307 / 1968718 .
  10. ^ Smith: New results and old problems in finite transformation groups. In: Bulletin of the American Mathematical Society. Volume 66, No. 6, 1960, pp. 401-415, doi : 10.1090 / S0002-9904-1960-10491-0 .
  11. ↑ Whenever I see a derivative I feel sick , Steven G. Krantz : Mathematical Apocrypha Redux. More Stories and Anecdotes of Mathematicians and the Mathematical. Mathematical Association of America, Washington DC 2005, ISBN 0-88385-554-2 , p. 76.
personal data
SURNAME Smith, Paul A.
ALTERNATIVE NAMES Smith, Paul Althaus (full name)
BRIEF DESCRIPTION American mathematician
DATE OF BIRTH May 18, 1900
DATE OF DEATH June 13, 1980