David Gabai

David Gabai (* July 7, 1954 in Philadelphia, Pennsylvania) is an American mathematician who deals with low-dimensional differential geometry and geometric topology.

Gabai studied at the Massachusetts Institute of Technology (MIT) and Princeton University ( master's degree, 1977). In 1980 he completed his doctorate there with William Thurston on foliations on 3-manifolds. After that, he was at Harvard University, the University of Pennsylvania and from 1986 at Caltech, where he became a professor. In 2001 he was a professor at Princeton. 1982/1983 and 1989 he worked at the Institute for Advanced Study.

Gabai turned that began in his dissertation study of foliations on 3-manifolds in the 1980s to study some previously open problems of the topology of 3-manifolds in (for example in the treatment of "Property R" in the knot theory, when stretching surgery on a knot in a 3- sphere results in a product of a 2-sphere and a circle homeomorphic 3-manifold ). His works were also in the proof of the "Property P" presumption of knot theory fundamentally, which was announced in 2004.

From the early 1990s, he was also involved in three-dimensional hyperbolic manifolds (whose importance for the topology of 3-dimensional manifolds, Thurston had worked out ). He proved the converse of a theorem of Thurston: irreducible 3-manifolds that are homotopic to a hyperbolic manifold (from the same homotopy type), also have a hyperbolic structure. Furthermore, he proved the Smale conjecture for hyperbolic 3-manifolds M on the homotopy type of the space of diffeomorphic mappings of M onto itself

Ian Agol, Danny Calegari and David Gabai received the 2009 Clay Research Award for the proof of Marden tameness Conjecture ( Zahm PRIME presumption of Marden ), a conjecture of Albert Marden. It states that a hyperbolic 3-manifold with finitely generated fundamental group is homeomorphic to the interior of a compact, possibly bounded 3-manifold (the manifold is then tame ). An equivalent formulation is that the ends have a local product structure. The conjecture was proved in 2004 by Agol and Calegari and Gabai whatever. For finite hyperbolic 3-manifolds it has been proved by Marden and partial results for some infinite hyperbolic manifolds were also already known. From it follows, among other things ( by the work of William Thurston and Canary ) even a guess by Lars Ahlfors on the invariant limit quantities small shear groups (namely those either measure have zero or full measure, in the latter case the effect of the group is ergodic throughout the room ).

In 2004 he was awarded the Oswald Veblen Prize -. In 1990 he was invited speaker at the ICM in Kyoto ( Foliations and 3- manifolds ) and 2010 in Hyderabad ( Hyperbolic 3- manifolds in the 2000 's). He is a Fellow of the American Mathematical Society.

Writings

  • Foliations and the topology of 3 - manifolds; I: J. Differential Geom 18 (1983 ), no 3, 445-503; II: J. Differential Geom 26 (1987 ), no 3, 461-478; III: J. Differential Geom 26 (1987 ), no 3, 479-536.
  • With U. Oertel: Essential laminations in 3- manifolds, Ann. of Math ( 2) 130 (1989 ), no 1, 41-73.
  • Convergence groups are Fuchsian groups, Ann. of Math ( 2) 136 (1992 ), no 3, 447-510.
  • With GR Meyerhoff, N. Thurston: Homotopy hyperbolic 3- manifolds are hyperbolic, Ann. of Math ( 2) 157 (2003 ), no 2, 335-431.
  • With D. Calegari: shrinkwrapping and the taming of hyperbolic 3- manifolds, J. Amer. Math Soc. 19 (2006 ), no 2, 385-446.
  • With GR Meyerhoff, P. Milley: Minimum volume cusped hyperbolic three- manifolds, J. Amer. Math Soc. 22 (2009), no.4, 1157-1215.
Tameness theorem Oswald Veblen Prize in Geometry R. H. Bing Integrated Authority File
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