David Gabai

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

Gabai studied at the Massachusetts Institute of Technology (MIT) and Princeton University (Masters Degree 1977). In 1980 he did his doctorate there under William Thurston on scrolling on 3-manifolds. He then went to Harvard University , the University of Pennsylvania and, from 1986, Caltech , where he became a professor. 1986 received a research grant ( Sloan Research Fellowship ) from the Alfred P. Sloan Foundation . From 2001 he was a professor at Princeton. 1982/1983 and 1989 he was at the Institute for Advanced Study .

Gabai applied the investigation of scrolls on 3-manifolds , which he had begun in his dissertation, in the 1980s to study some problems of the topology of 3-manifolds that had been open until then (for example in the treatment of the " property R " in knot theory , when expansion Surgery on a node in a 3-sphere yields a 3-manifold homeomorphic to the product of a 2-sphere and a circle). His work was also fundamental in the proof of the " Property P " conjecture of knot theory, which was announced in 2004.

From the beginning of the 1990s he also dealt with hyperbolic three-dimensional manifolds (whose importance for the topology of 3-dimensional manifolds Thurston had worked out). He proved with Meyerhoff and N. Thurston: irreducible 3-manifolds, which are homotopy-equivalent to a hyperbolic manifold (of the same homotopy type), also have a hyperbolic structure. Furthermore, he proved the Smale conjecture for hyperbolic 3-manifolds M over the homotopy type of the space of diffeomorphic mappings of M onto itself.

Independently of Andrew Casson and Douglas Jungreis, he set the keystone for the proof of the Seifert fiber space conjecture , building on the work of Geoffrey Mess , Pekka Tukia and others.

Ian Agol , Danny Calegari and David Gabai received the 2009 Clay Research Award for the proof of Marden tameness Conjecture ( Zahm Deities presumption of Marden ), a conjecture of Albert Marden . It says that a hyperbolic 3-manifold with a 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 proven in 2004 by Agol and independently by Calegari and Gabai. For geometrically finite hyperbolic 3-manifolds it was already proved by Marden and partial results for some geometrically infinite hyperbolic manifolds were also known. From it, among other things (by the work of following William Thurston and Richard Canary ) and a presumption of Lars Ahlfors on the invariant limit quantities small shear groups (namely that they either measure have zero or full measure, in the latter case, the effect of the group is ergodic on the entire edge at infinity ).

In 2004 he received 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 and an elected member of the National Academy of Sciences since 2011 and of the American Academy of Arts and Sciences since 2014 .

Fonts

  • 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.
  • The 4-dimensional light bulb theorem , arxiv : 1705.09989

Web links

References

  1. ↑ David Gabai in the Mathematics Genealogy Project (English)Template: MathGenealogyProject / Maintenance / id used
  2. ↑ A knot has “property P” if every (non-trivial) stretching surgery on the knot in the 3-sphere does not result in simply connected 3-manifolds. The presumption says that all nodes except the non-node (the untied loop) have "property P". The conjecture was established in the 1970s by RH Bing and Martin and independently of González-Acuña as a step towards proving the Poincaré conjecture .
  3. ↑ Gabai "Homotopy hyperbolic 3-manifolds are virtually hyperbolic '", Journal AMS, Vol. 7, 1994, p. 193, Gabai "On the geometric and topological rigidity of hyperbolic 3-manifolds", Journal AMS, Vol. 10, 1997 , P. 37, Gabai, Robert Meyerhoff, Nathaniel Thurston "Homotopy hyperbolic 3-manifolds are hyperbolic", Annals of Mathematics, Vol. 157, 2003, p. 335
  4. ↑ Gabai, Convergence groups are Fuchsian groups, Annals of Math., Volume 136, 1992, pp. 447-510
personal data
SURNAME Gabai, David
BRIEF DESCRIPTION American mathematician
DATE OF BIRTH July 7, 1954
PLACE OF BIRTH Philadelphia , Pennsylvania