William Thurston

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William Thurston 1991 in Oberwolfach

William Paul Thurston (born October 30, 1946 in Washington, DC - † August 21, 2012 in Rochester , New York ) - commonly known as Bill Thurston - was an American mathematician .

He came up with the idea of geometrizing to classify closed three-dimensional manifolds . For this he received the Fields Medal in 1982 . His theories linked mathematical areas previously viewed as separate with the theory of three-dimensional manifolds.


Thurston studied at New College in Sarasota in Florida ( Bachelor's degree in 1967) and then at Morris Hirsch and Stephen Smale at the University of Berkeley , where he received his doctorate in 1972 with a thesis on foliations special 3-manifolds (Foliations of 3-manifolds Which are circle bundles ) . In 1972/73 he was at the Institute for Advanced Study in Princeton . In 1973 he became an Assistant Professor at the Massachusetts Institute of Technology (MIT) and in 1974 Professor of Mathematics at Princeton University . Since 1991 he has been a professor at UC Berkeley , since 1996 at UC Davis and from 2003 at Cornell University . He was also director of the MSRI from 1992–97 .

In his laudatory speech on the awarding of the Fields Medal to Thurston in 1982, CTC Wall emphasized that Thurston revolutionized the field of topology of two- and three-dimensional manifolds with "fantastic geometric intuition and vision". In particular, Thurston clarified the central role of hyperbolic manifolds (that is, manifolds with a metric of constant negative curvature) also in the three-dimensional case (see geometrization of 3-manifolds ). In his geometrization conjecture (which he proved for the special case of hook manifolds) he classified three-dimensional manifolds into eight types. Previously, the classifications in dimensions two and greater than five were well understood, but not in dimensions three and four. The proof of the geometrization conjecture of Thurston (with the Poincaré conjecture as part) succeeded Grigori Perelman in 2003. The theory of discrete groups of isometries of hyperbolic space ( Klein's groups ), which was previously largely isolated from the topology , thus became a central topic of three-dimensional topology .

Before that, he also revolutionized the theory of scrolls in the early 1970s .

In 1982 Thurston formulated a series of 24 conjectures about 3-manifolds, all but one of which have now been solved. In addition to the geometrization conjecture, it has long been an open problem whether hyperbolic 3-manifolds are virtually fiberized ; this was proven in 2013 by Ian Agol and Daniel Wise . It is still open (2016): Show that there are two hyperbolic 3-manifolds whose volumes are not in a rational relationship to one another.

He pursued an intuitive, initially not uncontroversial approach to mathematical knowledge with the aim of understanding mathematical structures with an emphasis on ideas rather than details of formal evidence (which he underlined in an article in the Bulletin of the AMS 1994). According to his pupil Benson Farb , Thurston fundamentally changed the approach of modern topologists and geometers to mathematical objects, in contrast to which the previously used method seemed more like the pure manipulation of symbols. He also developed a Geometry and the Imagination course to make advanced geometric concepts accessible to mathematically inexperienced people such as artists and designers. He also worked with the Japanese fashion designer Dai Fujiwara on the best representation of three-dimensional "manifolds" in two dimensions (clothes).

Thurston was married twice, in his first marriage to Rachel Findley he had three children (including the mathematicians Dylan and Nathaniel Thurston), in his second marriage to Julian Thurston he had two children.

Awards (selection)

Thurston died in August 2012 at the age of 65 from melanoma diagnosed a year earlier.

His PhD students include Danny Calegari , Richard Canary , David Gabai , Steven Kerckhoff , Yair Minsky , Oded Schramm , Jeffrey Weeks , Richard Kenyon , Silvio Levy, Richard Evan Schwartz , William Goldman (second referee), Benson Farb , William Floyd , Igor Rivin .


  • Three dimensional geometry and topology , Volume 1. Princeton University Press, 1997 (editor Silvio Levy), covers only a small part of the Princeton lectures 1979-1981 , the latter at library.msri.org
  • Hyperbolic structures on 3-manifolds. I. Deformation of cylindrical manifolds. In: Annals of Mathematics. Volume 124, 1986, pp. 203-246.
  • Three dimensional manifolds, Kleinian Groups and Hyperbolic Geometry. In: Bulletin AMS. Volume 6, May 1982, pp. 357-381.
  • On the geometry and dynamics of diffeomorphisms of surfaces. In: Bulletin AMS. Volume 19, 1988, pp. 417-431.
  • with Steven Kerckhoff: Non continuity of the action of the modular group at Bers boundary of Teichmüller space. In: Inventiones Mathematicae. Volume 100, 1990, pp. 25-47.
  • with Yakov Eliashberg : Confoliations , American Mathematical Society, 1998.
  • Three manifolds, foliations and circles I . Preprint 1997
  • with Michael Gromow , H. Blaine Lawson : Hyperbolic 4-manifolds and conformally flat 3-manifolds. In: Publications Mathématiques de l'IHÉS. Volume 68, 1988, pp. 27-45. (numdam.org)
  • Existence of codimension one foliations. In: Annals of Mathematics. Volume 104, 1976, pp. 249-268.
  • The theory of foliations of codimension greater than one. In: Comm. Math. Helv. Volume 49, 1974, pp. 214-231.
  • with David B. Epstein , James W. Cannon , Derek F. Holt, Silvio Levy, Michael S. Paterson : Word processing in groups . Jones and Bartlett, Boston 1992.
  • On proof and progress in mathematics , Bull. Amer. Math. Soc. (NS), Vol. 30, 1994, pp. 161-177. Arxiv
  • with Jeffrey Weeks: The mathematics of three-dimensional manifolds. Scientific American, Volume 251, July 1984

See also


  • CTC Wall : On the work of W Thurston. In: Proceedings of the International Congress of Mathematicians. Warsaw 1983 1 (Warsaw, 1984), 11-14. (Laudation for Thurston's work on the occasion of the Fields Medal)
  • Albert Fathi , François Laudenbach , Valentin Poénaru : Travaux de Thurston. In: Asterisque. Volume 65/66, 1979.
  • Serge Lang : The beauty of doing mathematics . Springer Verlag, 1985, chapter Great problems of geometry and space (on Thurston's geometry program, popular science)
  • Jeffrey Weeks : The Shape of Space . Marcel Dekker, 1985 (on Thurston's geometry program)
  • Michael Gromow : Hyperbolic manifolds according to Thurston and Jorgensen. In: Seminaire Bourbaki. No. 546, 1979/80, (numdam.org)
  • Valentin Poenaru: Travaux de Thurston sur les difféomorphismes des surfaces et l'espace de Teichmüller. In: Seminaire Bourbaki. No. 529, 1979/80. (numdam.org)
  • Dennis Sullivan : Travaux de Thurston sur les groupes quasi-fuchsiens et les variétés hyperboliques de dimension 3 fibers sur . In: Seminaire Bourbaki. No. 554, 1979/80. (numdam.org)
  • Robert Roussarie: Construction of the feuilletages. In: Seminaire Bourbaki. No. 499, 1977. (numdam.org)
  • Richard Canary , David Epstein , P. Green: Notes on notes of Thurston. In: David Epstein, Albert Marden, Richard Canary (Eds.): Fundamentals of Hyperbolic Manifolds: Selected Expositions. (= London Mathematical Society Lecture Note. Series 328). Cambridge University Press, 2006. (first published 1987)
  • Donald J. Albers, GL Alexanderson, Constance Reid : More Mathematical People - Contemporary Conversations . Academic Press, 1994

Web links

Individual evidence

  1. ^ Terence Tao: What's new: Bill Thurston. Obituary. August 22, 2012, accessed August 30, 2019 .
  2. David Gabai , Steve Kerckhoff : William P. Thurston, 1946–2012 . In: Notices of the American Mathematical Society . tape 62 , no. December 11 , 2015, p. 1318–1332 ( ams.org [PDF; accessed May 17, 2016]).
  3. ^ Bulletin AMS. Volume 6, 1982, pp. 357-379.
  4. ^ Stefan Friedl: Thurston's Vision and the Virtual Fibering Theorem for 3-Manifolds. In: Annual report DMV. Issue 4, 2014. (pdf)
  5. ^ William Goldman: His intuitive style was pretty unconventional at the time, and a lot of the established mathematicians didn't appreciate him. That changed pretty quickly. This referred to the early 1970s, when a formalized representation in the Bourbaki style was common. Quoted from Evelyn Lamb: The mathematical legacy of William Thurston. Scientific American Blog 2003.
  6. ^ Benson Farb, quoted from Evelyn Lamb, Thurston Obituary in Scientific American Blog 2003.
  7. math.cornell.edu