William Thurston

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William Paul Thurston (born October 30, 1946) is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields medal for the depth and originality of his contributions to mathematics. He is currently a professor of mathematics and computer science at Cornell University (since 2003).

Contents

[edit] Mathematical contributions

[edit] Foliations

His early work, in the early 1970s, was mainly in foliation theory, where he proved, amongst other things, that any n-manifold with Euler characteristic zero has a codimension one foliation.

[edit] The geometrization conjecture

His later work, starting around the late 1970s, revealed that geometry, particularly hyperbolic geometry, played a fundamental role in the theory of 3-manifolds. Prior to Thurston, there were only a handful of known examples of hyperbolic 3-manifolds of finite volume, such as the Seifert-Weber space. The independent and distinct approaches of Robert Riley and Troels Jorgensen in the mid-to-late 1970s showed that such examples were less atypical than previously believed; in particular their work showed that the figure eight knot complement was hyperbolic. This was the first example of a hyperbolic knot.

Inspired by their work, Thurston took a different, more explicit means of exhibiting the hyperbolic structure of the figure eight knot complement. He showed that the figure eight knot complement could be decomposed as the union of two regular ideal hyperbolic tetrahedra whose hyperbolic structures matched up correctly and gave the hyperbolic structure on the figure eight knot complement. By utilizing Haken's normal surface techniques, he classified the incompressible surfaces in the knot complement. Together with his analysis of deformations of hyperbolic structures, he concluded that all but 10 Dehn surgeries on the figure eight knot resulted in irreducible, non-Haken non-Seifert-fibered 3-manifolds. These were the first such examples; previously it had been believed that except for certain Seifert fiber spaces, all irreducible 3-manifold were Haken. These examples were actually hyperbolic and motivated his next revolutionary theorem.

Thurston proved that in fact most Dehn fillings on a cusped hyperbolic 3-manifold resulted in hyperbolic 3-manifolds. This is his celebrated hyperbolic Dehn surgery theorem.

To complete the picture, Thurston proved a geometrization theorem for Haken manifolds. A particularly important corollary is that many knots and links are in fact hyperbolic. Together with his hyperbolic Dehn surgery theorem, this showed that closed hyperbolic 3-manifolds existed in great abundance.

The geometrization theorem has been called Thurston's Monster Theorem, due to the length and difficulty of the proof. Complete proofs were not written up until almost 20 years later. The proof involves a number of deep and original insights which have linked many apparently disparate fields to 3-manifolds.

Thurston was next led to formulate his geometrization conjecture. This gave a conjectural picture of 3-manifolds which indicated that all 3-manifolds admitted a certain kind of geometric decomposition involving eight geometries, now called Thurston model geometries. Hyperbolic geometry is the most prevalent geometry in this picture and also the most complicated.

[edit] Orbifold theorem

In his work on hyperbolic Dehn surgery, Thurston realized that orbifold structures naturally arose. Such structures had been studied prior to Thurston, but his work, particularly the next theorem, would bring them to prominence. In 1981, he announced the orbifold theorem, an extension of his geometrization theorem to the setting of 3-orbifolds. Two teams of mathematicians around 2000 finally finished their efforts to write down a complete proof, based mostly on Thurston's lectures given in the early 1980s in Princeton. His original proof relied partly on Hamilton's work on the Ricci flow.

[edit] Education and career

He was born in Washington, D.C and received his bachelors degree from New College (now New College of Florida) in 1968. For his undergraduate thesis he developed an intuitionist foundation for topology. Following this, he earned a doctorate in mathematics from the University of California, Berkeley, in 1972. His Ph.D. advisor was Morris W. Hirsch and his dissertation was on Foliations of Three-Manifolds which are Circle Bundles.

From 1972 to 1973, he visited the Institute for Advanced Study. Then he went to MIT where he was an assistant professor. After a year, in 1974, he became a full professor at Princeton University. Eventually he left Princeton and moved to Berkeley for several years. From 1996 to 2003, he was a professor at UC Davis. In 2003, he became a professor at Cornell University.

His Ph.D. students include Richard Canary, David Gabai, William Goldman, Benson Farb, Detlef Hardorp, Craig Hodgson, Steven Kerckhoff, Robert Meyerhoff, Yair Minsky, Lee Mosher, Igor Rivin, Oded Schramm, Richard Schwartz, Martin Bridgeman and Jeffrey Weeks.

Thurston has turned his attention in recent years to mathematical education and bringing mathematics to the general public. He has served as mathematics editor for Quantum magazine, a youth science magazine, and as head of The Geometry Center. As director of Mathematical Sciences Research Institute from 1992 to 1997, he initiated a number of programs designed to increase awareness of mathematics among the public.

In 2005 Thurston won the first AMS Book Prize, for Three-dimensional Geometry and Topology. The prize recognizes an outstanding research book that makes a seminal contribution to the research literature.[1]

[edit] Selected works

  • William Thurston, The geometry and topology of 3-manifolds, Princeton lecture notes (1978-1981).
  • William Thurston. Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. ISBN 0-691-08304-5
  • William Thurston, Hyperbolic structures on 3-manifolds. I. Deformation of acylindrical manifolds. Ann. of Math. (2) 124 (1986), no. 2, 203--246.
  • William Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), 357–381.
  • William Thurston. On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417--431
  • Epstein, David B. A.; Cannon, James W.; Holt, Derek F.; Levy, Silvio V. F.; Paterson, Michael S.; Thurston, William P. Word processing in groups. Jones and Bartlett Publishers, Boston, MA, 1992. xii+330 pp. ISBN 0-86720-244-0
  • Eliashberg, Yakov M.; Thurston, William P. Confoliations. University Lecture Series, 13. American Mathematical Society, Providence, RI, 1998. x+66 pp. ISBN 0-8218-0776-5

[edit] See also

[edit] External links


Fields Medalists

1936: Ahlfors • Douglas || 1950: Schwartz • Selberg || 1954: Kodaira • Serre || 1958: Roth • Thom || 1962: Hörmander • Milnor || 1966: Atiyah • Cohen • Grothendieck • Smale || 1970: Baker • Hironaka • Novikov • Thompson || 1974: Bombieri • Mumford || 1978: Deligne • Fefferman • Margulis • Quillen || 1982: Connes • Thurston • Yau || 1986: Donaldson • Faltings • Freedman || 1990: Drinfeld • Jones • Mori • Witten || 1994: Zelmanov • Lions • Bourgain • Yoccoz || 1998: Borcherds • Gowers • Kontsevich • McMullen || 2002: Lafforgue • Voevodsky || 2006: Okounkov • Perelman • Tao • Werner
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