William Thurston in 1991
|Born||William Paul Thurston
October 30, 1946
Washington, D.C., United States
|Died||August 21, 2012
Rochester, New York, United States
University of California, Davis
University of California, Berkeley
|Alma mater||New College of Florida
University of California, Berkeley
|Doctoral advisor||Morris Hirsch|
|Doctoral students||Richard Canary
|Notable awards||Fields Medal (1982)
Oswald Veblen Prize in Geometry (1976)
National Academy of Sciences (1983)
Leroy P. Steele Prize (2012).
William Paul Thurston (October 30, 1946 – August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds. From 2003 until his death he was a professor of mathematics and computer science at Cornell University.
His early work, in the early 1970s, was mainly in foliation theory, where he had a dramatic impact. His more significant results include:
- The proof that every Haefliger structure on a manifold can be integrated to a foliation (this implies, in particular that every manifold with zero Euler characteristic admits a foliation of codimension one).
- The construction of a continuous family of smooth, codimension one foliations on the three-sphere whose Godbillon–Vey invariant (after Claude Godbillon and Jacques Vey) takes every real value.
- With John Mather, he gave a proof that the cohomology of the group of homeomorphisms of a manifold is the same whether the group is considered with its discrete topology or its compact-open topology.
In fact, Thurston resolved so many outstanding problems in foliation theory in such a short period of time that, according to Thurston, it led to a kind of exodus from the field, where advisors counselled students against going into foliation theory because Thurston was "cleaning out the subject" (see "On Proof and Progress in Mathematics", especially section 6 ).
|This section requires expansion. (June 2008)|
The geometrization conjecture
His later work, starting around the mid-1970s, revealed that hyperbolic geometry played a far more important role in the general theory of 3-manifolds than was previously realised. 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 Jørgensen 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-manifolds 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 hyperbolization 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. The conjecture was proved by Grigori Perelman in 2002–2003.
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.
Education and career
Thurston was born in Washington, D.C. to a homemaker and an aeronautical engineer. He received his bachelors degree from New College (now New College of Florida) in 1967. 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.
After completing his Ph.D., he spent a year at the Institute for Advanced Study, then another year at MIT as Assistant Professor. In 1974, he was appointed Professor of Mathematics at Princeton University. In 1991, he returned to UC-Berkeley as Professor of Mathematics and in 1993 became Director of the Mathematical Sciences Research Institute. In 1996, he moved to University of California, Davis. In 2003, he moved again to become Professor of Mathematics at Cornell University.
His Ph.D. students include Richard Canary, Suyoung Choi, Renaud Dreyer, David Gabai, William Goldman, Benson Farb, Sergio Fenley, Detlef Hardorp, Craig Hodgson, Christopher Jerdonek, Richard Kenyon, Steven Kerckhoff, Silvio Levy, Robert Meyerhoff, Yair Minsky, Lee Mosher, Igor Rivin, Nicolau Saldanha, Oded Schramm, Richard Schwartz, Martin Bridgeman, William Floyd and Jeffrey Weeks. His son Dylan Thurston is an associate professor of mathematics at Indiana University.
In later years Thurston turned his attention 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".
Thurston has an Erdős number of 2, via John Horton Conway. Paths of length 3 are many; for example, Allan R. Wilks is a co-author, and Ronald Graham is a co-author with Wilks. Kenneth Steiglitz is a co-author, and Daniel J. Kleitman is a co-author with Steiglitz. Joel Hass is a co-author, and Laszlo Lovasz is a co-author with Hass. Graham, Lovasz and Kleitman are all co-authors with Erdős.
- William Thurston, The geometry and topology of three-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
- Thurston, William P. (April 1994). "On Proof and Progress in Mathematics". Bulletin of the American Mathematical Society 30 (2): pages 161–177. arXiv:math/9404236. doi:10.1090/S0273-0979-1994-00502-6.
- Institute for Advanced Study: A Community of Scholars
- "William P. Thurston Receives 2005 AMS Book Prize". Retrieved 2008-06-26.
- "AMS prize booklet 2012".
- "Department mourns loss of friend and colleague, Bill Thurston", Cornell University
- Obituary from American Mathematical Society
- Leslie Kaufman (August 23, 2012). "William P. Thurston, Theoretical Mathematician, Dies at 65". New York Times. p. B15.
- William Thurston at the Mathematics Genealogy Project
- O'Connor, John J.; Robertson, Edmund F., "William Thurston", MacTutor History of Mathematics archive, University of St Andrews.
- Thurston's page at Cornell
- Tribute and remembrance page at Cornell