William Thurston

William Thurston
William Thurston in 1991
Born	William Paul Thurston (1946-10-30)October 30, 1946 Washington, D.C., United States
Died	August 21, 2012(2012-08-21) (aged 65) Rochester, New York, United States
Nationality	American
Alma mater	New College of Florida University of California, Berkeley
Awards	Fields Medal (1982) Oswald Veblen Prize in Geometry (1976) National Academy of Sciences (1983) Leroy P. Steele Prize (2012).
Scientific career
Fields	Mathematics
Institutions	Cornell University University of California, Davis University of California, Berkeley Princeton University
Doctoral advisor	Morris Hirsch
Doctoral students	Richard Canary Benson Farb David Gabai William Goldman Steven Kerckhoff Yair Minsky Oded Schramm Danny Calegari

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. He was last a professor of mathematics and computer science at Cornell University (since 2003).

Mathematical contributions

Foliations

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 ).

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.

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.

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, Richard Kenyon, Steven Kerckhoff, 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 University of Indiana.

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".

In 2012 Thurston was awarded the Leroy P Steele Prize by the AMS for seminal contribution to research. The citation described his work as having "revolutionized 3-manifold theory".

He died on August 21, 2012 in Rochester, New York, of a melanoma that was diagnosed in 2011.

Thurston has an Erdős number of 2.

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

See also

• Milnor–Thurston kneading theory
• Misiurewicz–Thurston points
• Nielsen–Thurston classification
• Jørgensen–Thurston theorem
• confoliation
• automatic group
• Thurston norm
• Thurston's double limit theorem
• Thurston's geometrization conjecture
• Circle packing theorem

External links

• 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