William Thurston


William Paul Thurston 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.

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:
In fact, Thurston resolved so many outstanding problems in foliation theory in such a short period of time that 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".

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 Richard S. 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 bachelor's degree from New College 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 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. He and his first wife, Rachel Findley, had three children: Dylan, Nathaniel, and Emily. In 1991, he returned to UC-Berkeley as Professor of Mathematics and in 1993 became Director of the Mathematical Sciences Research Institute. In 1996, his wife Julian, who had earlier been his Ph.D. student at Princeton University, made a career switch to veterinary medicine, and began her studies at the UC Davis School of Veterinary Medicine. Bill and Julian moved to Davis, California, where Bill became Professor of Mathematics at UC Davis. In 2000, their first child Jade was born, and in 2003 their second child Liam was born. Bill and Julian had visited Ithaca in 1997 for a family celebration for his mother's 80th birthday. They were enchanted by the beauty of Ithaca, and in 2003 the family moved to Ithaca, New York, where Bill became Professor of Mathematics at Cornell University.
His Ph.D. students include Martin Bridgeman, Danny Calegari, Richard Canary, Suhyoung Choi, Renaud Dreyer, Julian Thurston, David Gabai, William Goldman, Benson Farb, Matt Grayson, 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, William Floyd, Biao Wang and Jeffrey Weeks. His son Dylan Thurston is a professor of mathematics at Indiana University.
In later years Thurston widened his attention to include mathematical education and bringing mathematics to the general public. He has served as mathematics editor for Quantum Magazine, a youth science magazine, and was one of the founders 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 sinus mucosal melanoma that was diagnosed in 2011.
Thurston and his family had been in the process of moving back to Davis, California, where he was to rejoin the mathematics faculty at UC Davis while his wife completed her veterinary medical degree. Thurston died before he could make the move to California. He had remained with his brother George in Rochester, New York, while his family went ahead of him to California to get settled, waiting for him to gain better physical strength for making the cross-country trip to California to join them. Bill's health declined rapidly, and the family returned to Rochester to be with him during his final days.
In Thurston's last days, he sometimes used American Sign Language to communicate with his children, Liam and Jade. Bill and Julian had spent a year studying ASL when Jade was an infant, and the family had become somewhat fluent. He also communicated by writing on one of his many pads of paper. One of his last written messages was, "Treasure Island," and this reference remains mysterious to his family.

Selected works