William Thurston

William Paul Thurston (October 30, 1946 – August 21, 2012) was an American mathematician.

He received his bachelor's degree from New College in 1967 as part of its inaugural class.

For his undergraduate thesis, he developed an intuitionist foundation for topology.

His early work, in the early 1970s, was mainly in foliation theory.

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 an 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 ).

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 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 hyperbolization theorem for Haken manifolds 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.

Following this, he received a doctorate in mathematics from the University of California, Berkeley under Morris Hirsch, with his thesis Foliations of Three-Manifolds which are Circle Bundles in 1972.

After completing his Ph.D., Thurston spent a year at the Institute for Advanced Study, then another year at the Massachusetts Institute of Technology as an assistant professor.

In 1974, Thurston was appointed a full professor at Princeton University.

He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.

Thurston was a professor of mathematics at Princeton University, University of California, Davis, and Cornell University.

He was also a director of the Mathematical Sciences Research Institute.

William Thurston was born in Washington, D.C., to Margaret Thurston (Martt), a seamstress, and Paul Thurston, an aeronautical engineer.

William Thurston suffered from congenital strabismus as a child, causing issues with depth perception.

His mother worked with him as a toddler to reconstruct three-dimensional images from two-dimensional ones.

He returned to Berkeley in 1991 to be a professor (1991-1996) and was also director of the Mathematical Sciences Research Institute (MSRI) from 1992 to 1997.

He was on the faculty at UC Davis from 1996 until 2003, when he moved to Cornell University.

Thurston was an early adopter of computing in pure mathematics research.

He inspired Jeffrey Weeks to develop the SnapPea computing program.

During Thurston's directorship at MSRI, the institute introduced several innovative educational programs that have since become standard for research institutes.

His Ph.D. students include Danny Calegari, Richard Canary, David Gabai, William Goldman, Benson Farb, Richard Kenyon, Steven Kerckhoff, Yair Minsky, Igor Rivin, Oded Schramm, Richard Schwartz, William Floyd, and Jeffrey Weeks.