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 his contributions to the study of 3-manifolds. He is currently a professor of mathematics and computer science at Cornell University (since 2003).

Contents

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 takes every real value.

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 [1]).

The geometrization conjecture

His later work, starting around the late 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.

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