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# The Mathematical Legacy of William Thurston (1946-2012)

William Thurston. Photo courtesy of Cornell University Department of Mathematics

William Thurston, whose geometrization conjecture changed the fields of geometry and topology and whose approach to mathematics and mathematics education has reverberated throughout the mathematical world, died on August 21 following a battle with cancer. He has appeared in the pages of Scientific American in the article The Mathematics of Three-Dimensional Manifolds, which he co-wrote with Jeffrey Weeks, and as a figure in many of our other articles about mathematics.

When Thurston comes up in mathematical conversations, so do the words “visionary” and “intuitive.” “When he knew something, he knew it intimately. His intuition knew where things were going,” says Bill Goldman, a mathematician at the University of Maryland and one of Thurston’s students. “He saw how to put everything together and saw much further than anyone else did at the time.”

Thurston earned his bachelor’s degree at New College in Florida and his Ph.D. at the University of California-Berkeley in 1972. His early work revolutionized the study of geometric objects called foliations in the early 1970′s, to the extent that mathematicians started advising their students not to go into the field because Thurston seemed to be singlehandedly proving all the possible theorems. Starting in the mid-1970′s, he also revolutionized the field of low-dimensional geometry and topology. One of Thurston’s most important contributions was the geometrization conjecture about the properties of three-manifolds. A three-manifold is an object in which every point looks locally like three-dimensional Euclidean space. (For a good example, look around you: objects in our world, and in fact the world itself, are three-manifolds.)

The geometrization conjecture states that three-manifolds that are closed and bounded can be decomposed into pieces, each of which has one of eight well understood geometric structures. The Poincaré conjecture, posed in 1904, was considered one of the most important unsolved problems in mathematics. It is just one case of Thurston’s geometrization conjecture. At the time he made the conjecture, the related questions in dimensions one and two and in dimensions five and greater were understood, but little was known about dimensions three and four, where he worked.

In 1982, Thurston received a Fields Medal, one of the highest honors in mathematics, in part for his proof of the geometrization conjecture for a large class of manifolds called Haken manifolds. Reclusive Russian mathematician Grigori Perelman completed the proof of the geometrization conjecture in 2003, more than 20 years after it was made. For this achievement, Perelman was also awarded a Fields Medal, which he famously refused.

Thurston’s instincts about what was true in mathematics were remarkable. “He kind of had a truth filter. I’d tell him my ideas, which were always wrong, and he’d just sort of stare off in space and then come back and turn it into something correct. His mind rejected false mathematics,” says Goldman.

Thurston’s intuitive approach to mathematics was in part a reaction to the formal style prevalent in the early 1970′s, which emphasized rigorous proofs at the expense of exposition. “His intuitive style was pretty unconventional at the time, and a lot of the established mathematicians didn’t appreciate him. That changed pretty quickly,” says Goldman, when Thurston’s ideas started to transform entire fields of mathematics.

Thurston embraced efforts to make mathematics more accessible and enjoyable for students and the general public, especially in later years. In a 1994 article for the Bulletin of the American Mathematical Society, he wrote that the fundamental question for mathematicians should not be, “How do mathematicians prove theorems?” but, “How do mathematicians advance human understanding of mathematics?” He believed that this human understanding was what gave mathematics not only its utility but its beauty, and that mathematicians needed to improve their ability to communicate mathematical ideas rather than just the details of formal proofs.

He worked on projects to increase public understanding of mathematics and saw the mathematical sides of art and design. He co-developed a course called “Geometry and the Imagination” designed to introduce deep geometric concepts to people who did not necessarily have an advanced background in math.  In 2010, he collaborated with designer Dai Fujiwara on mathematically-inspired fashion. About that project, he wrote, “[Fujiwara] observed that we are both trying to understand the best 3-dimensional forms of 2-dimensional surfaces, and he noted that we each, independently, had come around to asking our students to peel oranges to explore these relationships. This resonated strongly with me, for I have long been fascinated (from a distance) by the art of clothing design and its connections to mathematics.”

Thurston worked at Princeton, MIT, University of California-Berkeley and Cornell University, among other places. He advised 33 students and has 157 mathematical descendants, including this author. In addition to his mathematical family, he is survived by his mother Margaret Thurston; his first wife Rachel Findley, from whom he was divorced; three children from his first marriage, Nathaniel Thurston, Dylan Thurston and Emily Thurston; his wife Julian Thurston; his two children from his second marriage Hannah Jade Thurston and Liam Thurston; his siblings Robert Thurston, Jean Baker and George Thurston; and two grandchildren.

Benson Farb, a mathematician at the University of Chicago and a student of Thurston, said in an email, “in my opinion Thurston is underrated: his influence goes far beyond the (enormous) content of his mathematics. He changed the way geometers/topologists think about mathematics. He changed our idea of what it means to ‘encounter’ and ‘interact with’ a geometric object. The geometry that came before almost looks like pure symbol pushing in comparison.” His fundamental contributions to the field have influenced much of the mathematical world, and he will be greatly missed.

The views expressed are those of the author and are not necessarily those of Scientific American.

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