Bill Thurston passed away yesterday, at the age of 65, after a battle with melanoma. Thurston was for many years the dominant figure in the study of 3 dimensional topology and geometry, winning a Fields medal for this work in 1982. His “Geometrization Conjecture” classifying the topology of 3 manifolds was finally proved by Perelman as part of his work on the Poincaré Conjecture.
For an exposition of some of his work, see The Geometry and Topology of Three-manifolds, which exists as a set of unpublished notes here, and a book covering the first few chapters of the notes here. Thurston was sometimes criticized for not writing up full proofs of his results, making it difficult for others entering the field (and sometimes students were advised not to enter the field since Thurston was so good the danger was he would just solve all open problems). He wrote a truly wonderful essay On Proof and Progress in Mathematics, responding to this and laying out part of his vision of how to do mathematics.
My first encounter with Thurston was in the early eighties, when I was a physics graduate student at Princeton. I was working on the problem of defining the topological charge of a lattice gauge field, and it became clear that one approach to do this would require computing the volumes of “spherical tetrahedra”, the 3d analog of the problem of computing the areas of spherical triangles. I’d had some experience trying to talk to mathematicians about the problem I was working on, with the usual result a baffling response about principal bundles, sections, characteristic classes, and all sorts of what seemed to be abstract nonsense (which later on of course I learned was the right way to think about the problem…). So, I was pretty convinced that mathematicians were uniformly experts in a lot of abstract, high-powered technology, surely no longer conversant with the kind of more concrete formulas of the mathematics of earlier centuries.
This was before the days of the internet, so the answer to my problem couldn’t be found via Google, and a bit of library research got me nowhere. So, I stopped by to see a friend who was a math grad student and asked him my volume question. He said that while he didn’t know, he knew someone who could surely help me, and took me over to the math lounge, where Thurston could often be found. After I asked my question, Thurston immediately knew the answer, explained it to me on the blackboard, and gave me the proper reference of where to read more (you break them up in a certain way and then get an answer in terms of things called Schläfli functions, see here). I realized that my views of how much the best research mathematicians knew about concrete calculations and lore from previous centuries had been rather naive.
Thurston’s death at such a relatively young age is a loss for us all. My condolences to his family, including his son Dylan, a very talented topologist in his own right, who has been my colleague here for the last several years.
Update: Terry Tao has more about Thurston and his work here.
Update: Jordan Ellenberg has something here.
Update: The New York Times has an obituary here.
Update: Jonah Sinick has put together a memorial slideshow here.
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