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For an exposition of some of his work, see The Geometry and Topology of Three-manifolds, which exists as a set of unpublished notes here, and a book covering the first few chapters of the notes here.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: More here, here and here. Also worth the time is seeing what he had to say on MathOverflow.
Update: Jordan Ellenberg has something here.
Update: The New York Times has an obituary here.
Update: There’s a Cornell site here, Scientific American has a piece by Evelyn Lamb here, John Horgan here.
Update: Jonah Sinick has put together a memorial slideshow here.
Very sad news and a great loss to science. My condolences to all bereaved.
However, Peter, did you manage to define the topological charge of the gauge field for the general gauge group case ? Also, if you solved it in the lattice case, did your definition carry over to the continuum limit ?
The question of defining topological charge carried by gauge fields seems very important to me, because gauge field theory is the theory of interaction of these charges, and empirically the bound states of gauge charges, in the limit, yields gravitating bodies.
I didn’t use the volume formulas that Thurston told me about, that’s too hard. What I did do was just for SU(2), and was easy to compute, made sense in the continuum limit. It only worked for SU(2), where things come down to computing degrees of maps. Lots of other people have worked on this over the years, a long story, maybe for another time, since the only relation to Thurston I know of is our one conversation in the common room.
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OK, as a mathematician I fell somehow obliged to tell my Thurston story. I never actually met him, though I know several of his students pretty well. When I was an undergrad at Hampshire in the early 80’s Thurston came to give a talk at Smith about his work with 3-manifolds. My advisor, Ken Hoffman, took me to see him. Didn’t really understand all that much, but the way he explained things really made the basic ideas understandable, even to an undergraduate who was at the time a physics major and hadn’t even taken modern algebra yet. In grad school I spent hours (days, weeks…) with a xerox of his lecture notes, beautiful doesn’t really even begin to describe how wonderful they are. I spent some time a few years ago with Steve Kerchoff at a conference, he was one of Thurston’s first grad students, and was the advisor of a friend of mine. Kerchoff was at the lectures which turned into the notes, he described how amazing it was to see all this stuff when it was brand new, and totally out of the mainstream.
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In the era, 1982-1985 both Bill Thurston and Mike Freedman would often visit the University of Texas at Austin. The visits were so frequent, in fact, that it was difficult to do anything other than attend the seminars, dinners, and after dinner parties. Once, at Cameron Gordon’s house, some of started to play with a toy motor cycle (wind up, hand held thing) that Cameron’s son Andrew had. Bill reved the thing up (rrr-rrr-rrr) I think he even added his own sound effects, and he got the thing to fly across the diagonal of the den. This feat impressed me — not as much as his mathematics — but I always think of him as being playful.
I feel about Bill Thurston’s passing much as I feel about the death of great spiritual leaders. Profane though he was like the rest of us, his mind was sacred. Thurston conceived, manipulated and illustrated the texture of God.
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I did not have the privilege of knowing Bill Thurston personally, but his immense influence, the beauty of his ideas, and his special personality give me now a feeling of a great personal loss. Of course, it is a great loss for mathematics.
Bill and I were colleagues for many years at Princeton
and at the Geometry Center. The joke there was, Why is Bill
so involved in producing software to make high-dimensional geometry visible
when the last person in the world who needs any such tool is Bill?
His geometric intuition was beyond belief (I have fun anecdotes about that).
But I am too sad today to talk about anything that’s not, well, sad. His 60th b’day
conference at Princeton would be one such occasion. Marvelous talks and
a sweet personal note when Bill told me he followed my political
activism on the web and added “We have no choice but be active.”
Bill was the guy who turned down an endowed chair at Princeton
because the name attached to it was a rightwing militarist.
What’s so sad about all that? Well, the last two people I talked
to at the conference were Bill and his former student Oded Schramm,
both of them gone way too soon. Leaves a knot in one’s stomach.
Bill approached math with a touch of magic. A very sad day.
Danny Calegari wites.
Bill Thurston is ever so missed here at Cornell. There is a page up in tribute. It includes a place for posting remembrances and a selection of quotes assembled by Dylan.
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Brought to my attention via this blog, I always have in mind Thurston’s laudation on the Award of the Millennium Prize to Grigoriy Perelman For Resolution of the Poincaré Conjecture and hope that it is worth reviewing at this apposite time. Here it is in part: “Perelman’s aversion to public spectacle and to riches is mystifying to many. I have not talked to him about it and I can certainly not speak for him, but I want to say I have complete empathy and admiration for his inner strength and clarity, to be able to know and hold true to himself. Our true needs are deeper – -yet in our modern society most of us reflexively and relentlessly pursue wealth, consumer goods and admiration. We have learned from Perelman’s mathematics. Perhaps we should also pause to reflect on ourselves and learn from Perelman’s attitude toward life.”
We should learn too from Thurston’s own exemplary attitude toward life.
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