Bill (1946-2012)

I still remember the aura of mystery that surrounded Bill Thurston’s work when I first encountered it as a graduate student in a past life: a late afternoon seminar on the Travaux de Thurston sur les surfaces, a reading course on the notes The Geometry and Topology of Three-manifolds. Laminations with transverse invariant measure? A hyperbolic structure on the figure-eight knot complement? How many hours spent poring over those diagrams and drawing new ones, until I convinced myself (rightly or wrongly) that I could see what he was talking about. It was unlike anything I had ever seen in mathematics.

Later, sometime in the early 1980s, for a while I attended his group’s research seminar at Princeton (called “graduate course” there). The topic was compactifying the moduli space of postcritically finite rational maps, and he was developing it during the course, using the audience as a sounding board.  Laminations of the disk were involved, and at some point I joined his students and postdocs in computing examples.  My notes on this have been lost; I just checked Bill’s Mathscinet list, and did not find a publication corresponding to this. Since, like myself, the other participants had other main interests, it could be that some beautiful mathematics has vanished.

 That’s the risk with doing things in “Thurston mode”: for it to become a permanent part of mathematics, somebody else has to understand it well enough to be able to record it for people who operate at a “normal level”, and that didn’t always happen. Implementing parts of a "Thurston program" has been a nontrivial creative activity in itself, inspiring the careers of many.

Bill was aware of this problem; he appears to have given a lot of thought to the gulf between understanding a mathematical situation intuitively/geometrically and the need to communicate  (and record) it in a necessarily downgraded symbolic version:

Mathematics is commonly explained and recorded in symbolic and concrete forms that are easy to communicate, rather than in conceptual forms that are easy to understand once communicated. Translation in the direction conceptual --> concrete and symbolic is much easier than translation in the reverse direction, and symbolic forms often replace the conceptual forms of understanding. (Mathoverflow)

Here is Bill’s comment on the real reason mathematics is important:

The product of mathematics is clarity and understanding. Not theorems, by themselves. Is there, for example, any real reason that even such famous results as Fermat's Last Theorem, or the Poincaré conjecture, really matter? Their real importance is not in their specific statements, but their role in challenging our understanding, presenting challenges that led to mathematical developments that increased our understanding.

Elsewhere he said “the goal of mathematics is to develop ways for humans to see and think about the world”.  But, in that case, it is not enough for one human (or a handful) to reach such “increased understanding”; the work is not finished—not really useful as a new way to “think about the world”—until it reaches a form in which (at least) scientifically educated non-mathematicians can make sense of it; but by that time it is no longer research, and mathematicians have moved on.

(Think, for instance, of the recent proof of Thurston’s Geometrization Conjecture in the style of Hamilton and Perelman. How long until the “understanding” implicit in this approach can be put into a form that would make sense to a student--or an expert-- in Geometric Topology, or to a physicist? Who is going to do the work?)

Bill was a unique mathematician, and his passing at a point when he was still contributing so much saddens geometers everywhere. The memorial page set up by the Cornell mathematics department has links to sources including some of his views on mathematics, like this one:

Mathematics only exists in a living community of mathematicians that spreads understanding and breathes life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others.

Exactly.  You can do it (up to a point) while separated from your `tribe’, but it’s no fun. And while all mathematicians know this is true, I hadn't seen it stated so precisely and eloquently until now. We'll miss him.