A Man Out Standing In His Field

Two of our colloquia this semester were concerned with work very much in the public eye this year. The first was from Francis Bonahan of the Mathematics Department here at USC.

He talked about the work that won the Fields medal Рthe proof, by Grigori Perelman, of the Poincar̩ conjecture. Or better, I should say the work toward the proof, since the citation does not explicitly mention the conjecture, but his larger body of work. (MathWorld link, Wikipedia link.) In fact, Francis spoke about a lot more than just the Poincar̩ conjecture.

He talked about the larger setting in which that work fits, something mathematicians call the “geometrization conjecture”, which Mathematicians care a lot more about. Perelman’s work does more than just prove the Poincaré, it addresses the whole (3-)ball of wax, so to speak. He told us quite a bit about that in the talk, spending most time talking about what they were and how they fit into the scheme of things, rather than focusing on how Perelman solved the problem. (Here are links about the Geometrization conjecture of Thurston at Wikipedia and at MathWorld).

Here is his abstract:

The recent proof of the Poincaré Conjecture, a century-old problem in 3-dimensional geometry, has made many headlines this summer. I will explain what the Poincaré Conjecture was, and also introduce its less talked about (but perhaps more important) extension the Geometrization Conjecture. I will try to convey why mathematicians cared about these problems, why physicists may be interested in their solution, and what the current and future impact of these breakthroughs is likely to be. The talk will assume no mathematical background beyond calculus.

This turned out to be one of the most popular talks of the semester. We had people sitting on the floor, on the steps, standing in the back, etc.

One reason is the topic, and the other is that the speaker is known for giving excellent talks, and he did not fail us. He started off showing some newpaper headlines about Perelman and the prize, (and the fact that he was a recluse, lives with his mother, and turned down the prize), and contrasted this with the fact that there was zero coverage of the fact that a bright young fellow, Terence Tao, at UCLA also won the Fields medal and was not mentioned (even though some of those headlines were in local Los Angeles publications). His question: Why do mathematicians only make it to the news if they’ve done something weird!? This brought the house down, and the talk kept at least at that level of enjoyability all the way through.

It has been too long since I was at the talk for me to try and reconstruct any of the ideas without making a pig’s ear of it. Let me just say that it was extraordinarily well-presented (he assumed nothing but basic calculus and lived up to that), fascinating, and entertaining. I’ll let you look at other material on the web (linked above) and elsewhere for more information.

(In the picture above, he’s drawing an example of a two dimensional non-compact surface, and describing a result about their classification… yes, he did call those little loops or handles (indicating higher genus features -a torus (doughnut) is a sphere with a handle in topology) “Loch Ness monsters”. Of course he did.)

-cvj