I was going to talk about Grigory Perelman turning down his Fields medal, and that if he wants to avoid publicity this is probably not an ideal way of doing it, but that has been covered well enough already. In particular, Gooseania has all you probably want to know about it. Show off time: people involved in the media storm around this that I’ve met: Nigel Hitchin (my supervisor next year), John Ball (president of the IMU, and took a manifolds course with me last year, which was odd) and Marcus Du Sautoy (media spokesperson for mathematics, it seems).

Instead I thought I’d talk a bit about what he’s meant to have done, as this gives me a reason to find out myself. As anyone on the street can probably now tell you, it’s the Poincaré Conjecture, which asks if a 3-manifold with trivial fundamental group must be topologically a 3-sphere, or in plain talking that if you have a 3-dimensional space where every loop can be contracted to a point, then must your space be a deformed 3-dimensional sphere. So far so good. Many have tried, though, and many have failed, but they found out lots of cool stuff about 3-manifolds, and so probably started the field of Low Dimensional Topology, so it’s not all tears. Perelman got around the problem by instead solving a much harder one, Thurston’s Geometrization Conjecture which concerns, in broad strokes, the decomposition of 3-manifolds into pieces which have certain, well- or better-understood geometries (the Thurston model geometries).

How then did he do it? He used an idea of Richard Hamilton (so-called Ricci flow), whereby you distort a given 3-manifold by letting it’s Ricci curvature follow a differential-equation analogous to the heat equation. The idea is that this should smooth the Ricci curvature out until it is uniform over the whole manifold, and what you then have is some sort of canonical form for the manifold, which should be one of the Thurston model geometries. The interesting thing about dynamical systems is their singularities, but what needed to be shown was that these would be well-behaved ones that can be resolved. This is what Perelman did: he formulated a variant of the above (Ricci flow with surgery) in which singularities develop but can be removed and so controlled.

For the keen, his original papers on the arXiv (may be down due to demand):

• Perelman, Grisha (2002). “The entropy formula for the Ricci flow and its geometric applications.” November 11
• Perelman, Grisha (2003). “Ricci flow with surgery on three-manifolds.” March 10
• Perelman, Grisha (2003). “Finite extinction time for the solutions to the Ricci flow on certain three-manifolds.” July 17

John Milnor has an article on the Ricci flow approach.