:I need to understand the entropy angle better on this next.--[[User:Aschlafly|Andy Schlafly]] 00:06, 23 December 2009 (EST)
::The proof is not something I really know anything about, but I think the gist of what "entropy" is for is this. Perelman defines a certain functional (now everybody calls it the Perelman entropy), which can be proved to decrease under the Ricci flow. Remember that the basic idea of the proof, as described in this article, is to put a metric on our manifold, and then let it change according to some differential equation called Ricci flow. We can visualize it as the manifold "bending" in space as time passes, or something like that. And it turns out that this entropy decreases as it does. It also has the useful property of being scale-invariant, so we can globally rescale the metric without changing the Perelman entropy. Additionally it gives some degree of control over the geometry of the manifold under the Ricci flow, and helps prove that if we rescale the Ricci curvature under the flow, we get a lower bound on a standard gadget called the "injectivity radius", which means [[geodesic|geodesics]] must have a certain length before they can intersect each other. The bottom line is that the decreasing entropy ensures that singularities can develop only in very highly-curved regions. Then in the limit as time goes to infinity, after appropriate rescaling, we're supposed to end up with a stationary point of the Perelman entropy, and apply the Soul Theorem (which Perelman worked on earlier in his career). We conclude that what we have is either a sphere or a shrinking cylinder, and the proof is basically done.
::As you can tell from the above garbled gibberish, I don't understand the proof at all, so I don't think I'm in any position to help you! Terry Tao has posted some slides about the proof which I think is where I first heard this stuff, maybe you could start there. --[[User:JimR|JimR]] 00:28, 23 December 2009 (EST)