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In order to motivate the lengthy and detailed analysis of Ricci flow that will occupy the rest of this course, I will spend this lecture giving a high-level overview of Perelman’s Ricci flow-based proof of the Poincaré conjecture, and in particular how that conjecture is reduced to verifying a number of (highly non-trivial) facts about Ricci flow.

At the risk of belaboring the obvious, here is the statement of that conjecture:

Theorem 1. (Poincaré conjecture) Let M be a compact 3-manifold which is simply connected (i.e. it is connected, and every loop is contractible to a point). Then M is homeomorphic to a 3-sphere $S^3$.

[Unless otherwise stated, all manifolds are assumed to be without boundary.]

I will take it for granted that this result is of interest, but you can read the Notices article of Milnor, the Bulletin article of Morgan, or the Clay Mathematical Institute description of the problem (also by Milnor) for background and motivation for this conjecture. Perelman’s methods also extend to establish further generalisations of the Poincaré conjecture, most notably Thurston’s geometrisation conjecture, but I will focus this course just on the Poincaré conjecture. (On the other hand, the geometrisation conjecture will be rather visibly lurking beneath the surface in the discussion of this lecture.)