In 1904 Henri Poincare conjectured that the only three manifold which shares a few simple features with the 3-sphere is the 3-sphere itself. While a number of proofs of this conjecture have appeared in the literature, until now all have eventually been shown to have serious flaws -- but four years ago a Russian mathematician named Grigory Perelman announced the outline of a proof which, in just the last few months, seems to have been verified by a group of specialists. Perelman was this summer offered the Fields medal in recognition of his achievement, but he turned it down.
I will discuss some of this bizarre history but mostly spend time on the geometric and analytic ideas surrounding the conjecture. In particular, I'll talk about various geometric classification theorems in low dimensions and the topological, geometric and analytic tools invented to help with them, ending with indications of the method Perelman used -- the Ricci flow -- which has proven to be successful.
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