The Poincaré Conjecture

It is a great pleasure for me to report on the recent spectacular developments concerning the Poincaré Conjecture. Grigory Perelman has solved the Poincaré Conjecture. He has shown that, as Poincaré conjectured, any closed, simply connected 3-manifold is homeomorphic to the 3-sphere. The paper in which Poincaré posed this problem in 1904 ([14]) marked, in my view, the founding of topology as an independent discipline within pure mathematics. Over the intervening 100 years, the problem has been much studied and generalized, and many related problems have been solved. It has been linked, in one way or another, with most of the progress in topology in the last 100 years. While related problems have been solved, the original conjecture stood untouched, resisting all attempts. Before Perelman’s work, there had been no progress on toward solving the Poincaré Conjecture, and many viewed it as the siren song of Topology, for many a boat had foundered on the rocks trying to reach it. There have been innumerable proposed proofs and proposed counter-examples, but none, before Perelman’s, withstood scrutiny. Solving the Poincaré Conjecture is a signal achievement for Perelman, but it is also a signal achievement for all of mathematics, for it gives a measure of how far our understanding of the subject has advanced in the last 100 years. To paraphrase Newton, Perelman has seen far, but to do so he stood on the shoulders of giants who came before him. One giant, in particular, stands out. He is Richard Hamilton. Over a period of 25 years, Hamilton painstakingly built the solid and elaborate foundation upon which Perelman constructed the edifice of his proof. Without Hamilton’s work, Perelman’s would not have been possible. One of the most interesting aspects of the resolution of the Poincaré Conjecture is the nature of the solution. While the problem is purely topological in its formulation, the proof is not. The proof uses deep techniques and results from other areas of mathematics, namely analysis and differential geometry. It is not at all clear a priori that these ideas have any relevance to the Poincaré Conjecture, but in the end they turn out to be the only way (so far) to approach this question successfully. My goal in this article is to give you a sense of the importance, centrality, and the depth of the Poincaré Conjecture. Then I will discuss surfaces and 3-dimensional spaces and describe how topologists think about them. Next, I will formulate and