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Geometrization of 3-Manifolds via the Ricci Flow

184 NOTICES OF THE AMS VOLUME 51, NUMBER 2 Introduction The classification of closed surfaces is a milestone in the development of topology, so much so that it is now taught to most mathematics undergraduates as an introduction to topology. Since the solution of the uniformization problem for surfaces by Poincaré and Koebe, this topological classification is now best understood in terms of the geometrization of 2-manifolds: every closed surface Σ admits a metric of constant Gauss curvature +1, 0, or −1 and so is uniformized by one of the standard space-form geometries S2, R2, H2. Hence any surface Σ is a quotient of either the 2-sphere, the Euclidean plane, or the hyperbolic disc by a discrete group Γ acting freely and isometrically. The classification of higher-dimensional manifolds is of course much more difficult. In fact, due to the complexity of the fundamental group, a complete classification as in the case of surfaces is not possible in dimensions ≥ 4. In dimension 3 this argument does not apply, and the full classification of 3-manifolds has long been a dream of topologists. As a very special case, this problem includes the Poincaré Conjecture. In this article we report on remarkable recent work of Grisha Perelman [15]-[17], which may well have solved the classification problem for 3manifolds (in a natural sense). Perelman’s work is currently under intense investigation and scrutiny by many groups around the world. At this time, much of his work has been validated by experts in the area. Although at the moment it is still too soon to declare a definitive solution to the problem, Perelman’s ideas are highly original and of deep insight. Morever, his results are already being used by others in research on related topics. These circumstances serve to justify the writing of an article at this time, which otherwise might be considered premature. The work of Perelman builds on prior work of Thurston and Hamilton. In the next two sections we discuss the Thurston picture of 3-manifolds and the Ricci flow introduced and analyzed by Hamilton. For additional background, in particular on the Poincaré Conjecture, see Milnor’s Notices survey [14] and references therein. For much more detailed commentary and discussion on Perelman’s work, see [13].