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Geometry and Topology of 3-dimensional spaces

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When he introduced what is now known as Riemannian geometry, Riemann vastly generalized what had come before. He also explicitly separated geometry from the topology of the underlying space; geometry became an extra structure given to a topological space. Around the turn of the 20th century Poincare put topology on an independent footing as a subdiscipline of mathematics. He also formulated a question which he considered as central. That question was to characterize the simplest 3-dimensional space, the 3-sphere. Poincare’s conjecture was generalized to include a classification of all 3-dimensional spaces, technically, compact 3-manifolds, and even higher dimensional spaces (but that is another story). In the 1980s Thurston conjectured that 3-dimensional spaces could be classified, and Poincare’s original conjecture could be resolved, by uniting homogeneous Riemannian geometry and topology in dimension 3, undoing, in a sense for 3-dimensional spaces, Riemann’s separation of topology and geometry. Around the same time, Richard Hamilton proposed a method of attacking Thurston’s conjecture. His idea was to use a version of the heat equation for Riemannian metrics to evolve any starting Riemannian metric on the space under consideration to a nice Riemannian metric. Recently, Perelman has given a complete proof of Thurston’s conjecture along the general lines envisioned by Hamilton.

The talk will introduce ways of thinking about the topology 3-dimensional spaces and the homogeneous geometries that come into play. The talk will describe the version of the heat-type equation, called the Ricci flow equation, for Riemannian metrics. It will then discuss the analytic and geometric approaches and ideas and some of the difficulties that one must overcome in order to arrive at a positive resolution of all these conjectures by these methods.

This talk is part of the Rouse Ball Lectures series.

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