Curvature, Sphere Theorems, and the Ricci Flow

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• Prof. Simon Brendle (Stanford)
• Monday 19 November 2012, 17:00-18:00
• MR2, Centre for Mathematical Sciences.

If you have a question about this talk, please contact Shepherd-Barron.

To be followed by a reception in the central core

In 1926, Hopf proved that any compact, simply connected Riemannian manifold with constant curvature 1 is isometric to the standard sphere. Motivated by this result, Hopf posed the question of whether a compact, simply connected manifold with suitably pinched curvature is topologically a sphere. This question has been studied by many authors over the past six decades, a milestone being the Topological Sphere Theorem proved by Berger and Klingenberg in 1960.

In this lecture, I will discuss the history of this problem, and describe the proof (joint with R. Schoen) of the Differentiable Sphere Theorem. This theorem classifies all manifolds with 1/4-pinched curvature up to diffeomorphism. The distinction between homeomorphism and diffeomorphism is significant in light of the exotic spheres constructed by Milnor; the proof uses the Ricci flow technique pioneered by Hamilton.

This talk is part of the CMS Colloquia series.

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