SeminarsHamilton’s Ricci Flow and the Differentiable Sphere Theorem (I)
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A central problem in Riemannian geometry concerns the classification of manifolds of positive sectional curvature. In 1951, H.E. Rauch introduced the notion of curvature pinching for Riemannian manifolds and posed the question of whether a simply connected manifold Mn whose sectional curvatures all lie in the interval (1, 4] is necessarily homeomorphic to the sphere Sn. This was proven by using comparison techniques due to M. Berger and W. Klingenberg around 1960. However, this theorem leaves open the question of whether M is diffeomorphic to Sn. This conjecture is known as the Di§erentiable Sphere Theorem. The goal of this seminar is to present this work via Hamilton’s Ricci flow due to S. Brendle and R. Schoen around 2009.
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2015-01-08
14:00:00 - 17:00:00
308 , Mathematics Research Center Building (ori. New Math. Bldg.)
A central problem in Riemannian geometry concerns the classification of manifolds of positive sectional curvature. In 1951, H.E. Rauch introduced the notion of curvature pinching for Riemannian manifolds and posed the question of whether a simply connected manifold Mn whose sectional curvatures all lie in the interval (1, 4] is necessarily homeomorphic to the sphere Sn. This was proven by using comparison techniques due to M. Berger and W. Klingenberg around 1960. However, this theorem leaves open the question of whether M is diffeomorphic to Sn. This conjecture is known as the Di§erentiable Sphere Theorem. The goal of this seminar is to present this work via Hamilton’s Ricci flow due to S. Brendle and R. Schoen around 2009.
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