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Convergence of Riemannian manifolds and Metric Measure Spaces

Connections for Women on Optimal Transport: Geometry and Dynamics August 22, 2013 - August 23, 2013

August 22, 2013 (09:30AM PDT - 10:30AM PDT)
Speaker(s): Christina Sormani (CUNY, Graduate Center)
Location: MSRI: Simons Auditorium
Video

Abstract Recent advances in Geometric Analysis and in Optimal Transport have provided new insight into both fields. In Geometric Analysis we study the limits of sequences of Riemannian manifolds and produce limit spaces with a variety of structures. With lower bounds on Ricci curvature, Cheeger-Colding combined Gromov's Compactness Theorem and ideas of Fukaya to show that sequences of Riemannian manifolds with uniform lower Ricci curvature bounds have metric measure limits. They show these limits are metric measure spaces with a doubling measure that has many of the properties of a Riemannian manifold with a lower Ricci curvature bound. Sturm and Lott-Villani then generalized the notion of a lower Ricci curvature bound to metric measure spaces using notions from Optimal Transport. Sturm also defined a new notion of metric measure convergence of metric measure spaces based upon the Optimal Transport notion of a Wasserstein distance between probability measures. Other notions of convergence provide more structure on the limit spaces and do not require the lower Ricci curvature bound. One notion, with applications in General Relativity, is the intrinsic flat distance introduced in joint work with Stefan Wenger (building upon work of Ambrosio-Kirchheim) produces integer weighted countably $\mathcal{H}^m$ rectifiable limit spaces. A newer notion soon to be introduced in joint work with Guofang Wei (building upon work of Solorzano and on Sturm's metric measure convergence) preserves the structure of the tangent bundle. We provide a brief survey of these notions with examples. The details in the papers can be understood after the audience has attended the introductory workshop next week.
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