# Mathematical Sciences Research Institute

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# Seminar

OT Programmatic Seminar: Geometric analysis on the space of metric measure space September 05, 2013 (03:45 PM PDT - 04:45 PM PDT)
Parent Program: Optimal Transport: Geometry and Dynamics MSRI: Simons Auditorium
Speaker(s) Karl-Theodor Sturm (Universität Bonn)
Description No Description
Video
The space $\mathbb X$ of all metric measure spaces $(X,d,m)$ plays an important r\^ole in image analysis, in the investigation of limits of Riemannnian manifolds and metric graphs as well as in the study of geometric flows that develop singularities. We show that the space $\mathbb X$ -- equipped with the $L^2$-distortion distance $\Delta\!\!\!\!\Delta$ -- is a challenging object of geometric interest in its own. In particular, we show that it has nonnegative curvature in the sense of Alexandrov. Geodesics and tangent spaces are characterized in detail. Moreover, classes of semiconvex functionals and their gradient flows on $\mathbb X$ are presented.