# Heat kernel approach to geometric analysis on metric measure spaces with Ricci curvature bounded below

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Shouhei Honda, Tohoku University

Let $(X, d, m)$ be a compact metric measure space with Ricci curvature bounded below in a synthetic sense, so-called an RCD space. The heat kernel allows us to embed the space into $L^2$ for any time $t>0$, and the pull-back $g_t$ defines a geometric flow on the space. The geometric flow $g_t$ has various applications to metric measure geometry, including a resolution of a conjecture raised by De Philippis-Gigli. In this talk we discuss Sobolev maps between RCD
spaces via $g_t$, instead of using Nash's embedding in the smooth setting. In particular, we discuss a compatibility with Korevaar-Schoen theory, and a nonlinear analogue of Cheeger's differentiability theorem for Sobolev functions.