*Preprint*

**Inserted:** 15 may 2012

**Last Updated:** 15 may 2012

**Year:** 2012

**Notes:**

To the memory of Enrico Magenes, whose exemplar life, research and teaching shaped generations of mathematicians.

**Abstract:**

We provide a quick overview of various calculus tools and of the main results concerning the heat flow on compact metric measure spaces, with applications to spaces with lower Ricci curvature bounds.

Topics include the Hopf-Lax semigroup and the Hamilton-Jacobi equation in metric spaces, a new approach to differentiation and to the theory of Sobolev spaces over metric measure spaces, the equivalence of the $L^2$-gradient flow of a suitably defined ``Dirichlet energy'' and the Wasserstein gradient flow of the relative entropy functional, a metric version of Brenier's Theorem, and a new (stronger) definition of Ricci curvature bound from below for metric measure spaces. This new notion is stable w.r.t. measured Gromov-Hausdorff convergence and it is strictly connected with the linearity of the heat flow.

**Keywords:**
Optimal transport, entropy, Ricci curvature, Heat Flow

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