## Some relations between $CD^*(K,N)$, the dimensional Bakry-Emery condition and the EVI property of gradient flows

#### Andrea MondinoScuola Normale Superiore

The reduced curvature dimension condition $CD*(K,N)$, with $K \in {\mathbb R}$ and $N\geq 1$ was introduced by Bacher-Sturm as a local modification of the $CD(K,N)$ codition of Lott-Sturm-Villani. It defines, by analyzing convexity properties of a special functional called entropy, what does it mean for a non smooth metric space to have Ricci curvature bounded from below by $K$ and dimension bounded above by $N$. Another way, called Bakry-Emery condition, for defining this concept is via the Bochner identity. A third way is to analyze the contractivity properties of the gradient flow of the entropy, or more precisely the EVI property. In the seminar we discuss some relations between these three points of view.

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