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A colloquium on discrete curvature

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Discrete Ricci curvature via convexity of the entropy

Jan Maas

Slides are available by following the link.

We present a notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm and Villani for geodesic measure spaces. In the discrete setting the role of the Wasserstein metric is taken over by a different metric, having the property that continuous time Markov chains are gradient flows of the entropy. Among the results are discrete analogues of results by Bakry--Emery and Otto--Villani.

This is joint work with Matthias Erbar.