Discrete Curvature and Applications

Inspired by exciting developments in optimal transport and Riemannian geometry, several independent groups have formulated notions of (Ricci) curvature in discrete spaces. I will mention briefly some of these approaches, results, examples and open problems. An interesting by-product is the result (obtained jointly with Klartag, Kozma, and Ralli) that the Cheeger inequality — relating the spectral gap to the edge-isoperimetric constant — is tight for the class of abelian Cayley graphs.