Abstract: After a brief introduction to the synthetic notions of Ricci curvature lower bounds in terms of optimal transportation, due to Lott-Sturm-Villani, I will discuss some applications to smooth Riemannian manifolds (joint work with Fabio Cavalletti, SISSA).

These include: rigidity/stability of the Levy-Gromov inequality and an almost euclidean isoperimetric inequality motivated by the celebrated Perelman’s Pseudo-Locality Theorem for Ricci flow.