Appears in collection : Summer School 2021 - Curvature Constraints and Spaces of Metrics

The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the '80ies and was pushed by Cheeger-Colding in the ‘90ies, who investigated the structure of spaces arising as Gromov-Hausdorff limits of smooth Riemannian manifolds satisfying Ricci curvature lower bounds. A completely new approach based on Optimal Transport was proposed by Lott-Villani and Sturm around ten years ago; via this approach, one can give a precise sense of what means for a non-smooth space (more precisely for a metric measure space) to satisfy a Ricci curvature lower bound and a dimensional upper bound. This approach has been refined in the last years by a number of authors (most notably Ambrosio-Gigli- Savarè) and a number of fundamental tools have now been established, permitting to give further insights in the theory and applications which are new even for smooth Riemannian manifolds. The goal of the lectures is to give an introduction to the theory and discuss some of the applications.