Aim of the course is to provide an introduction to the world of synthetic description of lower Ricci curvature bounds, which has seen a tremendous amount of activity in the last decade: by the end of the lectures the student will have a clear idea of the backbone of the subject and will be able to navigate through the relevant literature.

We shall start by studying Sobolev functions on metric measure spaces and the notion of heat flow. Then following, and generalizing, the intuitions of Jordan-Kinderlehrer-Otto we shall see that such heat flow can be equivalently characterized as gradient flow of the Cheeger-Dirichlet energy on L2 and as gradient flow of the Boltzmann-Shannon entropy w.r.t. the optimal transportation metric W2. This provides a crucial link between the Lott-Villani-Sturm (LSV) condition and Sobolev calculus on metric measure spaces and, in particular, it justifies the introduction of `infinitesimally Hilbertian' spaces as those metric measure structures for which W1;2(X) is a Hilbert space. By further developing calculus on these spaces we shall see that on infinitesimally Hilbertian spaces satisfying the LSV condition (these are called Riemannian curvature dimension spaces, or RCD for short) the Bochner inequality holds.

We shall then discuss more sophisticated calculus tools, such as the concept of differential of a Sobolev function, that of vector field on a metric measure spaces and the notion of Regular Lagrangian Flow on RCD spaces.

We shall finally see how these are linked to the lower Ricci curvature bound - most notably we shall prove the Laplacian comparison theorem - and finally how they can be used to prove a geometric rigidity result like the splitting theorem for RCD spaces. It is worth to notice that such statement gives new information - compared to those available through Cheeger-Colding's theory of Ricci-limit spaces - even about the structure of smooth Riemannian manifolds.