This advanced course aims at giving an introduction to the theory of non-smooth spaces with Ricci curvature bounded from below. Focus will be on basic definitions and stability results. In particular, the following concepts will be defined during the course:
- Sobolev functions on metric measure spaces
- Gradient flows in metric spaces
- Heat flow on metric measure spaces
- CD and RCD spaces
- Differential calculus on metric measure spaces
and for each of these corresponding stability statements will be given. Such stability statements will be interpreted mostly via a combination of De Giorgi’s Gamma convergence and Gromov’s convergence of geometric structures (both will be defined in the course), whence the title. I do not aim at giving full proofs of every statement, as the focus is instead to show how the theory of RCD is built. For this reason, often I shall either present the proof under simplifying assumptions (eg compactness) or just give a rough idea of the argument. The course is based on these lecture notes, where detailed bibliographical references can be found.