Some properties of non-smooth spaces with Ricci curvature lower bounds

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 and Colding in the ’90ies who investigated the structure of the spaces arising as Gromov Hausdorff limits of smooth Riemannian manifolds satisfying Ricci curvature lower bounds.

A completely new approach via optimal transportation was proposed by Lott-Villani and Sturm almost ten years ago; with this approach one can a give a precise meaning of what means for a non smooth space to have Ricci curvature bounded from below by a constant. This approach has been refined in the last years by a number of authors (let me quote the fundamental work of Ambrosio-Gigli-Savaré among others) and a number of fundamental tools have now been established (for instance the Bochner inequality, the splitting theorem, etc.), permitting to give further insights in the theory. In the seminar I will give an overview of the topic.