Mathematics

In the first part of my seminar I will explain what it means for metric measure spaces to have Ricci curvature bounded from below and how to make calculus on them. In particular I will introduce the first order differential structure of general metric measure spaces.In the second part I will focus on the problem I am studying, which is the generalization in the setting of metric measure spaces of a result true in Riemannian geometry. This theorem, due to Bochner, allows to characterize the flat torus of dimension N among all the Riemannian manifolds with non negative Ricci curvature and dimension N via the study of the first cohomology group.