Cohomology and other analytical aspects of RCD spaces

This thesis is primarily devoted to the study of analytic and geometric properties of metric measure spaces with a Ricci curvature bounded from below. The first result concerns the study of how a hypothesis on the Hodge cohomology affects the rigidity of a metric measure space with non negative Ricci curvature and finite dimension: we prove that if the dimension of the first cohomology group of a RCD∗(0,N) space is N, then the space is a flat torus. This generalizes a classical result in Riemannian geometry due to Bochner to the non-smooth setting of RCD spaces. The second result provides a direct proof of the strong maximum principle on finite dimen- sional RCD spaces mainly based on the Laplacian comparison of the squared distance.