Functional and geometric rigidities of RCD spaces and bi-Lipschitz Reifenberg's theorem in metric spaces

This thesis is divided in two independent parts. In the first one we will present the results concerning the theory of metric measure spaces satisfying synthetic Ricci curvature lower bounds, obtained in [131, 132, 185]. The focus will be on the extension of some analytical tools to this setting and on the derivation on both geometric and analytical rigidities and almost-rigidities. In the second part instead we will present the works in [130] and [209] about the bi-Lipschitz version of Cheeger-Colding’s intrinsic Reifenberg’s theorem in abstract metric spaces