Existence and uniqueness of parallel transport on non-collapsed RCD(K,N) spaces

The thesis mainly contains the construction of parallel transport on non-collapsed spaces, obtained in the work "Parallel transport on non-collapsed RCD(K,N)", done in collaboration with my supervisor N. Gigli and E. Pasqualetto. We obtain both existence and uniqueness results. Our theory covers the case of geodesics and, more generally, of curves obtained via the flow of sufficiently regular time dependent vector fields. In this generality, we don't study parallel transport along a single such curve, but along a generic collection of such integral curves. The notion of flow under consideration is the one of regular Lagrangian flow, after the axiomatization in the nonsmooth setting by Ambriosio and Trevisan. A preliminary introduction on calculus on metric measure spaces and the theory of flows of Sobolev vector fields, both in euclidean and nonsmooth setting, is also included.