Mathematics

Aim of this talk is to propose a notion of parallel transport for  spaces, which are metric measure spaces satisfying lower Ricci curvature bounds in a generalised sense. The first part of the seminar will be devoted to the description of the calculus tools that are already available in such nonsmooth framework and to the development of some new functional spaces that will constitute the setting for our theory. In the second part we will give the definition of parallel transport and prove both its well-posedness and uniqueness. Some geometric consequences of the existence of this concept of parallel transport will be shown to hold, in primis the constant dimension of the underlying   spaces. As a last step, we will provide existence of the parallel transport for a special class of finite-dimensional spaces. The existence for general  spaces is still an open problem. This is a joint work with Nicola Gigli.