Mathematics

We review a construction we proposed in the setting of metric measure spaces of parallel transport, for which we obtained both existence and uniqueness results. We work in the setting on non-collapsed RCD(K,N) spaces, which is a synthetic generalization of the class of Riemannian manifolds with a lower bound by K on the Ricci curvature and upper bound on the dimension by N, with the N-dimensional Hausdorff measure as reference measure. The problem is formulated as a transport of a measurable vector field (in the sense of normed modules as introduced by Gigli) along 'almost every' integral curve of a time-dependent Sobolev vector fields. The flow of such non-smooth vector fields is expressed in terms of the notion of Regular Lagrangian flows, defined in such a nonsmooth setting by Ambrosio and Trevisan.The seminar is based on a joint with N. Gigli and E. Pasqualetto.