Mathematics

In this seminar, we give an overview of the functional analytic objects involved in the problem of the parallel transport on metric measure spaces satisfying lower Ricci bounds.
In Riemannian geometry, given a smooth curve and a tangent vector at the initial point, one can construct a smooth vector field along the curve having zero covariant derivative along the curve and the prescribed initial condition.
In the first part of the talk, we introduce the objects that play the role of the non smooth counterpart of curves and tangent vectors, namely test plans and vector fields ($L^2$^{^^2} integrable with respect to the reference measure).
In the second part, we discuss what are the non smooth counterpart of vector fields along a curve in this context, that are Sobolev vector fields along a test plan.
We discuss two possible definitions of such objects, either in a distributional way or as the closure of  “test” vector fields along a test plan in the Sobolev norm.