An Overview on Calculus and Heat Flow in Metric Measure Spaces and Spaces with Riemannian Curvature Bounded from Below

摘要

In this paper we provide a short overview of the recent work in collaboration with N. Gigli and G. Savaré on the development of a differential calculus in metric measure spaces based on optimal transportation tools. This leads to a new approach to the theory of Sobolev spaces on metric measure spaces. In the second part we provide an application of these ideas, still obtained in collaboration with N. Gigli and G. Savaré, on the study of a synthetic notion of "Riemannian Ricci bound from below" which enjoys several nice properties including stability with respect to measured Gromov - Hausdorff convergence.