Analysis and Mathematical Physics

There is a celebrated connection between minimal (or constant mean curvature) hypersurfaces and Ricci curvature in Riemannian Geometry, often boiling down to the presence of a Ricci term in the second variation formula for the area. The first goal of the talk will be to discuss the validity of an analogous principle for non smooth metric measure spaces with Ricci bounded from below in synthetic sense. In the second part, I will illustrate some applications of Geometric Measure Theory in ambient spaces with low regularity to classical questions in Geometric Analysis on smooth Riemannian manifolds.