Geometric measure theory on singular spaces with lower Ricci bounds and the isoperimetric problem

The aim of this talk is to present some recent developments of Geometric Measure Theory on non smooth spaces with lower Ricci Curvature bounds, mainly related to the first and second variation formula for the area, and their applications to the isoperimetric problem on non compact manifolds. The reinterpretation of some classical results in Geometric Analysis in a low regularity setting, combined with the compactness and stability theory for spaces with lower curvature bounds, leads to a series of new geometric inequalities for smooth, non compact Riemannian manifolds. The talk is based on joint works with Andrea Mondino, Gioacchino Antonelli, Enrico Pasqualetto and Marco Pozzetta.