Mathematics

The course is devoted to the presentation of some classical and modern results on the deep relation between lower bounds on the Ricci curvature on a Riemannian manifold, the geometry of the space, and its isoperimetry.
Isoperimetry here is intended in a broad sense, comprising the study of isoperimetric problems on Riemannian manifolds in connection with the validity of functional, isoperimetric and spectral inequalities.

Basic facts and preliminaries in Riemannian Geometry are recalled, together with the basic theory of functions of bounded variations and of submanifolds on Riemannian manifolds. Then some classical theorems in comparison geometry are presented, focusing on the Bishop-Gromov volume comparison theorem. Next, selected topics on the isoperimetry of manifolds with Ricci lower bounds are presented, discussing the relations between geometric, variational, and functional analytic aspects. Time permitting, treated topics include: Poincaré and isoperimetric-type inequalities, differential inequalities on the isoperimetric profile, Lévy-Gromov isoperimetric inequality, isoperimetry of manifolds with nonnegative Ricci, Cheeger-Buser isoperimetric and spectral inequalities on the first eigenvalue of the Laplacian, relations with Optimal Transport and Brunn-Minkowski and isoperimetric inequalities.