30 may 2017 -- 14:30
Aula Dal Passo, Dip.Matematica, Università "Tor Vergata", Roma
Abstract.
Lower Ricci curvature bounds play a crucial role in many geometric and functional inequalities. In addition to the direct characterization in terms of the curvature tensor in Riemannian geometry, two other important approaches have been introduced to capture the analytic properties of the Ricci curvature: a first one, due to Bakry-Emery, is based on the notion of Gamma-calculus and is strictly related to the behaviour of the Heat flow. A second, more recent, approch have been proposed by Lott-Villani and Sturm: it involves the geometric properties of optimal transport and Entropy functionals. We will present the basic ideas behind the two viewpoints in a simple Euclidean setting, and we wil show how the metric-variational theory of gradient flows provides a key point to prove their equivalence, leading to a beautiful and rich theory in general metric-measure spaces.