After a brief introduction to the synthetic notions of Ricci curvature lower bounds in terms of optimal transportation, due to Lott-Sturm-Villani, I will discuss some applications to smooth Riemannian manifolds. These include: rigidity and stability of Levy-Gromov inequality, an almost euclidean isoperimetric inequality motivated by the celebrated Perelman’s Pseudo-Locality Theorem for Ricci flow, and some geometric properties of quotients of smooth manifolds having Ricci curvature bounded below.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/7138