Georgia Institute of Technology College of Sciences

 It is well known that isoperimetric type inequalities can imply concentration inequalities, but the reverse is not true generally. However, recently E Milman and M Ledoux proved that under some convex assumption of the Ricci curvature, the reverse is true in the Riemannian manifold setting. In this talk, we will focus on the semigroup tools in their papers. First, we introduce some classic methods to obtain concentration inequalities, i.e. from isoperimetric inequalities, Poincare's inequalities, log-Sobolev inequalities, and transportation inequalities. Second, by using semigroup tools, we will prove some kind of concentration inequalities, which then implies linear isoperimetry and super isoperimetry.