Event № 823

In this talk I will discuss a general method for obtaining sharp lower bounds for the constants associated with certain functional inequalities on weighted Riemannian manifolds, whose (generalized) Ricci curvature is bounded from below. Using this method we prove new sharp lower bounds for the Poincar\'{e} and log-Sobolev constants.The first major ingredient of the method is the localization theorem which has recently been developed by B. Klartag. The proof of this theorem is based on optimal transport techniques; it leads to a characterization of the constants associated with the pertinent functional inequalities as solutions to a mixed optimization problem over a set of functions and a set of measures, both of which are supported in R.The second major ingredient of the method is a reduction of the optimization problem to a subclass of measures, which are referred to as 'model-space' measures. This reduction is based on functional analytical arguments, in particular an important characterization of the extreme points of a subset of measures, and corollaries of the Krein-Milman theorem.The third ingredient of the method is an explicit solution to the reduced optimization problem over the subset of 'model-space' measures; this solution is approached by ad-hoc methods.In my talk I will discuss each of the ingredients of the method, with an emphasis on solving the problem of finding the sharp lower bound for the Poincar\'{e} constant.Our results show that within a certain range of the pertinent parameters (specifically 'the effective dimension N'), the characterization of the sharp lower bound for the Poincar\'{e} constant is of an utterly different nature from what has been known to this date.