Tel-Aviv University
School of Mathematical Sciences

Department Colloquium

Monday, November 23, 2015

Schreiber 006, 12:15

Emanuel Milman


Curvature-Dimension Condition for Non-Conventional Dimensions

Given an n-dimensional Riemannian manifold endowed with a probability density, we are interested in studying its isoperimetric, spectral and concentration properties. To this end, the Curvature-Dimension condition CD(K,N), introduced by Bakry and Emery in the 80's, is a very useful tool. Roughly put, the parameter K serves as a lower bound on the weighted manifold's "generalized Ricci curvature", whereas N serves as an upper bound on its "generalized dimension".  Traditionally, the range of admissible values for the generalized dimension N has been confined to [n,\infty]. In this talk, we present some recent developments in extending this range to N < 1, allowing in particular negative (!) generalized dimensions.

We will mostly be concerned with obtaining sharp isoperimetric inequalities under the Curvature-Dimension condition, identifying new one-dimensional model-spaces for the isoperimetric problem. Of particular interest is when curvature is strictly positive, yielding a new single model space (besides the previously known N-sphere and Gaussian measure): the sphere of (possibly negative) dimension N<1, which enjoys a spectral-gap and improved exponential concentration.

Time permitting, we will also discuss the case when curvature is only assumed non-negative. When N is negative, we confirm that such spaces always satisfy an N-dimensional Cheeger isoperimetric inequality and N-degree polynomial concentration, and establish that these properties are in fact equivalent. In particular, this renders equivalent various weak Sobolev and Nash inequalities for different exponents on such spaces.

Coffee will be served at 12:00 before the lecture
at Schreiber building 006