Featured Speaker: Masha Gordina (UConn)

Title: Uniform volume doubling and functional inequalities on Lie groups

Abstract: On a compact Lie group with a left-invariant Riemannian metric, many important functional inequalities for the Laplacian (such as Poincar\'e inequality, parabolic Harnack inequality, etc.)  can be proved using only the volume doubling property.  That is,  constants in these inequalities can be controlled by the doubling constant of the metric; this can be strictly more powerful than classical techniques involving Ricci curvature lower bounds.  It can happen that there is a uniform bound on the doubling constants of all left-invariant metrics on a given Lie group; such a group is called uniformly doubling.  In such a case, the implicit constants in the functional inequalities will also be uniformly bounded over all left-invariant metrics.  We show that this happens for the special unitary group SU(2), via explicit uniform volume estimates and describe the consequences (heat kernel estimates, Weyl counting function etc)

This is joint work with Nate Eldredge and Laurent Saloff-Coste.