PIMS/UBC Mathematical Sciences Early Career Award Colloquium: Mathav Murugan

The energy image density property arises in two seemingly distinct lines of research. It first appeared in Malliavin’s proof of Hörmander’s hypoellipticity theorem and has since played a central role in the study of regularity properties of probability laws of solutions to stochastic differential equations. More recently, it has emerged in a different context: the study of almost-everywhere differentiability of Lipschitz functions on metric measure spaces, inspired by Rademacher’s theorem and initiated by Cheeger’s seminal work on PI spaces.

The goal of this talk is twofold: to provide a gentle introduction to these two research directions and to explain how a unified perspective on the energy image density property leads to the resolution of a conjecture of Bouleau and Hirsch from 1986. This is joint work with Sylvester Eriksson-Bique.