# The Hypoelliptic Laplacian

Date: Fri, Sep 23, 2011

Location: PIMS, University of British Columbia

Conference: PIMS/UBC Distinguished Colloquium Series

Subject: Mathematics

Class: Scientific

### Abstract:

If X is a Riemannian manifold, the Laplacian is a second order elliptic operator on X. The hypoelliptic Laplacian L_b is an operator acting on the total space of the tangent bundle of X, that is supposed to interpolate between the elliptic Laplacian (when b -> 0) and the geodesic flow (when b -> \infty). Up to lower order terms, L_b is a weighted sum of the harmonic oscillator along the fibre TX and of the generator of the geodesic flow. In the talk, we will explain the underlying algebraic, analytic and probabilistic aspects of its construction, and outline some of the applications obtained so far.