The purpose of the lectures is to give an introduction to the hypoelliptic Laplacian and to the results which have been obtained so far by this method.

The hypoelliptic Laplacian gives a universal interpolation between the Laplacian and the generator of the geodesic flow through a family of hypoelliptic operators acting on a bigger space than the original Laplacian. There are as many hypoelliptic Laplacians as there are Laplacians.

To this deformation of operators corresponds an interpolation of dynamics, that interpolates between Brownian motion, with the most irregular paths possible, and the geodesics which have the most regular trajectories.

Explicit geometric formulas for certain orbital integrals will appear as consequences. These give in arbitrary dimensions the analogues of the geometric formula obtained by Selberg in the case of Riemann surfaces.

The lectures are given by Jean-Michel Bismut.  Additionally, he will give a Colloquium talk for the general audience.