The hypoelliptic Laplacian in real and complex geometry

IMS Distinguished Lecture Series
The hypoelliptic Laplacian in real and complex geometry

Jean-Michel Bismut
Université Paris-Saclay, France

The purpose of the talk is to explain the construction of the hypoelliptic Laplacian in real and complex geometry, and to review some applications that have been obtained so far.

The hypoelliptic Laplacian is a deformation of the Laplacian by a family of operators acting on a bigger space (which is often the total space of the tangent bundle) that interpolates between the Laplacian and the generator of the geodesic flow. These new operators are geometric Fokker-Planck operators. The geometric content of the deformation is specific to the geometry that is considered.

There are are different motivations for this construction: either because of the interpolation property itself, and also because moving away from the original space allows for the lifting of obvious geometric obstructions, like the existence of a Kähler metric.

In the talk, we will illustrate the construction in the context of de Rham theory and of Dolbeault theory.