Hypoelliptic Laplacian, Brownian motion and the geodesic flow (Jean-Michel Bismut, Université Paris-Sud, Orsay)

16.03.2018 10:00

If $X$ is a compact Riemannian manifold, and if $\mathcal{X}$ is the total space of its tangent bundle, there is a canonical interpolation between the classical Laplacian of $X$ and the generator of the geodesic flow by a family of hypoelliptic operators $L^{X}_{b}\vert_{b>0}$ acting on $\mathcal{X}$. This interpolation extends to all the classical geometric Laplacians. There is a natural dynamical system counterpart, which interpolates between Brownian motion and the geodesic flow.

The hypoelliptic deformation preserves certain spectral invariants. like the Ray-Singer torsion, the holomorphic torsion and the eta invariants. In the case of locally symmetric spaces, the spectrum of the original Laplacian remains rigidly embedded in the spectrum of its deformation. This property has been used in the context of Selberg's trace formula.

In this lecture, I will give the structure of the hypoelliptic Laplacian. I will also describe a natural construction of the hypoelliptic Laplacian as a nonstandard Hodge Laplacian, and explain its connections with dynamical systems and an equation by Langevin.

Lieu

Bâtiment: Battelle

Séminaire "Groupes de Lie et espaces des modules"

entrée libre