Event

Talk
The hypoelliptic Laplacian
Prof. Jean-Michel Bismut (Orsay)
1 Feb 2011, 15:45 – 16:45

On a Riemannian manifold, the hypoelliptic Laplacian is a canonical second order operator acting on the total space of the tangent bundle, which is supposed to interpolate between the classical Laplacian and the generator of the geodesic flow. By Laplacian, we mean here the Hodge de Rham Laplacian, the Hodge Dolbeault Laplacian, and more generally the square of any geometrically defined Dirac operator.

Up to lower order terms, the hypoelliptic Laplacian is a scaled sum of the harmonic oscillator along the fibre and of the generator of the geodesic flow. On Euclidean vector spaces, a version of this operator is known as the Kolmogorov operator, the model of the hypoelliptic operators studied by Hörmander.

On locally symmetric spaces, the hypoelliptic deformation is essentially isospectral. It leads to a geometric proof of the evaluation of semisimple orbital integrals, by a method unifying index theory and the trace formula. More generally, because the hypoelliptic deformation gives new degree of freedom unavailable in the elliptic theory, it leads to new results in index theory, unaccessible in the classical elliptic world.

Related to:

• Jean-Michel Bismut