# Geometric Analysis and Topology Seminar

#### The Hypo Elliptic Laplacian in Real and Complex Geometry

**Speaker:**
Jean-Michel Bismut, Orsay

**Location:**
Warren Weaver Hall 517

**Date:**
Friday, April 18, 2014, 11 a.m.

**Synopsis:**

The hypoelliptic Laplacian is supposed to be a deformation of a standard elliptic Laplacian, that acts on the total space of the tangent bundle of a Riemann manifold, and interpolates between the elliptic Laplacian and the generator of the geodesic flow. Its construction involves a deformation of the underlying geometric and analytic structures. The hypoelliptic Laplacian is neither elliptic nor self-adjoint in the classical sense, but it is self-adjoint with respect to a Hermitian form of signature \((\infty,\infty)\). On locally symmetric spaces, the hypoelliptic deformation preserves the spectrum of the elliptic Laplacian. I will explain its construction in the case of the circle, its applications to Selberg's trace formula, and also to the proof of a Riemann-Roch-Grothendieck theorem in Bott-Chern cohomology.