# Geometric Analysis and Topology Seminar

#### Hypoelliptic Laplacian and the trace formula

**Speaker:**
Jean-Michel Bismut, Institut de Mathematique d'Orsay

**Location:**
Warren Weaver Hall 1302 (Note the nonstandard location)

**Date:**
Friday, April 12, 2024, 10 a.m.

**Synopsis:**

The heat equation method in index theory gives an explicit local formula for the index of a Dirac operator. Its Lagrangian counterpart involves supersymmetric path integrals.

Similar methods can be developed to give a geometric formula for semisimple orbital integrals associated with the Casimir operator of a reductive group, this computation being related to Selberg’s trace formula. The analogue of the heat equation method is now a suitable deformation of the Laplacian by a family of Fokker-Planck operators L_b, b>0, that interpolates between the Casimir operator and the geodesic flow.

We will also explain results obtained by Shu SHEN and ourselves when the Casimir operator is replaced by an arbitrary element of the center of the Lie algebra.