Geometric hypoelliptic Laplacian and orbital integrals (after Bismut, Lebeau and Shen)

TL;DR

This paper discusses recent advances in the theory of hypoelliptic Laplacians, including explicit formulas for orbital integrals and a solution to Fried's conjecture relating analytic torsion and dynamic zeta functions.

Contribution

It presents new developments in hypoelliptic Laplacian theory, notably Bismut's explicit orbital integral formula and Shen's proof of Fried's conjecture for symmetric spaces.

Findings

Explicit formula for orbital integrals by Bismut

Solution of Fried's conjecture by Shen for symmetric spaces

Establishment of the equality between analytic torsion and dynamic zeta function value

Abstract

About 15 years ago, Bismut gave a natural construction of a Hodge theory for a hypoelliptic Laplacian acting on the total space of the cotangent bundle of a Riemannian manifold. This operator interpolates between the classical elliptic Laplacian on the base and the generator of the geodesic flow. We will describe recent developments of the theory of hypoelliptic Laplacians, in particular the explicit formula obtained by Bismut for orbital integrals and the recent solution by Shen of Fried's conjecture (dating back to 1986) for locally symmetric spaces. The conjecture predicts the equality of the analytic torsion and the value at 0 of the dynamic zeta function.