Veranstaltung

Vortrag
Analytic torsion, the hypoelliptic Laplacian and the trace formula
Jean-Michel Bismut (Paris Sud)
9.12.2016, 9:00 Uhr – 10:00 Uhr

On a compact odd dimensional manifold equipped with a flat unimodular vector bundle, the Reidemeister torsion is a combinatorial invariant. Analytic torsion is a spectral invariant of the corresponding Hodge Laplacian. The Cheeger-Müller theorem asserts that these two invariants coincide.
A proof was given by Zhang and ourselves based on the Witten deformation of Hodge theory that is associated with a Morse function f. In a first part of the talk, we will review some aspects of the proof.
In a second part, when replacing f by the energy functional on the loop space, the corresponding Hodge Laplacian is a hypoelliptic operator acting on the total space of the tangent bundle. A result by Lebeau and ourselves asserts that elliptic and hypoelliptic torsion coincide.
If one pushes the deformation parameter to infinity, on manifolds with negative curvature, one obtains a formal version of the Fried conjecture, that relates the values at 0 of the Ruelle zeta function associated with the geodesic flow and analytic torsion.
In a last part of our talk, we will describe our approach to Selberg’s trace formula as a form of a Lefschetz formula. Using the corresponding formulas, Shu Shen was able to provide a complete proof of the Fried conjecture on compact locally symmetric spaces.