Geometric Analysis and Topology Seminar

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Hypoelliptic Laplacian and the Fried conjecture

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

Location: Warren Weaver Hall 512

Date: Friday, January 23, 2026, 11 a.m.

Synopsis:

If X is a compact manifold with negative curvature, Fried introduced his zeta function, a dynamical zeta function associated with the geodesic flow. The Fried conjecture asserts that its value at 0 can be properly defined, and is equal to the Reidemeister torsion, a combinatorial invariant of the manifold.

The purpose of the talk will be to give the proper perspective to this conjecture. Among others, I will explain the results by Giulietti-Liverani-Pollicott, and Dyatlov-Zworski, who showed that the Fried zeta function has the required properties.

I will also describe the results that have been recently been obtained by Shu Shen and myself, which show that the zeta functions of the hypoelliptic Laplacian (a deformation. of the classical Hodge Laplacian) converge to the Fried zeta function as the deformation parameter tends to +∞.