Analytic torsion and dynamical zeta function on locally symmetric spaces

Analytic torsion and dynamical zeta function on locally symmetric spaces

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Shu Shen, Humboldt-Universität
Fine Hall 314

The relation between the spectrum of the Laplacian and the closed geodesics on a closed Riemannian manifold is one of the central themes in differential geometry. Fried conjectured that the analytic torsion, which is an alternating product of regularized determinants of the Hodge Laplacians, equals the zero value of the dynamical zeta function. In the first part of the talk, we will give a formal proof of this conjecture based on the path integral and Bismut-Goette's V-invariants. In the second part, we will give the rigorous arguments in the case where the underlying manifold is a closed locally symmetric space. The proof is based on the Bismut's formula for semisimple orbital integrals.