Geometry & Topology Seminar: The Fried conjecture and Hecke operators

Abstract: Ray-Singer analytic torsion is a secondary invariant of a compact manifold, in the sense that it is well-defined if a primary invariant, cohomology, is zero. The Ruelle dynamical zeta function associated to a flow on such a manifold encodes information on the periodic orbits of the flow. The Fried conjecture is the statement that the absolute value of the Ruelle zeta function at zero equals the analytic torsion of the manifold. If the manifold is a locally symmetric space, i.e. the double quotient of a real reductive Lie group by a maximal compact subgroup and a discrete, torsion-free subgroup, then a relevant notion of symmetry on this space is given by Hecke operators. In work in progress with Yanli Song and Polyxeni Spilioti, we investigate a version of Fried’s conjecture that encodes information on Hecke operators.

Host: Yanli Song