Analysis and PDE Seminar: Dynamical zeta functions at zero on surfaces with boundary

Seminar | November 25 | 4:10-5 p.m. | 939 Evans Hall

 Charles Hadfield, Rigetti Quantum Computing

 Department of Mathematics

The Ruelle zeta function counts closed geodesics on a Riemannian manifold of negative curvature. Its zeroes are related to Pollicott-Ruelle resonances which have been heavily studied in the setting of Anosov dynamical systems. In 2016, Dyatlov-Zworski proved an unexpected result relating the structure of the zeta function near the origin to the topology of the manifold. This extended a formula previously only known to hold in the constant curvature setting.

This talk will consider the situation where the manifold has boundary. A similar story can be told and the ultimate result extends the constant curvature setting (understood in 2001) to the variable curvature setting.

The microlocal tools required to consider this problem had been well developed in earlier papers (principally Dyatlov-Guillarmou 2016) and it remained to manipulate correctly relative cohomology (in this case à la Bott-Tu) in order to understand the space of 1-form Pollicott-Ruelle resonances.