The APDE seminar on Monday, 11/25 will be given by Charles Hadfield in Evans 939 from 4:10 to 5pm.

Title: Dynamical zeta functions at zero on surfaces with boundary

Abstract: 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.

The APDE seminar on Monday, 11/18 will be given by Dean Baskin in Evans 939 from 4:10 to 5pm.

Title: Asymptotics of the radiation field on cones

Abstract:
Radiation fields are rescaled limits of solutions of wave equations near “null infinity” and capture the radiation pattern seen by a distant observer.  They are intimately connected with the Fourier and Radon transforms and with scattering theory.  We consider the wave equation on a product cone and show that the associated radiation field has an asymptotic expansion; the exponents seen in this expansion are the resonances of the hyperbolic cone with the same link.  This talk is based on joint work with Jeremy Marzuola (building on prior work with Andras Vasy and Jared Wunsch).