Linear Partial Differential Equations on Singular Spaces

I am interested in the rate of decay of waves near their source in different geometric settings. These rates are closely related to resolvent estimates: here we seek estimates on outgoing solutions to the Helmholtz equation. Relevant properties of the resolvent include bounds when the spectral parameter lies on the real axis, as well as estimates on existence or nonexistence of resonance poles when one continues to a non-physical half-plane. Many of my current and recent projects are centered around situations where non-smooth geometry creates diffractive effects, which interestingly complicates the asymptotic behavior of waves. With Jeffrey Galkowski, I recently obtained sharp estimates on the constants arising in the resolvent bounds on the real axis, in the smooth setting when the classical flow is non-trapping (all trajectories escape to infinity). In an ongoing project with Euan Spence and David Lafontaine, we consider the contrasting situation in which trajectories may be quite badly trapped, but nonetheless find that for a large subset of the real axis, the estimates are not too badly affected: growth there is polynomial, even though the overall estimate may have exponential growth in the spectral parameter. In an ongoing project with Oran Gannot, we have obtained results on where resonances engendered by trapping arising from diffractive effects may lie, in the case where semiclassical singularities encounter a non-smooth potential; we hope to refine these estimates in future work to show existence of such resonances in a variety of geometries. In previous work with Luc Hillairet, I had obtained a number of results about resonances arising from diffractions by cone points (or corners of polygons), showing that they arise along logarithmic curves in the complex plane. Gannot and I hope to establish analogous results in the semiclassical world. In a related direction of future work (with Dean Baskin and Oran Gannot) I will study the effects of the singularity of the Coulomb potential on the structure of the Dirac propagator; here again, diffractive effects play an important role. Another recent project in the area of diffraction of waves (joint with Galkowski) focused on the question of which cones do not diffract waves at all. We conjecture that the only examples are orbifolds, i.e., cones over spherical space forms. We are able to prove this for analytic cones in two or three dimensions; in higher dimensions we show that any analytic cone that fails to diffract must be a cone over a Zoll manifold. I will also continue a research direction joint with Dean Baskin and Andras Vasy to study the asymptotic behavior of waves propagating on certain curved spacetimes such as arise in the theory of general relativity. The goal is to understand the long-time behavior of the radiation pattern observed far away from a source.