This project will study partial differential equations on singular spaces, with an emphasis on spectral and scattering theory. The propagation of waves on smoothly varying spaces is well understood in many respects, but the interaction with singularities--which might range from boundaries, to corners, to structures "at infinity"--presents many open problems. One of the project's components is related to the understanding of wave propagation on Kerr spacetimes (i.e., near rotating black holes). The principal investigator will study the distribution of quasi-normal modes, the ways that the black hole may "ring" with damped oscillations. The initial goal of this project is to obtain a rigorous description of the exponential decay rate of high frequency modes. The project also includes work on problems of local energy decay on Riemannian manifolds, for which the geometry of the infinite ends turns out to have a profound effect on the low frequency phenomena that may dominate energy decay. Another aspect of wave propagation of interest to the principal investigator is the infinite-speed propagation occurring in solutions to the Schrodinger equation. In settings in which geometric rays are trapped in a bounded region, little is known about the regularity of solutions. The principal investigator is intent on studying the effects of these trapped rays, as well as the effects of geometric singularities such as cone points on the propagation.
Geometry influences the behavior of solutions to wave equations in many interesting and subtle ways. Following Newton, we know that light behaves in many regimes as if made of tiny particles. On the other hand, we also know that light can turn corners ("diffract") and that it tends to disperse. The effect of changes in geometry to changes in propagation of waves (be they light or sound or water or gravity waves, or the wave-functions describing quantum particles) is the central focus of this project's research. In particular, the principal investigator's work on quasi-normal modes for Kerr spacetimes is closely related to problems of intense interest in the physics community, as these modes are part of the signature of gravitational waves. The principal investigator's study of the linear Schrodinger equation is related to applications not just to the physics of nonrelativistic quantum particles, but also to the nonlinear Schrodinger equation, which models such disparate phenomena as laser pulses and superconductivity.
In work with Maciej Zworski, the PI explored the rate at which waves (for instance, of light) disperse in the presence of a rotating ("Kerr") black hole. The presence of trapped orbits, on which a point particle might circulate forever, is a potential obstruction to the decay of light waves in this setting. However, Zworski and the PI were able to show that the unstable, "normally hyperbolic," character of these trapped orbits is such that the waves disperse fast nonetheless near the trapping. We were able to show this not only in the exact setting of the Kerr black hole but also in any system in which the behavior of particle orbits is qualitatively similar. With Dean Baskin, the PI showed a similar decay result in which the potential obstruction to decay of waves is given by diffraction, in which light rays may follow trajectories into "shadow regions" inaccessible to particle motion. In this case too, we were able to show that the trapping effect is sufficiently weak as to not obstruct the rapid decay of waves, and indeed we were able to show that the decay gets better and better for higher frequency waves, in a certain sense. With Dean Baskin and Andras Vasy, the PI studied the long-range behavior of waves in a wide variety of spacetimes of the sort dictated by the theory of General Relativity. We were able to understand the behavior of the "radiation field," or pattern of radiation observed by a far-away observer, in a novel manner.
