Linear Partial Differential Equations on Singular Spaces

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This project focuses on the study of 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 or cone points, to large-scale structures "at infinity"--presents many open problems. The principal investigator will 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. Recent work of the principal investigator has yielded results in spacetimes with a compactification similar to that of Minkowski space, and he will continue to pursue such results in a wider range of geometries. The project will also study the decay of waves near their source in different geometric settings. In particular, in the presence of corners or cone points, diffraction of waves is a potential obstruction to the rapid decay of waves; the principal investigator will continue his ongoing investigations in this area, especially into the question of resonances generated by cone points. Additionally, he will continue to study the behavior of eigenfunctions and quasimodes of the Laplace operator associated to completely integrable systems, in an effort to understand to what degree an eigenfunction can concentrate on a sub-torus of an Arnold-Liouville torus. These techniques should also yield new estimates for solutions to the linear Schrodinger equation in completely integrable settings.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. In particular, the principal investigator's work on decay of waves in curved spacetimes is closely related to problems of intense interest in the physics community, for instance as it gives a much simplified model of gravitational waves. His closely related work on 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.