Linear Partial Differential Equations on Singular Spaces

Northwestern University, Evanston IL

Investigators

Abstract

The principal investigator will study the subtle correspondence between the movement of particles and the propagation of waves in a variety of geometric and physical situations. In the quantum world, this goes by the name of the "correspondence principle" and has long been understood in the simplest settings. However, many mysteries remain in the application of the correspondence principle to problems involving "bumpy" geometries or large scales. One of the principal investigator's projects addresses how waves (e.g., light waves or gravity waves) propagate on space-times of interest in Einstein's theory of general relativity, and in particular, how their decay is viewed by distant observers. This decay is influenced by the geometry of space-time on very large scales. Another part of the research will study how the familiar phenomenon of diffraction of waves influences the waves' rates of decay. This applies, for instance, to sound waves in the presence of sharp corners or cone points, and it is of practical importance in acoustics and in a variety of computational problems involving wave propagation. The project will study the asymptotic behavior of waves propagating on certain Lorentzian space-times such as arise in the theory of general relativity. A main goal is to understand the asymptotics of the Friedlander radiation field for a range of Lorentzian geometries, including asymptotically static space-times with asymptotically Euclidean ends (with long-range perturbations). The principal investigator will also study the decay of waves near their source in different geometric settings, especially in singular geometries. In particular, he will investigate the distribution of quantum resonances in a variety of geometries involving cone points and corners, as well as continue collaborations on more applied problems motivated by numerical analysis for high-frequency asymptotics of the Helmholtz equation. Finally, the principal investigator will study the closely related problem of the structure of the propagator for the Dirac-Coulomb equation, where the singularity of the potential diffracts the singularities of the solution, producing an outgoing spherical wave.