Linear partial differential equations on singular spaces

Northwestern University, Evanston IL

Investigators

Abstract

In this project, the PI will investigate several different aspects of the theory of linear partial differential equations associated to manifolds with singular Riemannian metrics. These include the propagation and diffraction of singularities for the wave equation on manifolds with conic and edge singularities, and the structure of the fundamental solution for the time-dependent Schrodinger equation on manifolds with scattering metrics. Tools such as the Fourier-Bros-Iagolnitzer transform will be studied and refined as needed. Linear partial differential equations (PDEs) that describe wave propagation (such as the classical wave equation and the Schrodinger equation) are of fundamental importance in physics and geometry. The propagation of waves on curved spaces often involves the geometry of the underlying space in a subtle manner. When the space is `singular,' either in the sense of being infinite in extent or of having degenerate properties in a finite region, the relationship between geometry and the properties of solutions to PDEs is especially fascinating and poorly understood.