Linear partial differential equations on singular spaces

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