Linear Partial Differential Equations on Singular Spaces

Project: Research project

Project Details

Description

Overview
The PI proposes to 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—roughness of geometry which might range from boundaries, to corners or cone points, to large-scale structures “at infinity”—presents many open problems, and is relevant to a wide variety of practical applications.
Intellectual Merit:
The PI 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.
The PI 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 PI will investigate the strength of this obstruction by resonant states of problems with cone points. He will also analyze the decay of waves in the presence of incomplete damping.
Broader Impacts:
The proposed work on wave propagation on curved spacetimes is closely related to problems of interest in the physics community involving the behavior of light and other waves in the vicinity of black holes, as well as the propagation of gravitational waves. The study of wave propagation on conic and other singular geometries has relevance to inverse problems of practical interest in medical and seismic imaging. The PI’s work on the exterior Helmholtz problem has consequences for the computer modeling of wave propagation problems.
The PI will integrate the training of undergraduate and graduate students into the
proposed research, and will continue to be active in organizing seminars and conferences
in the field.
StatusFinished
Effective start/end date8/1/167/31/19

Funding

  • National Science Foundation (DMS-1600023)

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