CAREER: Wave Evolution on Singular Spacetimes
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Many phenomena in science and engineering can be modeled by wave-type equations, including electromagnetic and gravitational waves as well as the behavior of the electron. The study of these equations is central to at least two areas of mathematics and mathematical physics: scattering theory and general relativity. This project will study the (often subtle) relationship between the underlying geometry of a space and the long- and short-time behavior of waves propagating on it. Complementing the research in this proposal are educational and outreach activities, with a focus on encouraging the development of communication skills in graduate and undergraduate students. A central component of this work is the development of a series of programs to help graduate students develop their writing and presentation skills.There are two main research components of this project: The first aims to provide a full microlocal treatment of the Dirac equation, with an emphasis on establishing the phenomenon of diffraction for systems. In this direction, the project pursues an explanation of diffraction for an electron in the presence of singularities (such as those provided by the Coulomb potential). Such theorems reflect observable phenomena. One example of observable short-time behavior can be seen in a freshman physics laboratory: by shining a light at the tip of a needle, one sees a radial diffractive pattern. The second focuses on determining the asymptotic behavior of waves on curved spacetimes with conic singularities such as those arising in the theory of cosmic strings.