Propagation Phenomena for Waves and Scattering

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The object of the proposed research is the study of propagation phenomena in different contexts. Topics in the proposal encompass such diverse areas as the study of diffractive phenomena for the wave equation at edges, as well as the asymptotic behavior of waves on curved space-times, such as in general relativity. The project will employ geometric microlocal, or phase space, techniques to explore new problems in these areas. Some of these techniques are constructive, such as the proposed construction of the forward fundamental solution and the scattering operator on a class of asymptotically Anti-de Sitter (AdS) spaces, while others rely on phase-space energy estimates, such as the microlocal positive commutator estimates that are expected to be of use in the study of the asymptotic behavior of the wave equation on curved space-times (such as Kerr-de Sitter space, describing rotating black holes), of wave equations with global conditions replacing initial conditions, and in wave propagation on curved space-times with asymptotic edges and cusps. Even elliptic problems, which correspond to stationary (no propagation) problems, can sometimes benefit from a different point of view: recent work on the meromorphic continuation of the resolvent of the Laplacian on asymptotically hyperbolic spaces, and on its high energy behavior, provides an example.Wave propagation is ubiquitous in the physical world, with electromagnetic waves (e.g., light waves) being one of the most prevalent examples. A typical question is how waves interact with structures such as edges, generalizing the well-known law of reflection from mirrors. Scattering theory is another subject governed by propagation phenomena. For instance, scattering phenomena enter into the description of quantum waves on large scales, mathematically modelled as "infinity." As propagation phenomena are ubiquitous in the physical world, their more precise understanding has many potential applications. A concrete physical application that is expected to be of interest is developing scattering theory (i.e., investigating how light waves or quantum particles behave globally, in the context of curved space-times, which give a description of our universe in the framework of general relativity). Some of the techniques involved are also expected to benefit the understanding of imaging problems, for instance, in the geophysical context.