Microlocal Analysis of Linear and Nonlinear Problems

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This project develops and applies tools from the field of mathematics known as microlocal analysis. Roughly speaking, microlocal analysis devises methods to keep track of the position and frequency (or momentum) of waves (or, more generally, functions) simultaneously. The project's applications are to wave propagation and other related phenomena, as well as to inverse problems for determining a function from integrals along curves (the X-ray transform) and related problems for determining the structure of a material from boundary measurements. Although the project itself concerns their mathematical theory, the problems under investigation are closely connected to the physical world. Wave propagation is ubiquitous in nature, with electromagnetic waves, such as light, being one of the most prevalent examples. The theory of general relativity is another important physical example via (the recently detected) gravitational waves. Scattering theory for quantum particles (such as protons and electrons) is another subject governed by microlocal analysis, aspects of which enter into the description of both quantum waves at large distances and semiclassical phenomena (those in which Planck's constant can be regarded as small, often the case in chemistry). The inverse problems under study are also of broad significance. One application of the theory under development here is the determination of an unknown variable sound speed in an object via the measurement of travel times of waves, which for instance is relevant to imaging to interior of Earth using the travel times of earthquake waves.Parts of this project describe the long-time or far-field behavior of waves on curved space-times. Physically these arise in scattering theory and general relativity, including electromagnetic waves on a curved background. The microlocal approach to analysis on these spaces has made breakthroughs possible in work on linear and nonlinear problems on asymptotically (real) hyperbolic spaces as well as on Kerr-de Sitter space. The projects here aim to improve the understanding of Lorentzian scattering spaces, which include asymptotically Minkowski spaces (asymptotically flat spaces, which our universe approximates, at least locally, even if there is a small positive cosmological constant). Other projects concern the behavior of waves at edges -- specifically, the diffraction of the Rayleigh (surface) waves of elasticity. Yet another main area is inverse problems, building on recently-developed tools for spatially localized inversion of the geodesic X-ray transform, including the solution of fixed conformal class boundary rigidity: under suitable assumptions, one can determine a variable sound speed inside an object from the travel times of waves between points on the surface. The parts of the project in this area aim to extend the foregoing result to tensors, which describe anisotropic sound speeds in an appropriate sense; namely, the project will investigate boundary rigidity, for example, recovering a Riemannian metric from its boundary distance function.