Microlocal analysis for waves and inverse problems

Stanford University, Stanford CA

Investigators

Abstract

The planned research develops and applies tools of the field of microlocal analysis. Roughly speaking, this field keeps track of the position and frequency, or momentum, of waves (or more generally, functions) simultaneously. The planned applications are to wave propagation and other related phenomena, as well as inverse problems for determining a function from integrals along curves (X-ray transform) and related problems for determining the structure of a material from boundary measurements. Although the proposal concerns their mathematical theory, these problems 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 being another important physical example. Scattering theory of quantum particles (such as protons and electrons) is another subject governed by microlocal analysis: these aspects enter both into the description of quantum waves at large distances, and into semiclassical phenomena, i.e. when Planck's constant can be regarded as small, which happens often in chemistry. The inverse problems under study are also of broad significance: an application of the theory developed 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. Some of the proposed projects describe the long-time or far field behavior of waves on curved space-times. Physically these arise in general relativity, including electromagnetic waves on a curved background. However, there are also examples of purely mathematical origin, such as asymptotically complex hyperbolic (ACH) space and asymptotically Anti de Sitter spaces (AdS); the latter does have a different connection with physics via string theory. The microlocal approach to analysis on these spaces has made breakthroughs possible in the author's work on linear problems asymptotically (real) hyperbolic (AH) spaces as well as Kerr-de Sitter space. The projects here aim to extend these tools to the ACH, AdS-type spaces, improve the understanding of Lorentzian scattering spaces (which include asymptotically Minkowski spaces), as well as to quasilinear PDE. Other projects concern the behavior of waves at edges, concretely the diffraction of the Rayleigh (surface) waves of elasticity and a refined, second-microlocal, description of diffraction of scalar or electromagnetic waves. Yet another main area is inverse problems, where the author, together with Uhlmann, has introduced new tools for spatially localized inversion of the geodesic X-ray transform. The projects aim to extend this to tensors, and to investigate boundary rigidity, i.e. recovering a Riemannian metric from its boundary distance function.