Microlocal Analysis and Applications

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 is another important physical example via (the not long ago detected) gravitational waves: the PI’s recent work with Hintz showed that in a universe with a (possibly small, as our universe is currently understood) positive cosmological constant, perturbations of black holes decay to a slightly different black hole, emitting gravitational waves in the process. 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. The inverse problems under study are also of broad significance: an application of the theory developed here is the determination of an unknown variable speed of elastic waves 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. Many of the projects are suitable for research by doctoral students, and the PI strives to contribute to the education of a new generation of mathematicians and scientists. 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. The microlocal approach to analysis on these spaces has made breakthroughs possible in the PI's work on linear and non-linear (with his former PhD student, P. Hintz) problems on asymptotically hyperbolic (AH) spaces as well as Kerr-de Sitter (KdS) space (rotating black holes in a cosmological spacetime), culminating in the proof of the stability of slowly rotating KdS spaces with Hintz. More recently, with Hafner and Hintz the PI extended some of these tools to the vanishing cosmological constant case (Minkowski, Kerr). The projects here aim to extend these tools to further spaces, such as perturbations of Kerr spacetimes, and also to study other equations on cosmological spacetimes. Other projects study basic objects in quantum field theory, in particular the Feynman propagator. Another main area is inverse problems, where the PI, together with Uhlmann, has introduced new tools for the spatially localized inversion of the geodesic X-ray transform, and with Stefanov and Uhlmann extended this to the boundary rigidity problem. One project here aims to extend this to anisotropic elasticity which plays an important role in the interior of the Earth. Another project with the PI's former postdoc Wang studies the light ray transform with potential applications to imaging by the cosmic background radiation. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.