Microlocal Analysis and Geometry

This project develops and applies methods in the area of microlocal analysis. Roughly speaking, microlocal analysis keeps track of the position and frequency, or momentum, of waves (or more generally, functions, such as the amplitudes and phases of waves) simultaneously. The planned applications are to wave propagation and other related phenomena, as well as inverse problems related to determining the structure of a material from surface measurements as well as to imaging by cosmic background radiation. Although the project concerns their mathematical theory, these problems are closely connected to the physical world. Wave propagation is ubiquitous in nature, with light and gravitational waves being important examples. Scattering theory of quantum particles is another subject governed by microlocal analysis: these aspects enter into the description of quantum waves at large distances. The inverse problems under study are also of broad significance: applications of the theory developed here include the determination of an unknown variable speed of elastic waves in an object via the measurement of travel times of waves, as well as the development of the universe through cosmic microwave background data. 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. Parts of the project describe the long-time or far field behavior, including existence, of waves, such as electromagnetic or gravitational waves, on curved space-times. The microlocal approach to analysis on these spaces has made breakthroughs possible in the PI's (in part collaborative) work on linear and nonlinear problems on asymptotically hyperbolic 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 aim here is to extend these tools to further spaces, such as fast rotating KdS and perturbations of Kerr spacetimes. Other parts of the project study basic objects in quantum field theory, in particular the Feynman propagator. A novel direction, with Tripathy and Zimet, is construction of Ricci flat metrics on K3-type surfaces. 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. A 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.