Microlocal Analysis and Wave Phenomena in Geometric Settings

This award supports the development and applications of methods in the field of microlocal analysis. Roughly speaking, this field 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, such as the analysis of long-time or far field behavior, are to wave propagation and other related phenomena such as quantum fields on curved spaces; curvature of the space itself is a feature of mathematical general relativity. 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, and the latter (gravitation) giving rise to curved spacetimes. Scattering theory of quantum particles is another subject governed by microlocal analysis: these aspects enter into the description of quantum waves at large distances. Many of the projects are suitable for research by doctoral students, and the Principal Investigator (PI) strives to contribute to the education of a new generation of mathematicians and scientists. Some of the projects describe the long-time or far field behavior, including existence, of waves, such as electromagnetic or gravitational waves, on curved spacetimes. The microlocal approach to analysis on these spaces has made breakthroughs possible in the PI's (in part collaborative) work on linear and non-linear 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 (as solutions of Einstein's equation) 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, namely Minkowski and Kerr spaces - so far only to the linearized result in the Kerr case but including fast (subextremal) rotation. With Hintz and Petersen, the PI also extended the nonlinear stability result to fast rotating (subextremal) KdS spacetimes, though in a conditional manner. The projects here aim to extend these tools to further spaces, such as perturbations of subextremal Kerr spacetimes. Other projects study basic objects in quantum field theory, in particular the Feynman propagator, on curved spacetimes, and their uses for spectral theory, including for a spectral action principle. Two particular area of focus for this are radiative spacetimes that arise by solving Einstein's equation in general relativity with perturbed Minkowski initial data (which gives rise to gravitational radiation as the spacetime settles down to Minkowski space) and perturbations of de Sitter spacetimes, which are the underlying spacetimes for positive cosmological constant problems. The microlocal techniques involved in understanding the latter also relate to wave diffraction from boundaries or edges, the quintessential near field phenomena on which the PI had worked extensively; a longer-term project is to use the new insights to gain a better understanding of these phenomena. 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. NSF Award ID: 2553664 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Andras Vasy | Institution: Stanford University, STANFORD, CA | Award Amount: $199,999 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2553664 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2553664.html