Geometric Inverse Problems and Dynamics

This project will center on the study of inverse problems, particularly those involving transport-type partial differential equations. The theory of inverse problems lies at the borderline between pure and applied mathematics, with connections to statistics, physics, engineering, and biology. The diverse array of applications which can be addressed via this theory include X-ray computed tomography, geophysical prospection, and parameter identification for partial differential equations. An effective approach to tackle such questions often involves the use of a geometric framework and geometric tools, facilitating the reconstruction of internal structure from local or boundary measurements. This project focuses specifically on geometric inverse problems. The relevant equations feature velocity fields capable of generating chaotic dynamics, an aspect which presents new challenges for the analysis. The project will also generate opportunities for research and professional training for graduate and undergraduate students. The project addresses several novel strands of research situated at the interface of geometric inverse problems, dynamical systems, microlocal analysis, and complex geometry. At its core, the planned research is motivated by the desire to comprehend and characterize distinguished solutions to transport problems, a pursuit with potentially far-reaching consequences including the resolution of certain longstanding geometric inverse problems. Among those are the determination of the range of the scattering relation (the first return map of the geodesic flow), deciphering the information encoded within the Ruelle zeta function at zero, and determining topological features of the underlying space from the periods of closed trajectories of the velocity field. The study of distinguished solutions to transport equations necessitates an in-depth analysis of X-ray transforms and the spectral theory of hyperbolic flows. 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.