Inverse Problems in Geometry and Partial Differential Equations

University Of Kentucky Research Foundation, Lexington KY

Investigators

Abstract

DMS-0408419 Title: Inverse problems in geometry and partial differential equations PI: Peter A. Perry, University of Kentucky ABSTRACT This project involves inverse spectral and scattering theory in three areas of mathematical investigation: (1) the spectral theory of two-step nilpotent groups and compact nilmanifolds, (2) the inverse resonance problem for exterior domains and scattering manifolds, and (3) inverse scattering for singular potentials with applications to nonlinear dispersive equations. Two-step nilpotent lie groups play an important role in pure mathematics as models of sub-Riemannian geometry and as a rich source of examples of manifolds with identical Laplace spectra but distinct geometries; a detailed investigation of the trace formula for these manifolds will be carried out using analysis on nilpotent Lie groups. Resonances are discrete scattering data for non-compact manifolds analogous to the eigenvalues of the Laplacian on a compact manifold; the inverse resonance problem will be investigated in the contexts of conformal and asymptotically flat geometries, and invariants such as the determinant will be studied. Nonlinear dispersive equations with singular initial data may be viewed, via the inverse scattering method, as linearized flows for the scattering data of a very singular potential. We hope to extend the inverse scattering picture for the KdV and mKdV equations to such singular data and obtain greater insight into these dynamical systems--by constructing the flow on Hilbert spaces of initial data which are singular but very natural from a dynamical point of view. Inverse spectral theory is the mathematical discipline that underlies important applications of mathematics to medical imaging, geophysical prospection, non-destructive testing, and many other areas. In these applications, properties of a physical system (a human body, the earth, or an industrial material) are deduced from its response to externally imposed stimuli (electromagnetic radiation, seismic waves, or ultrasound). The properties deduced may loosely be described as the "geometry" of the system and its response to external stimuli the "spectral data" (or "normal modes"). A deep result of the study of completely integrable systems is that certain physical phenomena, such as the propagation of waves in shallow water, can be solved using an associated inverse spectral problem. Thus advances in inverse spectral theory lead to a better understanding of how such nonlinear waves propagate. The impact of this project will be twofold: first, it will elucidate, by studying carefully chosen geometric contexts, the relation between speectrum and geometry. Secondly, it will deepen our understanding of nonlinear dispersive waves by extending tools of inverse scattering theory to study nonlinear wave propagation with very singular waves.