Spectral Problems in Geometry and Partial Differential Equations

Intellectual Merit. The Principle Investigator will continue his research in three areas of geometric analysis: (1) resonances in geometric scattering, (2) nonlinear dispersive equations, and (3) harmonic analysis on two-step nilpotent Lie groups. Asymptotically real- and complex-hyperbolic manifolds are complete, non-compact Riemannian manifolds with "simple geometry at infinity." We will study the fundamental questions related to the existence, distribution, and geometric content of the scattering resonances, exploiting recent progress in scattering theory for these geometric classes. We will analyze invariants of the scattering data such as determinants of Laplacians and Selberg's zeta function. In the past decade, global well-posedness has been shown for nonlinear dispersive equations such as the Korteweg-de Vries (KdV) equation for initial data which are highly singular or unbounded at infinity. By studying the inverse problem for Schrodinger operators with singular and unbounded potentials, we will extend the inverse scattering method to the KdV equation and the closely related mKdV equation with these classes of initial data. Our aim is to obtain a more fully geometric picture of the KdV flow and develop new tools for qualitative analysis of the solutions. Two-step nilpotent Lie groups play an important role in spectral geometry (compact nilmanifolds provide interesting examples of non-isometric Riemannian manifolds with the same spectral data), in control theory (certain two-step nilpotent Lie groups are model spaces for control problems), and in harmonic analysis (the sub-Laplacian is a natural model for hypoelliptic differential operators). We will study the sub-Riemannian geometry of certain of these spaces and the associated behavior of heat and wave kernels for geometric operators on these spaces. Broader Impact. Inverse spectral theory is the mathematical discipline that underlies important applications of mathematics to medical imaging, geophysical prospection, and non-destructive testing. In these applications, the "geometry" of a physical system (a human body, the earth, or an industrial product) is reconstructed from the scattering of electromagnetic, seismic, or acoustic waves. Advances in geometric scattering theory clarify the relationship between scattering data and geometry. The inverse scattering method has produced powerful tools to study the propagation of nonlinear waves in hydrodynamics and nonlinear optics; extending these methods to a richer set of initial conditions will deepen our understanding of mathematical models for nonlinear waves.