The principal investigator will study spectral geometry in three basic contexts. For smooth infinite-volume hyperbolic manifolds (without cusps), he will study the question of finiteness of classes of surfaces with the same resonance set, refine techniques involving determinants that he and his co-authors have already developed, and study resonances as functions of the deformation space of a hyperbolic structure. For the more general class of asymptotically hyperbolic manifolds, the main goal is to analyze the determinants and relative determinants and use them to derive geometric constraints from resonance data. The techniques developed will be applied to define determinants and obtain constraints for exterior domains in two and three dimensions as well. A final goal is to understand the basic spectral theory of a broader class of negatively curved surfaces. Here the objectives are to determine the essential spectrum and prove a limiting absorption principle which will characterize the essential spectrum and lead towards the establishment of a scattering theory.
Geometric spectral theory lies at the interface of the fields of differential geometry and mathematical physics. In physical theories such as quantum mechanics or wave propagation, one can draw a natural distinction between geometric properties of a system, meaning the underlying structure, and analytic properties, which reflect how the physics of the system behaves. Analytic properties, of which the spectrum is a prime example, are generally the aspects of a system most readily determined by experiment (for example, component colors of light emitted by stars, or frequency spikes in a radar scan). In many physical applications, the basic goal is to derive information about the geometric structure from the spectral data. The PI will pursue this goal in settings for which one already has good sources of conjectures and mathematical tools. Understanding the spectral theory of these cases will provide new geometric invariants of interest in differential geometry, while developing intuition for problems of a more applied nature.