Spectral Geometry of Non-Compact Domains and Riemannian Manifolds

The principal investigator will study the spectral geometry of non-compact domains and Riemannian manifolds in order to elucidate the geometric content of scattering poles. First, the PI will continue his study of the spectral geometry of hyperbolic manifolds and their perturbations. He will study resonances as functions on the deformation space of the underlying discrete group, and define and analyze a determinant of the Laplacian. Secondly, the PI will study scattering theory for the wave equation on two-step nilpotent Lie groups and their quotients by discrete subgroups. New parametrices or the wave equation on the Heisenberg and Heisenberg-type groups will be derived, and trace formula for certain quotients obtained. Riemannian submersion techniques of Gordon, Wilson, and others will be used to obtain pairs and families of `isoscattering' manifolds which will help determine the limits of geometric information which may be deduced from a knowledge of the scattering poles. Thirdly, the PI will study the isoscattering problem for exterior domains in Euclidean space. The fundamental problem of spectral geometry is to elucidate the geometric content of the Laplace spectrum on a Riemannian manifold. For so-called scattering manifolds, the eigenvalues of the Laplacian together with scattering resonances constitute the spectral data for the manifold. Elucidating the geometric content of such spectral data advances our understanding of quantization, produces new analytic tools for the study of geometric objects, and provides insight into inverse problems of a more `applied' nature where the eigenvalues and scattering poles are measurable quantities. The present work aims to begin with geometrically natural examples where techniques of Lie theory, automorphic functions, and harmonic analysis may be used, and progress to harder problems such as target identification by radar where such techniques are not available but the underlying mathematical problems are very similar.