Mathematical Sciences: Some Spectral and Isospectral Problems in Riemannian Geometry

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Peter Perry will work on two areas of spectral theory on Riemannian manifolds. These involve the scattering theory of manifolds of constant negative curvature and inverse spectral problems on compact Riemannian manifolds. These are geometric problems but the techniques to be employed will involve methods of mathematical physics. The first project will make use of the stationary scattering theory of elliptic operators invented for for the Schrodinger equation. The inverse spectral theory will employ techniques of partial differential equations. The first set of problems to be addressed concerns the spectral theory of the Laplace operator on a real hyperbolic manifold and its connections with the theory of discrete groups of hyperbolic isometries. Perry will concentrate on those manifolds which are quotients of hyperbolic space by a group of hyperbolic isometries. The work will involve investigations of the resolvent, eigenfunctions and scattering matrix of the Laplace operator on the hyperbolic space. It is hoped that this will be helpful in studying the Selberg trace formula and the Selberg zeta function of the isometry group. He will also continue his study of isospectral metrics on a Riemannian manifold. By restricting attention to metrics in a conformal class, Perry, in joint work with Brooks and Yang has, extended some two-dimensional results to the three- and four- dimensional case. Attempts will be made to remove the restriction to conformal classes and to find higher dimensional results.