Mathematical Sciences: Spectral Geometry of Compact Riemannian Manifolds and Kleinian Groups

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9707051 Perry This project deals with several problems in spectral geometry: the inverse spectral problem for compact surfaces; trace formulas and dynamical zeta functions for the Laplacian on hyperbolic manifolds and associated vector bundles; the scattering operator and its determinant as a function on the deformation space of a Kleinian group. Techniques to be employed include perturbation theory, global analysis, and Teichmuller theory. One of the aims of this project is to show that the isopectral set of a compact surface is finite for metrics close to constant curvature. Spectral geometry is concerned with the interaction of differential-geometric properties of a Riemannian manifold (a curved space with a metric) with the spectra of natural differential operators associated with it. The spectrum of an operator often captures various analytic properties of an otherwise intractable differential operator in a discrete and computable manner. Elucidating the geometric content of various spectral data then is the main concern of spectral geometry.