Mathematical Sciences: Some Spectral and Isospectral Problems in Riemannian Geometry

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The principal investigator will continue his research in two areas of spectral geometry. The first involves isospectral three dimensional manifolds with curvature assumptions but no restriction on conformal class. In the second project he will develop a trace formula and Selberg zeta function for arbitrary geometrically finite discrete groups whose fundamental regions have infinite hyperbolic volume. The principal investigator will extend his theories of isospectral manifolds which are not necessarily isometric. In two dimensions, isospectral manifolds are surfaces which sound alike when struck. In higher dimensions, examples are known of hypersurfaces which sound alike but which are not isometric; that is, they do not have identical shapes.