Mathematical Sciences: Scattering Theory

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The goals of this project are the analysis of certain problems in number theory, function theory and scattering theory. Techniques to be used involve spectral theory of the Laplace-Beltrami operator acting on automorphic functions over a fundamental domain with a finite number of sides and of either finite or infinite volume. The Laplace-Beltrami operator is central to all three fields representing an important concept worthy of concentrated study. This work as well as that recently completed on explicit expanders and Ramanujan graphs gives the best possible expanders; these are the building blocks for switching networks as well as sorting schemes. A sharpened form of the Moser inequality for the projective plane and detailed estimates on the destruction of cusp forms under perturbation of the Laplacian (and thus the winding number of the determinant of the scattering matrix) on surfaces of finite volume are expected.