Arithematic Manifolds: Geodesics, Spectra and L-Functions

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Professor Wolpert will investigate questions about negatively curved space, especially shortest paths and high-energy oscillations. The goal is to make new applications to problems in analytic number theory. Geodesics on hyperbolic manifolds have special properties. Professor Wolpert will continue his work with A. Basmajian on describing geodesics in hyperbolic surfaces and three-manifolds. The spectrum of the modular group serves as a model case for the study of automorphic forms and even questions in Quantum Chaos. The spectrum is a major subject of study in modern analytic number theory. Several open questions including "uniqueness of quantum limits," "the spectral decomposition of nonarithmetic groups," and the Ramanujan-Peterson conjecture' are actually questions about the magnitude of certain expansion coefficients. Professor will use discoveries made under a previous grant to establish new and better estimates of these coefficients.This is research in the general area of number theory. The historical roots of number theory are in the study of the whole numbers, considering questions of divisibility of one whole number by another. Number theory is one of the oldest fields in mathematics and has at times guided the direction of the development of the entire field. In current times it has served as an indispensable tool in computer science, communications, and electronic security.