Heat kernel methods applied to zeta functions of Ihara, Rankin-Selberg, and Selberg

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The Principal Investigator proposes to use techniques from analysis, in particular from the study of heat and wave kernels and distribution theory, to study zeta functions of number theory. In the field of regular graphs, both finite and infinite, the project will systematically develop a heat kernel approach to number theory on graphs, including the Ihara zeta function. In previous work, the PI has developed a succinct relation between Fourier coefficients of weight zero Maass forms and weight two holomorphic forms, both associated to any finite volume hyperbolic Riemann surface. The proposal involves a continuation of this work, striving toward bounds of the Fourier coefficients.Both the heat kernel and the wave kernel are mathematical objects which admit interpretations from many mathematical fields. The heat kernel, in particular, can be viewed as a solution of a partial differential equation, as a one-parameter family of positive integrable functions, and as a function associated to probability theory and random walks. As a result, the heat kernel provides a means by which one can employ ideas and results from one mathematical discipline in order to approach problems in another field. The PI plans to continue this approach to study problems in number theory, analysis, and graph theory. In addition to research interests, the PI has undertaken a number of educational endeavors striving to enhance opportunities for students. As a faculty member at The City College of New York, the PI has developed courses in statistics and in financial mathematics for the Mathematics Department, and he works closely with the School of Education in their course offerings in teacher training programs. In October 2010, the PI, together with J. Kramer, taught a graduate level course at the University of Sarajevo. The proposed research includes a component of effort by the PI to enhance and further develop these teaching aspects of his mathematical interests.