Mathematical Sciences: Regularized Products, Heat Kernal Analysis and Analytic Number Theory

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The problems that are to be addressed in this project lie in the general area of spectral theory and number theory, with particular interest in trace formulae, explicit formulae and prime number theorems, and various analytic and arithmetic aspects of Arakelov theory. The proposed research is very broad, involving analysis, number theory, and algebraic geometry. This research uses the techniques of algebraic geometry, one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.