Heat Kernel Analysis and Zeta Functions on Quotients of Symmetric Spaces

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Abstract for proposal of JorgensonJay Jorgenson proposes to continue ongoing research with his co-authors into applications of heat kernel analysis to number theory. Jorgenson and Serge Lang will study spectral theory on finite volume quotients of symmetric spaces using generalizations of Eisenstein series defined using heat kernels, as opposed to classical Eisenstein series which are defined using automorphic forms. Jorgenson and Lang plan to investigate questions involving analytic continuation, functional equations, and special values of their heat Eisenstein series. Upon completion of this component of their work, Jorgenson and Lang then will investigate applications of heat Eisenstein series to Weyl's law as well as to constructions of zeta functions which will yield higher rank generalizations of Selberg zeta functions. In collaboration with Jurg Kramer, Jorgenson will study analytic aspects of Arakelov theory. In recent work, Jorgenson and Kramer study bounds on special values of Selberg's zeta function and asymptotic bounds through covers. Jorgenson and Kramer propose to extend these results to study asymptotic behavior of Faltings's delta function through covers, thus extending results first obtained by Jorgenson in his 1989 Stanford Ph.D. thesis. In collaboration with Jozek Dodziuk, Jorgenson plans to use Lang's definition of degenerating number fields (from Lang's 1971 Inventiones paper) to study spectral theory on the corresponding sequence of degenerating Hilbert modular varieties. The proposed methods to employ involve generalizations of research previously obtained by Jorgenson with Dodziuk and with Huntley and Lundelius. As time permits, Jorgenson plans to study proposed additional research problems with Carol Fan, Edward Jenvey and Peter Grabner.Classically, the heat kernel is a function defined for positive values of time t and points x and y in a domain D, and the heat kernel measures the amount of heat at point x in D at time t when a unit burst of heat is introduced at point y in D at time zero. Although the origin of the heat kernel lies in physics, the mathematics surrounding the function known as the heat kernel manifests itself in virtually every area of pure and applied mathematics. In addition, the heat kernel is present in the theoretical foundations of many fields of statistics as well as econometrics and, more specifically, financial mathematics. Part of the research undertaken by Jay Jorgenson and his collaborators involves understanding various ways in which the heat kernel appears in one area of mathematics and then translate the questions, theorems and techniques to other areas of mathematics, statistics, and economics. Going beyond mathematical research, Jay Jorgenson proposes to extend his research endeavors to incluce applications of heat kernels to practical problems of financial mathematics and economics, which naturally includes developing ways in which one can program constructions of heat kernels in order to obtain precise, numerical evaluations.