Applications of Heat Kernel Techniques to Zeta Functions of Quotients of Symmetric Spaces and of Graphs

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The grant proposal describes several directions ofresearch to be undertaken by Jay Jorgenson and his co-authorswhich, somewhat generally, build on his prior investigation ofnumber theoretic applications of heat kernel and heat kernelanalysis. Previous work by Jorgenson with J\"urg Kramer focus onanalytic aspects of Arakelov theory. Continuing this work,Jorgenson and Kramer recently established a relation betweencertain L-functions attached to non-holomorphic Maass forms andL-functions attached to holomorphic forms of weight two. Part ofthe proposed research describes work to be carried out byJorgenson and Kramer which studies the application of theirL-function relation to questions involving bounds of Fouriercoefficients of Maass forms. Additionally, the Jorgenson andKramer collaboration has resulted in projects involving theinter-relations between hyperbolic, parabolic, and ellipticEisenstein series. The research proposal describes a number ofproblems which remain in this direction. In collaboration withGautam Chinta and Anders Karlsson, Jorgenson has initiated aprogram of study of heat kernels and analytic number theoryassociated to totally disconnected spaces, including discrete toriand Cayley graphs associated to finite groups. The proposaloutlines further questions to be investigated by Jorgenson in hisjoint research efforts with Chinta and Karlsson.The heat kernel, and heat kernel analysis, is presentin many areas of mathematics and related fields. For example, onecomponent of the joint research undertaken by Chinta, Jorgensonand Karlsson is to study determinants of certain matricesassociated to graphs associated to finite groups. Going back asfar as Kirckhoff in 1847, connections have been known whichestablish relations between these determinants and otherdisciplines, such as electrical engineering and chemistry. Morerecently, one can say that a considerable amount of financialmathematics and financial engineering is built on heat kernelanalysis. In addition to specific research problems posed byJorgenson which relate to finance, his teaching duties includecurricular development activities involving financial engineeringand bio-statistics courses, all of which benefit from Jorgenson'sresearch program. Finally, the proposal describes a wide range ofproblems which are assessable to students of many levels, rangingfrom advanced undergraduate students to recent Ph.D.'s.