The grant proposal describes several directions of research to be undertaken by Jay Jorgenson and his co-authors which, somewhat generally, build on his prior investigation of number theoretic applications of heat kernel and heat kernel analysis. Previous work by Jorgenson with J""urg Kramer focus on analytic aspects of Arakelov theory. Continuing this work, Jorgenson and Kramer recently established a relation between certain L-functions attached to non-holomorphic Maass forms and L-functions attached to holomorphic forms of weight two. Part of the proposed research describes work to be carried out by Jorgenson and Kramer which studies the application of their L-function relation to questions involving bounds of Fourier coefficients of Maass forms. Additionally, the Jorgenson and Kramer collaboration has resulted in projects involving the inter-relations between hyperbolic, parabolic, and elliptic Eisenstein series. The research proposal describes a number of problems which remain in this direction. In collaboration with Gautam Chinta and Anders Karlsson, Jorgenson has initiated a program of study of heat kernels and analytic number theory associated to totally disconnected spaces, including discrete tori and Cayley graphs associated to finite groups. The proposal outlines further questions to be investigated by Jorgenson in his joint research efforts with Chinta and Karlsson.
The heat kernel, and heat kernel analysis, is present in many areas of mathematics and related fields. For example, one component of the joint research undertaken by Chinta, Jorgenson and Karlsson is to study determinants of certain matrices associated to graphs associated to finite groups. Going back as far as Kirckhoff in 1847, connections have been known which establish relations between these determinants and other disciplines, such as electrical engineering and chemistry. More recently, one can say that a considerable amount of financial mathematics and financial engineering is built on heat kernel analysis. In addition to specific research problems posed by Jorgenson which relate to finance, his teaching duties include curricular development activities involving financial engineering and bio-statistics courses, all of which benefit from Jorgenson's research program. Finally, the proposal describes a wide range of problems which are assessable to students of many levels, ranging from advanced undergraduate students to recent Ph.D.'s.
