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

该拨款提案描述了几个方向的研究将进行杰伊乔根森和他的co-authorswhich，有点一般，建立在他以前的调查数论应用的热核和热核分析。Jorgenson和Jurg克雷默以前的工作集中在Arakelov理论的分析方面。 继续这项工作，Jorgenson和克雷默最近建立了一个关系betweentain L-函数重视非全纯的马斯形式和L-函数重视全纯形式的重量二。 部分拟议的研究介绍了工作进行乔根森和克雷默研究的应用他们的L-函数关系的问题，涉及边界的傅立叶系数的马斯形式。此外，Jorgenson和Kramer的合作导致了涉及双曲，抛物线和椭圆艾森斯坦级数之间的相互关系的项目。 研究建议描述了一些问题仍然在这个方向。 在合作与高塔姆Chinta和安德斯Karlsson，Jorgenson发起了一个程序的研究热核和解析数论相关的完全不连通的空间，包括离散tori和凯莱图相关的有限群。 Jorgenson在他与Chinta和Karlsson的联合研究工作中，提出了进一步的问题。热核和热核分析在数学和相关领域的许多领域都存在。 例如，一个组成部分的联合研究所承担的钦塔，乔根森和卡尔森是研究决定因素的某些matrices相关的图形相关联的有限群体。 追溯到1847年Kirckhoff，这些决定因素与其他学科（如电气工程和化学）之间的联系已经被人们所知。 最近，人们可以说相当数量的金融数学和金融工程是建立在热核分析的基础上的。 除了由乔根森提出的与金融有关的具体研究问题外，他的教学职责还包括涉及金融工程和生物统计学课程的课外发展活动，所有这些都受益于乔根森的研究计划。 最后，该建议描述了一系列广泛的问题，这些问题对许多层次的学生都是可评估的，从高年级的本科生到最近的博士生。's.

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.