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

拨款提案描述了Jay Jorgenson和他的合著者将进行的几个研究方向，从某种程度上说，这些方向建立在他之前对热核和热核分析的数论应用的研究基础上。Jorgenson和J‘urg Kramer之前的工作集中在Arakelov理论的分析方面。在这一工作的基础上，Jorgenson和Kramer最近建立了赋于非全纯Maass形式的某些L-函数与附于二重全纯形式的L-函数之间的关系。该研究的一部分描述了Jorgenson和Kramer将要开展的工作，他们研究了他们的L-函数关系在涉及Maass形式的傅立叶系数的界的问题上的应用。此外，Jorgenson和Kramer的合作还导致了涉及双曲、抛物线和椭圆Eisenstein级数之间的相互关系的项目。研究提案描述了仍在这一方向上存在的一些问题。在Gautam Chinta和Anders Karlsson的合作下，Jorgenson发起了一个研究与完全不连通空间相关的热核和解析数理论的计划，包括与有限群相关的离散Tori图和Cayley图。该提议概述了约根森在他与钦塔和卡尔森的联合研究工作中需要研究的进一步问题。热核和热核分析存在于数学和相关领域的许多领域。例如，Chinta、Jorgenson和Karlsson进行的联合研究的一个组成部分是研究与有限群的图相关的某些矩阵的行列式。追溯到1847年的基尔克霍夫，人们已经知道在这些决定因素和其他学科之间建立联系，如电气工程和化学。最近，可以说相当多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Jorgenson提出的与金融相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课程的课程开发活动，所有这些都受益于Jorgenson的研究计划。最后，该提案描述了一系列广泛的问题，可供多个层次的学生评估，从高级本科生到新近获得博士学位的学生。S说。

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.