Heat Kernel Analysis and Zeta Functions on Quotients of Symmetric Spaces

乔根森（jay Jorgenson）提议与他的合作者继续研究热核分析在数论中的应用。Jorgenson和Serge Lang将使用热核定义的爱森斯坦级数的推广来研究对称空间有限体积商的谱理论，而不是使用自同构形式定义的经典爱森斯坦级数。Jorgenson和Lang计划研究涉及解析延拓、泛函方程和他们的热爱森斯坦级数的特殊值的问题。在完成这部分工作后，Jorgenson和Lang将研究热爱森斯坦级数在Weyl定律以及zeta函数构造中的应用，这将产生Selberg zeta函数的更高阶推广。与Jurg Kramer合作，Jorgenson将研究Arakelov理论的分析方面。在最近的工作中，Jorgenson和Kramer研究了Selberg的zeta函数的特殊值的界和通过覆盖的渐近界。Jorgenson和Kramer提出将这些结果扩展到通过盖研究Faltings函数的渐近行为，从而扩展了Jorgenson在1989年斯坦福大学博士论文中首次得到的结果。Jorgenson与Jozek Dodziuk合作，计划使用Lang关于退化数域的定义（来自Lang 1971年的《发明》论文）来研究退化希尔伯特模变的相应序列的谱理论。所提出的方法涉及对Jorgenson与Dodziuk以及Huntley和Lundelius先前获得的研究的概括。如果时间允许，Jorgenson计划与Carol Fan， Edward Jenvey和Peter Grabner一起研究提出的其他研究问题。经典地，热核是一个函数，定义为时间t和点x和y在D域中的正值，热核测量时间t在D点x处的热量，当时间0在D点y处引入单位热量时。虽然热核的起源在于物理学，但围绕着被称为热核的函数的数学几乎在纯数学和应用数学的每个领域都表现出来。此外，热核存在于许多统计学领域的理论基础以及计量经济学，更具体地说，是金融数学。Jay Jorgenson和他的合作者所进行的部分研究包括理解热核在一个数学领域出现的各种方式，然后将这些问题、定理和技术转化为数学、统计学和经济学的其他领域。除了数学研究之外，Jay Jorgenson建议扩展他的研究工作，将热核应用于金融数学和经济学的实际问题，这自然包括开发人们可以对热核结构进行编程的方法，以获得精确的数值评估。

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.