Heat Kernel Analysis and Zeta Functions on Quotients of Symmetric Spaces

对称空间商的热核分析和 Zeta 函数

• 批准号: 0071363
• 负责人: Jay Jorgenson
• 金额: $ 9.3万
• 依托单位: CUNY City College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0071363&HistoricalAwards=false
• 关键词: Heat; Kernel; Analysis; Zeta; Functions

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.

JorgensonJay Jorgenson的提案摘要Jorgenson建议继续与他的合著者进行研究，将热核分析应用于数论。 Jorgenson和Serge Lang将研究对称空间的有限体积导数的谱理论，使用热内核定义的Eisenstein级数的推广，而不是使用自守形式定义的经典Eisenstein级数。 乔根森和朗计划调查的问题，涉及解析延续，功能方程，和特殊价值观的热爱森斯坦系列。 在完成这一部分的工作，乔根森和郎然后将调查应用热爱森斯坦系列外尔定律以及建设的zeta函数将产生更高的排名概括塞尔伯格zeta函数。 与Jurg克雷默合作，乔根森将研究阿拉克洛夫理论的分析方面。 在最近的工作中，Jorgenson和克雷默研究了Selberg的zeta函数的特殊值的界和通过覆盖的渐近界。 Jorgenson和克雷默建议将这些结果推广到研究Faltings δ函数通过覆盖的渐近行为，从而推广了Jorgenson在1989年斯坦福大学博士学位论文中首次获得的结果。论文 在与Jozek Dodziuk的合作中，Jorgenson计划使用Lang的退化数域定义（来自Lang 1971年的Inventiones论文）来研究退化希尔伯特模簇的相应序列的谱理论。 建议采用的方法包括对乔根森与多齐乌克以及亨特利和伦德利乌斯以前所做研究的概括。 在时间允许的情况下，乔根森计划与Carol Fan、Edward Jenvey和Peter Grabner一起研究提出的其他研究问题。经典上，热核是定义为时间t和区域D中的点x和y的正值的函数，热核测量当在时间零时在D中的点y处引入单位热爆发时在时间t处D中的点x处的热量。 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实际上在纯数学和应用数学的每一个领域都表现出来。此外，热核存在于许多统计学和计量经济学领域的理论基础中，更具体地说，是金融数学。 Jay Jorgenson和他的合作者进行的部分研究涉及理解热核在数学的一个领域中出现的各种方式，然后将问题，定理和技术翻译到数学，统计学和经济学的其他领域。 超越数学研究，Jay Jorgenson提出将他的研究努力扩展到将热核应用于金融数学和经济学的实际问题，其中自然包括开发方法，可以对热核的构造进行编程，以获得精确的数值评估。