Heat Kernal Analysis and Analytic Number Theory on Symmetric Spaces, Calabi-Yau Varieties and Moduli Spaces

对称空间、Calabi-Yau簇和模空间的热核分析和解析数论

• 批准号: 9622535
• 负责人: Jay Jorgenson
• 金额: $ 2.86万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622535&HistoricalAwards=false
• 关键词: Heat; Kernal; Analysis; Analytic; Number

This award supports research in the general area of spectral theory and number theory, with particular interest in trace formulae, explicit formulae and prime number theorems, spectral theory on degenerating Riemannian manifolds, and various analytic and arithmetic aspects of Arakelov theory on Calabi-Yau varieties. Jorgenson is continuing his collaboration with Serge Lang. They are studying the heat kernel on symmetric spaces to provide further examples of their general theory. Specific examples such as Hilbert modular and Picard modular varieties will be studied deeply. Jorgenson is continuing resarch with Andrey Todorov in the study of the spectral theory and arithmetic of Calabi-Yau varieties. The long term goal of this study is the development of a theory of arithmetic of Calabi-Yau varieties which would be similar to the existing theory of arithmetic of elliptic curves. With Jozek Dodziuk, Jorgenson is studying the spectral theory of degenerating families of finite volume Riemannian manifolds which possess metrics that have constant negative sectional curvatures near the developing singularities. Jorgenson is continuing ongoing research with Jonathon Huntley in the study of Weyl's laws for the counting function of cuspidal eigenvalues on quotients of symmetric spaces of arbitrary rank, with particular emphasis on the symmetric spaces associated to GL(n,R). In collaboration with Jurg Kramer, Jorgenson is continuing the investigation of the Arakelov theory of Calabi-Yau varieties. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data tran smission and processing, and communication systems.

该奖项支持谱理论和数论一般领域的研究，特别是对迹公式，显式公式和素数定理，退化黎曼流形的谱理论，以及关于卡拉比-丘氏变种的Arakelov理论的各种分析和算术方面的研究。乔根森继续与塞尔日·朗合作。他们正在研究对称空间上的热核，为他们的一般理论提供进一步的例子。具体的例子，如希尔伯特模和皮卡德模品种将深入研究。Jorgenson正在与Andrey Todorov继续研究Calabi-Yau品种的光谱理论和算法。本研究的长期目标是发展一种类似于现有椭圆曲线的算法理论的Calabi-Yau变种算法理论。Jorgenson与Jozek Dodziuk一起研究有限体积黎曼流形退化族的谱理论，这些流形具有在发展奇点附近具有恒定负截面曲率的度量。Jorgenson正在与Jonathon Huntley继续进行关于任意秩对称空间商的逆特征值计数函数的Weyl定律的研究，特别强调与GL（n，R）相关的对称空间。在与Jurg Kramer的合作下，Jorgenson正在继续对Calabi-Yau品种的Arakelov理论进行研究。本研究属于数论的一般数学领域。数论的历史根源在于对整数的研究，解决的问题是一个整数能被另一个整数整除的问题。它是数学中最古老的分支之一，人们为了纯粹的美学原因而追求了许多世纪。然而，在过去的半个世纪里，它已经成为数据传输和处理以及通信系统等各种应用领域不可或缺的工具。