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、关于退化黎曼流形的谱理论以及关于Calabi-Yau簇的Arakelov理论的各种分析和算术方面的研究。乔根森正在继续他与Serge Lang的合作。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，如希尔伯特模变元和Picard模变元将被深入研究。Jorgenson正在继续与Andrey Todorov一起研究Calabi-Yau变种的谱理论和算法。这项研究的长期目标是发展一种类似于现有椭圆曲线算术理论的Calabi-Yau变种算术理论。Jorgenson和Jozek Dodziuk正在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常负截面曲率的度量。Jorgenson和Jonathon Huntley正在继续研究Weyl定律，研究任意秩对称空间的商上的尖角特征值的计数函数，特别是与GL(n，R)相关的对称空间。Jorgenson与Jurg Kramer合作，继续研究Calabi-Yau品种的Arakelov理论。这项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，它解决了一些问题，比如一个整数被另一个整数整除的问题。它是数学中最古老的分支之一，出于纯粹的美学原因，人们追寻了许多个世纪。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领域的各种应用中不可或缺的工具。