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.

该奖项支持研究一般领域的谱理论和数论，特别感兴趣的迹公式，明确的公式和素数定理，谱理论退化黎曼流形，以及各种分析和算术方面的阿拉克洛夫理论的卡拉比-丘品种。乔根森正在继续与Serge Lang合作。他们正在研究对称空间上的热核，以提供他们一般理论的更多例子。具体的例子，如希尔伯特模和皮卡德模品种将深入研究。Jorgenson继续与Andrey Todorov一起研究Calabi-Yau簇的谱理论和算法。本研究的长期目标是发展一种类似于现有的椭圆曲线算术理论的Calabi-Yau算法。与Jozek Dodziuk，乔根森正在研究谱理论的退化家庭的有限体积黎曼流形拥有度量，有恒定的负截面曲率附近的发展奇点。 乔根森是继续进行中的研究与乔纳森亨特利在研究外尔的法律的计数功能尖特征值的concurents对称空间的任意秩，特别强调对称空间相关的GL（n，R）。在与Jurg克雷默，乔根森正在继续调查的阿拉克洛夫理论的卡拉比-丘品种。 本文的研究属于数论中的一般数学领域福尔斯。数论有其历史根源，在研究整个数字，解决这样的问题，如那些处理整除一个整数由另一个。它是数学最古老的分支之一，几个世纪以来出于纯粹的美学原因而受到人们的追求。 然而，在过去的半个世纪，它已成为数据等领域的各种应用中不可或缺的工具 传输和处理以及通信系统。