Analytic questions motivated by L functions, Eisenstein series, automorphic forms, and trace formulae

由 L 函数、爱森斯坦级数、自同构形式和迹公式引发的分析问题

• 批准号: 0503669
• 负责人: Jay Jorgenson
• 金额: --
• 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: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0503669&HistoricalAwards=false
• 关键词: Analytic; questions; motivated; functions; Eisenstein

The research undertaken by Jay Jorgenson involves collaboration with numerous co-authors on a range of questions in analysis which are motivated by questions from number theory, such as L-functions, automorphic forms and trace formulae. The work with Jurg Kramer focuses on questions which arise in certain aspects of Arakelov theory, and extends into the study of theta functions, automorphic forms, and Selberg's zeta function. With Cormac O'Sullivan, Jorgenson is developing the theory of higher order modular forms which to date has produced analogues of Kronecker's limit formula and Dedekind sums. The research with Serge Lang involves zeta function constructions for general symmetric spaces, beginning with SL(n,C). Certain methods of proof are common to all investigations, such as techniques from analytic number theory, algebraic geometry and heat kernel analysis, which provides for an interesting mixture of ideas and consolidation of results. The mathematical gadget known as the heat kernel appears in many diverse areas of mathematics research, either explicitly or implicitly, and one aspect of Jay Jorgenson's research activities is the investigation of the many manifestations of the heat kernel. In the mathematics of finance, the classical heat kernel in one dimension can be used to describe the Black-Scholes-Merton formula, which is used to price certain stock options. In number theory, the heat kernel is used to construct many basic objects of study, such as theta and zeta functions. In geometry, analysis and probability, the heat kernel is both an object of primary study as well as a tool to understand specific questions of research interest. By studying the many realizations of heat kernels and heat kernel techniques, Jay Jorgenson is able to utilize proven techniques in one field of mathematics and develop ideas in another discipline. In addition, this method of study has proven useful in providing students, both undergraduate and graduate, with new ways in which they have access to certain areas of mathematical research.

Jay Jorgenson的研究涉及到与众多合著者在分析中的一系列问题上的合作，这些问题的动机来自数论中的问题，如L函数、自同构形和迹公式。与Jurg Kramer的工作集中于在Arakelov理论的某些方面出现的问题，并扩展到对theta函数、自同构形和Selberg的Zeta函数的研究。Jorgenson与Cormac O‘Sullivan一起发展了高阶模形式理论，到目前为止已经产生了类似于Kronecker极限公式和Dedekind和的理论。Serge Lang的研究涉及一般对称空间的Zeta函数构造，从SL(n，C)开始。某些证明方法对所有研究都是通用的，例如来自解析数论、代数几何和热核分析的技术，它提供了有趣的想法和结果的巩固。被称为热核的数学小工具出现在数学研究的许多不同领域，或明或暗，而Jay Jorgenson的研究活动的一个方面是对热核的许多表现形式的调查。在金融数学中，一维经典热核可以用来描述布莱克-斯科尔斯-默顿公式，该公式被用来为某些股票期权定价。在数论中，热核被用来构造许多基本的研究对象，如theta和zeta函数。在几何学、分析学和概率学中，热核既是主要研究的对象，也是理解研究兴趣的具体问题的工具。通过研究热核和热核技术的许多实现，Jay Jorgenson能够利用一个数学领域中经过验证的技术，并在另一个学科中发展思想。此外，事实证明，这种学习方法在为本科生和研究生提供进入某些数学研究领域的新途径方面很有用。