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

杰伊·乔根森进行的研究涉及与许多合著者合作，研究了一系列由数论问题激发的分析问题，如L-函数，自守形式和迹公式。 工作与Jurg克雷默的重点问题出现在某些方面的Arakelov理论，并延伸到研究θ函数，自守形式，和塞尔伯格的zeta函数。 与科马克奥沙利文，乔根森是发展理论的高阶模的形式，迄今已产生类似克罗内克的极限公式和戴德金总结。 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。 某些方法的证明是共同的所有调查，如技术从解析数论，代数几何和热核分析，这提供了一个有趣的混合物的想法和巩固的结果。 数学小工具被称为热核出现在许多不同领域的数学研究，无论是明确或隐含的，一个方面的杰伊乔根森的研究活动是调查的许多表现形式的热核。 在金融数学中，经典的一维热核可以用来描述Black-Scholes-Merton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、分析和概率中，热核既是主要研究的对象，也是理解研究兴趣的特定问题的工具。 通过研究热核和热核技术的许多实现，Jay Jorgenson能够在一个数学领域中利用经过验证的技术，并在另一个学科中发展思想。 此外，这种研究方法已被证明是有用的，为学生提供本科和研究生，以新的方式，他们有机会获得某些领域的数学研究。

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.