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

点击翻译按钮获取中文摘要

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.