Constructions of L-functions, Eigenvalue Bounds and Statistics

这是一个研究自同构形式、表示理论和酉矩阵统计的项目。Bump教授将利用二重狄利克雷级数的新方法研究对称立方和对称四次l函数。该方法将l -函数视为二重狄利克雷级数的残数，其泛函方程由较简单的l -函数的泛函方程衍生而来。其他类型的l -函数将使用Rankin-Selberg方法进行研究。在相关工作中，Bump教授还将通过舒尔函数研究随机酉矩阵的统计。研究者希望一种新颖的新方法能进一步推进这一经典研究。这是数论的一个课题。数论的核心是研究数数1、2、3、4....这些最基本的数字具有复杂、有趣和神秘的性质，在科学和现代技术的许多分支中都有应用。数字的许多重要性质可以用数学方法概括为l函数。这些l函数起源于微积分的研究，可以用几种不同的公式来表示。巴普教授的项目集中在某些l函数的研究上。他希望他对l函数的研究能增加我们对这些函数所编码的数字的性质的理解。

Daniel BumpStanford University UniversityDMS 9970841 This is a project in the study of automorphic forms, representation theory, and the statistics of unitary matrices. Professor Bump will study the symmetric cube and the symmetric fourth power L-functions by means of a new technique of double Dirichlet series. In this new approach L-functions are treated as residues of double Dirichlet series whose functional equations arise from functional equations of simpler L-functions. Other types of L-functions will be studied using the methods of Rankin-Selberg. In related work Professor Bump will also study the statistics of random unitary matrices via Shur functions. The investigator hopes that a novel new approach will further this classical study. This is a project in Number Theory. At its heart, Number Theory is the study of the counting numbers 1, 2, 3, 4.... These most basic numbers have elaborate, interesting, and mysterious properties with applications in many branches of science and modern technology. Many of the important properties of numbers can be mathematically summarized in objects called L-functions. These L-functions have their origins in the study of calculus and can be represented by several different formulas. Professor Bump's project concentrates on the study of certain kinds of L-functions. He expects his investigation of L-functions to increase our understanding of the properties of numbers that these functions encode.