Constructions of L-functions, Eigenvalue Bounds and Statistics

Daniel Bump斯坦福大学DMS 9970841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教授将利用双重Dirichlet级数的新技巧来研究对称立方函数和对称四次L函数。在这一新方法中，L函数被视为二重狄里克莱级数的残数，其函数方程由更简单的L函数的函数方程导出。其他类型的L函数将用Rankin-Selberg的方法进行研究。在相关工作中，Bump教授还将通过Shur函数研究随机酉阵的统计量。研究人员希望，一种新的方法将推动这一经典研究。这是数论中的一个项目。数论的核心是研究计数1、2、3、4……这些最基本的数字具有精细、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称为L函数的对象。这些L函数起源于微积分的研究，可以用几个不同的公式来表示。Bump教授的项目集中于对某些类型的L函数的研究。他希望自己对L函数的研究能加深我们对这些函数所编码的数的性质的理解。

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.