FRG: Collaborative Research: Combinatorial representation theory, multiple Dirichlet series, and moments of L-functions

FRG：协作研究：组合表示理论、多重狄利克雷级数和 L 函数矩

• 批准号: 0652529
• 负责人: Benjamin Brubaker
• 金额: $ 9.84万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0652529&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Combinatorial; representation

Great progress has been made in recent years in the theory of multiple Dirichlet series. A variety of previously studied examples have been organized into a coherent framework. The emergent structures serve to both suggest natural generalizations---often with applications to analytic number theory---and point towards unexpected connections with such diverse areas of mathematics as the spectral theory of automorphic forms, arithmetic of function fields, the geometry of affine root systems and combinatorial representation theory. Many applications in analytic number theory have been found and many more are expected. These include moment estimates and convexity breaking for L-functions over an arbitrary number field, nonvanishing results for L-functions over number fields and function fields and results on the nature of the mysterious Whittaker coefficients of metaplectic Eisenstein series and higher order theta functions. Moreover, during the past several years the combined efforts of the investigators have demonstrated that Weyl group multiple Dirichlet series have a beautiful structure that was previously unknown, and by elucidating this structure, new connections with other areas of mathematics are rapidly emerging. The grant will fund continued investigation of these rapidly developing areas. In addition, two workshops are planned for the dissemination of these results and new techniques to research mathematicians and graduate students.Number theory began thousands of years ago and was initially inspired by questions about prime numbers. Dirichlet series are infinite series, such as the Riemann zeta function, and are a primary tool in the study of prime numbers. More recently they have come to fore by providing interconnections between many diverse areas of pure mathematics and physics. Multiple Dirichlet series are simply Dirichlet series in several variables -- they have the merit that the number theoretic quantities they measure can themselves be Dirichlet series, in particular L-functions, which are fundamental objects that can be associated with many classes of number-theoretic data, such as elliptic curves, representations of Galois groups, or modular forms.

近年来，在多个Dirichlet系列的理论中取得了巨大进展。许多先前研究的例子已被组织成一个连贯的框架。紧急结构既提示自然概括 - 通常都应用于分析数理论，并指向与自动形式的光谱理论，功能领域的算术，仿射根系和组合代表理论等数学的多样化领域的意外联系。已经发现了分析数理论的许多应用，并且期望更多。这些包括在任意数字字段上L功能的力矩估计和凸度破坏，l功能在数字字段和功能字段上的非呈现结果以及对Mystapious Whittaker系数的性质的性质以及更高顺序的THETA功能的性质。此外，在过去的几年中，调查人员的综合努力表明，Weyl Group多个Dirichlet系列具有以前未知的美丽结构，并且通过阐明这种结构，与其他数学领域的新联系正在迅速发展。该赠款将资助对这些快速发展的地区的持续调查。此外，还计划了两个研讨会，以传播这些结果，并为研究数学家和研究生进行新技术。名称理论始于数千年前，最初是受到有关质数问题的启发。 Dirichlet系列是无限序列，例如Riemann Zeta函数，并且是研究质数的主要工具。最近，它们通过在纯数学和物理学的许多不同领域之间提供互连来提起诉讼。 多个Dirichlet系列仅在几个变量中只是Dirichlet系列 - 它们的优点是，他们测量的数量理论数量本身可以是Dirichlet系列，特别是L功能，这些函数是基本的对象，这些对象可以与许多类别的数字理论数据相关联，例如椭圆形曲线，Galois组的表示，Galois组的表示，或模块化形式。