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

近年来，多重狄利克雷级数的理论研究取得了很大的进展。各种先前研究的例子被组织成一个连贯的框架。这些涌现的结构既暗示了自然的推广——通常应用于解析数论——也指向了与各种数学领域的意想不到的联系，如自同构形式的谱理论、函数场的算术、仿射根系统的几何和组合表示理论。在分析数论中已经发现了许多应用，并且期望有更多的应用。其中包括任意数域上l -函数的矩估计和凸破，数域和函数域上l -函数的非消失结果以及关于元爱森斯坦级数和高阶函数的神秘惠特克系数性质的结果。此外，在过去几年中，研究人员的共同努力已经证明，Weyl群多重狄利克雷级数具有一个以前未知的美丽结构，并且通过阐明这个结构，与其他数学领域的新联系正在迅速出现。这笔赠款将资助对这些快速发展地区的持续调查。此外，计划举办两次讲习班，向研究数学家和研究生传播这些成果和新技术。数论始于几千年前，最初是受到质数问题的启发。狄利克雷级数是无穷级数，如黎曼ζ函数，是研究素数的主要工具。最近，它们通过提供纯数学和物理的许多不同领域之间的相互联系而脱颖而出。多重狄利克雷级数就是几个变量的狄利克雷级数——它们的优点是它们测量的数论量本身可以是狄利克雷级数，特别是l函数，它是可以与许多类型的数论数据相关联的基本对象，如椭圆曲线、伽罗瓦群的表示或模形式。

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.