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

多重狄里克莱级数的理论近年来取得了很大的进展。以前研究过的各种例子被组织成一个连贯的框架。新出现的结构既用于暗示自然概括-通常用于解析数论-并指向与诸如自同构形式的谱理论、函数域的算术、仿射根系的几何和组合表示理论等不同数学领域的意想不到的联系。在解析数论中已经发现了许多应用，并期待着更多的应用。这些结果包括任意数域上L函数的矩估计和凸破缺，数域和函数域上L函数的非零化结果，以及关于亚二次Eisenstein级数和高阶theta函数的神秘惠特克系数的性质的结果。此外，在过去的几年中，研究人员的共同努力表明，Weyl群多重狄利克莱级数具有以前未知的美丽结构，通过阐明这种结构，与其他数学领域的新联系正在迅速出现。这笔赠款将用于继续调查这些快速发展的地区。此外，还计划举办两个研讨会，向研究数学家和研究生传播这些结果和新技术。数字理论始于数千年前，最初是受到有关质数的问题的启发。狄里克莱级数是无穷级数，如Riemann Zeta函数，是研究素数的主要工具。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相互联系而脱颖而出。多重Dirichlet级数是简单的多变量Dirichlet级数--它们的优点是它们测量的数论数量本身可以是Dirichlet级数，特别是L函数，这些函数是基本对象，可以与许多类数论数据联系在一起，如椭圆曲线、伽罗华群的表示或模形式。

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.