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

重Dirichlet级数理论近年来取得了很大的进展。各种以前研究的例子已经组织成一个连贯的框架。新兴的结构都建议自然的概括-经常与应用程序解析数论-并指向意想不到的连接等不同领域的数学作为频谱理论的自守形式，算术的功能领域，几何的仿射根系统和组合表示理论。许多应用在解析数论中已经发现，并期待更多。这些包括矩估计和凸性破坏的L-函数在任意数域，非零的结果L-函数在数域和功能领域和结果的性质神秘的惠特克系数的metaplectic爱森斯坦系列和高阶theta函数。此外，在过去的几年里，研究人员的共同努力证明了外尔群多重狄利克雷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。这笔赠款将资助对这些迅速发展的地区的继续调查。此外，还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。数论始于数千年前，最初受到有关素数问题的启发。狄利克雷级数是无穷级数，例如黎曼zeta函数，并且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互连而脱颖而出。 多重狄利克雷级数是简单的狄利克雷级数在几个变量-他们有一个优点，他们测量的数论数量本身可以是狄利克雷级数，特别是L-函数，这是基本对象，可以与许多类数论数据，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.