Analysis of Whittaker periods and applications to automorphic forms

这一提议是由自同构形式和l -函数的理论驱动的。重点是谐波分析和遍历理论之间的相互作用。主要的重点是惠特克时期的分析。自同构形式的惠特克周期在许多问题中经常出现。我们的目标是有一个完整的理论，并证明明确的结果。该提案包含三个密切相关的项目。一个由数字几何驱动的项目涉及欧几里得空间中积分旗的均匀偏度分布。一个相关的分析问题是估计全球惠特克周期。我们将使用遍历理论的全局方法显著改进文献中先前的结果。第三个问题涉及局部惠特克函数。我们要建立惠特克函数在无穷远处的猜想渐近性质。有几个结果证明了这个猜想，以及应用到自同构形式。PI将应用几何和表示理论的方法。数论是数学中最古老的分支之一，小学就开始教授基础算术。对技术的应用是普遍的：通信系统，数据处理，密码算法。l -函数捕获了随处可见的素数的基本信息。朗兰兹计划是一个巨大的猜想和结果网络，由表征理论和数论之间的相互作用所激发。提出的研究将在朗兰兹计划和分析和表示理论的几个主题之间提供一个新的桥梁。这将通过识别深刻的类比加深我们的理解和知识，并促进不同领域专家之间的合作。由于它的历史，分析和数论之间的接口存在许多长期存在的问题；周期和l函数的分析是一个中心主题和驱动力。所提出的研究提供的理论结果可作为许多不同问题的前瞻性工具：不同团队的数值研究，特殊值和算术循环的消失，矩，周期界和次凸性问题。PI将继续教授和指导学生的研究项目：初级论文，高级和博士论文，传播知识和发现，同时通过调查开放问题促进学习。

This proposal is motivated by the theory of automorphic forms and L-functions. The emphasis is on the interplay between harmonic analysis and ergodic theory. The primary focus is the analysis of Whittaker periods. Whittaker periods of automorphic forms occur very frequently in many problems. Our goal is to have a complete theory and to prove sharp results. The proposal contains three closely related projects. A project motivated by the geometry of number concerns the uniform distribution of the skewness of integral flags in euclidean space. A related analytic problem is to estimate global Whittaker periods. We shall improve significantly previous results in the literature using global methods from ergodic theory. The third problem concerns local Whittaker functions. We want to establish the conjectural asymptotic behavior of Whittaker functions at infinity. There are several outcomes of proving this conjecture as well as applications to automorphic forms. The PI will apply methods from geometry and representation theory.Number theory is among the oldest branches in mathematics and basic arithmetic is taught in elementary school. Applications to technology are prevalent: communication systems, data processing, cryptographic algorithms. L-functions capture fundamental information about prime numbers which appear everywhere. The Langlands program is a vast network of conjectures and results motivated by the interplay between representation theory and number theory. The proposed research will provide a new bridge between the Langlands program and several topics in analysis and representation theory. This will deepen our understanding and knowledge through identifying profound analogies and stimulate collaboration between experts in different fields. Because of its history the interface between analysis and number theory has plenty of longstanding problems; the analysis of periods and L-functions is a central theme and driving force. The proposed research provides theoretical results which can be used as prospective tools in many different problems: numerical investigations by different teams, vanishing of special values and arithmetic cycles, moments, period bounds and subconvexity problems. The PI will continue to teach and mentor students research projects: junior papers, senior and PhD thesis, disseminating knowledge and discoveries while promoting learning through the investigation of open problems.