Analysis of Whittaker periods and applications to automorphic forms

这个建议的动机是自守形式和L-函数的理论。重点是调和分析和遍历理论之间的相互作用。主要重点是惠特克时期的分析。自守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和证明尖锐的结果。该提案包括三个密切相关的项目。一个由数的几何学激发的项目涉及欧几里得空间中整数标志的偏斜度的均匀分布。一个相关的分析问题是估计全球惠特克周期。我们将显着改善以前的结果在文献中使用全局遍历理论的方法。第三个问题涉及当地惠特克职能。我们要建立Whittaker函数在无穷远处的渐近性质。有几个结果证明这个猜想以及应用自守形式。PI将应用几何和表示论的方法。数论是数学中最古老的分支之一，小学教授基本算术。技术应用很普遍：通信系统、数据处理、加密算法。L-函数捕捉了关于到处出现的素数的基本信息。朗兰兹纲领是一个由表示论和数论之间的相互作用所激发的庞大的理论和结果网络。拟议的研究将提供一个新的桥梁之间的朗兰兹程序和几个主题的分析和表征理论。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有很多长期存在的问题;周期和L函数的分析是一个中心主题和驱动力。所提出的研究提供了理论结果，可以作为前瞻性的工具，在许多不同的问题：不同的团队的数值调查，消失的特殊值和算术周期，时刻，周期界限和次凸性问题。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，同时通过开放问题的调查促进学习。

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.