CAREER: Trace Formula and Geometric Analysis of Automorphic Forms

本研究计划的许多方面与建立数论中的随机性实例密切相关。数论是数学最古老的分支之一；它在技术上的应用对于通信系统、数据处理和计算算法来说是普遍和至关重要的。这个项目的目标是由算术族的里程碑问题驱动的。当把具有共同特征的物体聚集在一起研究时，家庭就产生了。即使一个人对单一对象先验地感兴趣，家庭也常常是至关重要的，因此对最近解决某些困难的代数和渐近问题至关重要。这些项目目标的补充是针对本科生和研究生教育的具体举措，以培养有效的写作和沟通技能为中心。在与康奈尔大学写作研究所的合作下，PI将组织每月一次的写作研讨会，定期的写作小组，以及一个在线维基，作为一个交流平台，为公众提供资源。PI将继续指导本科生的研究项目，传播知识和发现，同时通过对开放性问题的调查促进学习。本研究计划旨在发展特殊函数渐近性的定量理论，例如表征的特征。长期目标是解决自同构周期、l函数的次凸性和非消失性、族的算术统计等问题。迹公式是数论的基本工具，特别是朗兰兹程序的发展。尽管已经取得了巨大的进步，但重要的问题仍然存在，特别是对许多应用至关重要的分析方面。在亚瑟和其他人的作品之后，这些问题现在已经成熟了。本研究的直接成果是GL(n)上质量形式族的Sato-Tate等分布定理，解决了一个长期存在的问题。对轨迹公式几何边的绝对收敛性的认识是目前该学科中最紧迫的问题之一。第二个重点是痕迹字符，这是表征理论的核心问题，如局部朗兰兹对应和功能转移。PI将致力于定量方面的工作，自哈里什-钱德拉的开创性工作以来，这些工作几乎没有取得进展。与此相关，PI将继续研究惠特克周期，特别是关于无穷远处渐近行为的祖克曼猜想。提议的活动是将分析，几何，表示理论和数学物理的方法充分发挥其优势，特别是辛几何和可积系统；一个直接的目标是系统地研究伴随轨道的数量方面。

Many aspects of this research project are intimately related to establishing instances of randomness in number theory. Number theory is among the oldest branches of mathematics; its applications to technology are prevalent and vital for communication systems, data processing, and computational algorithms. The goals of this project are driven by landmark problems on arithmetic families. Families arise when assembling and studying together objects that share common features. Families are often crucial even if one is a priori interested in a single object and thereby are central to the recent resolution of certain difficult algebraic and asymptotic questions. These goals of the project are complemented by concrete initiatives targeted at undergraduate and graduate education that are centered on developing effective writing and communication skills. In collaboration with the Institute for Writing at Cornell University, the PI will organize a monthly seminar on writing, regular writing groups, and an online wiki that will serve as a communication platform and access to resources for the general public. The PI will continue to mentor undergraduate research projects, disseminating knowledge and discoveries while promoting learning through the investigation of open problems.This research project aims to develop a quantitative theory of the asymptotics of special functions, such as characters of representations. The long-term goal is to solve problems on automorphic periods, subconvexity and non-vanishing of L-functions, and arithmetic statistics of families. The trace formula is a fundamental tool in number theory and the development of the Langlands program in particular. Even though there has been enormous progress, important questions remain open, notably analytic aspects that are critical for many applications. These questions are now ripe for investigation following the works of Arthur and others. An immediate outcome of this research is a Sato-Tate equi-distribution theorem for families of Maass forms on GL(n), resolving a long-standing problem. The understanding of the absolute convergence of the geometric side of the trace formula is currently one of the most urgent problems in the subject. A second focus is on trace characters, which are a central concern in representation theory, such as the local Langlands correspondence and functorial transfers. The PI will work on quantitative aspects that have seen little progress since the seminal work of Harish-Chandra. Related to this, the PI will continue work on Whittaker periods, notably towards a conjecture of Zuckerman on the asymptotic behavior at infinity. The proposed activity is to bring methods from analysis, geometry, representation theory, and mathematical physics in their full strength, notably symplectic geometry and integrable systems; an immediate goal is the systematic study of the quantitative aspects of coadjoint orbits.