CAREER: Trace Formula and Geometric Analysis of Automorphic Forms

职业：自守形式的迹公式和几何分析

• 批准号: 1454893
• 负责人: Nicolas Templier
• 金额: $ 48万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1454893&HistoricalAwards=false
• 关键词: CAREER; Trace; Formula; Geometric; Analysis

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.

这个研究项目的许多方面都与建立数论中随机性的实例密切相关。数论是数学中最古老的分支之一;它在技术上的应用是普遍的，对于通信系统，数据处理和计算算法至关重要。这个项目的目标是由算术家庭的里程碑式的问题。当将具有共同特征的物体组装在一起并进行研究时，就会出现家庭。家庭往往是至关重要的，即使是一个先验感兴趣的一个单一的对象，从而是中央最近解决某些困难的代数和渐近问题。该项目的这些目标得到了针对本科生和研究生教育的具体举措的补充，这些举措以发展有效的写作和沟通技能为中心。PI将与康奈尔大学写作研究所合作，每月组织一次写作研讨会，定期组织写作小组，并建立一个在线wiki，作为公众交流和获取资源的平台。PI将继续指导本科生研究项目，传播知识和发现，同时通过研究开放性问题促进学习。本研究项目旨在发展特殊函数的渐近性的定量理论，如表征的特征。长期目标是解决自守周期、L-函数的次凸性和非零性以及族的算术统计等问题。 迹公式是数论中的一个基本工具，特别是朗兰兹纲领的发展。尽管已经取得了巨大的进展，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。在亚瑟和其他人的著作之后，这些问题现在已经成熟，可以进行调查了。这项研究的一个直接成果是GL（n）上Maass形式族的Sato-Tate等分布定理，解决了一个长期存在的问题。 几何边迹公式的绝对收敛性的认识是目前该学科最迫切需要解决的问题之一。第二个重点是迹字符，这是一个中心关注的表示理论，如本地朗兰兹对应和函子转移。PI将致力于自Harish-Chandra开创性工作以来几乎没有进展的定量方面。与此相关，PI将继续研究惠特克周期，特别是Zuckerman关于无穷大渐近行为的猜想。拟议的活动是把分析，几何，表示论和数学物理的方法，在他们的全部力量，特别是辛几何和可积系统;一个直接的目标是系统的研究coadjoint轨道的定量方面。