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.

该研究项目的许多方面与建立数量理论中的随机性实例密切相关。数字理论是数学最古老的分支之一。它对技术的应用对于通信系统，数据处理和计算算法至关重要。该项目的目标是由算术家庭中的地标问题驱动的。组装和研究共同特征的物体时会出现家庭。即使一个人对单个物体感兴趣，家庭也通常是至关重要的，因此对于最近解决某些困难的代数和渐近问题的解决至关重要。该项目的这些目标是针对本科和研究生教育的具体计划，这些计划集中在发展有效的写作和沟通能力上。与康奈尔大学的写作学院合作，PI将每月组织有关写作，常规写作小组和在线Wiki的每月研讨会，该研讨会将用作通信平台并为公众提供资源。 PI将继续指导本科研究项目，通过调查开放问题促进学习，在促进学习时传播知识和发现。本研究项目旨在开发有关特殊功能的渐近学的定量理论，例如表示形式的字符。长期的目标是解决自动型时期，l功能的次数和非逐渐播出的问题以及家庭的算术统计。 痕量公式是数字理论的基本工具，尤其是兰兰兹计划的发展。即使取得了巨大进展，重要的问题仍然是开放的，尤其是对于许多应用程序至关重要的分析方面。这些问题现在已经在亚瑟和其他人的作品之后成熟了调查。这项研究的直接结果是针对GL（n）上Maass形式家庭的Sato-Tate Equi-Distripution定理，解决了一个长期存在的问题。 当前，对痕量公式的几何侧的绝对收敛是当前最紧迫的问题之一。第二个重点是痕量字符，这是表示理论中的核心问题，例如本地兰兰兹信函和功能传输。自哈里什·坎德拉（Harish-Chandra）开创性工作以来，PI将致力于定量方面的进步。与此相关的是，PI将继续在Whittaker时期工作，特别是Zuckerman对Infinity的渐近行为的猜想。所提出的活动是将分析，几何形状，表示理论和数学物理学的方法全部带入其全部强度，尤其是象征性的几何形状和整合系统。一个直接的目标是对共同轨道的定量方面的系统研究。