Automorphic Representations and L-Functions

自守表示和 L 函数

• 批准号: 2200890
• 负责人: Dihua Jiang
• 金额: $ 23.8万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-15 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200890&HistoricalAwards=false
• 关键词: Automorphic; Representations; Functions

Symmetries can be found in various basic structures of the universe and provide indispensable guidelines for human beings to understand intrinsic structures of fundamental objects. In mathematics, those symmetries provide a common ground for many different theories such as Geometry, Number Theory, Mathematical Physics, Algebra, and Analysis, especially through functions known as Automorphic Forms. Hence, the modern theory of automorphic forms provides the organizing principle for further research in these areas. This project will establish basic structures and yield substantial contributions to the modern theory of automorphic forms, and the related Langlands program. For broader impacts, the PI will continue to train graduate students and postdocs, give lectures on his research to a broader community, including public lectures, primary lectures and research talks in various settings, and organize research programs and zoom-seminars for the international mathematics community. The PI will continue his research on the discrete spectrum of square-integrable automorphic forms, L-functions, and the Langlands functoriality conjectures. The basic problems that the PI will investigate are refined structures of the discrete spectrum of automorphic forms on classical groups, analytic and arithmetic properties of automorphic L-functions, and explicit Langlands functorial transfers for square-integrable automorphic forms via automorphic integral transforms, in particular, using twisted automorphic descents. On the one hand, the PI intends to study refined structure based on the existence of endoscopy. On the other hand, the PI intends to construct explicit modules for the cuspidal automorphic forms via integral transform with automorphic kernel functions, so that the endoscopic transfers can be realized via automorphic integral transforms. Meanwhile, the PI will also develop the local theory, relating basic problems in harmonic analysis of groups over a local field to the arithmetic data that are given by the local Langlands conjecture. The long-term research goal of this project is to understand the general local-global-automorphic principles in the theory of automorphic forms, which reflects one of the basic principles in arithmetic and number theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

对称性存在于宇宙的各种基本结构中，为人类理解基本物体的内在结构提供了不可或缺的指导方针。在数学中，这些对称性为许多不同的理论提供了共同的基础，如几何学，数论，数学物理学，代数和分析，特别是通过称为自守形式的函数。 因此，自守形式的现代理论为这些领域的进一步研究提供了组织原则。该项目将建立基本结构，并对自守形式的现代理论和相关的朗兰兹纲领做出重大贡献。为了更广泛的影响，PI将继续培训研究生和博士后，向更广泛的社区讲授他的研究，包括公开讲座，主要讲座和各种环境的研究会谈，并为国际数学界组织研究计划和放大研讨会。PI将继续研究平方可积自守形式的离散谱，L函数和朗兰兹函数性。PI将研究的基本问题是经典群上自守形式的离散谱的精细结构，自守L-函数的分析和算术性质，以及通过自守积分变换，特别是使用扭曲自守下降的平方可积自守形式的显式Langlands函子转移。一方面，PI打算根据内窥镜检查的存在研究精细结构。另一方面，PI打算通过具有自守核函数的积分变换来构造尖点自守形式的显式模块，从而可以通过自守积分变换来实现内窥镜传输。同时，PI还将发展局部理论，将局部域上群的调和分析中的基本问题与局部朗兰兹猜想给出的算术数据联系起来。该项目的长期研究目标是理解自守形式理论中的一般局部-全局-自守原理，这反映了算术和数论的基本原理之一。该奖项反映了NSF的法定使命，通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。