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.

可以在宇宙的各种基本结构中找到对称性，并为人类提供必不可少的准则，以了解基本对象的内在结构。在数学中，这些对称性为许多不同理论（例如几何学，数理论，数学物理学，代数和分析）提供了共同的基础，尤其是通过称为自动形式的功能。 因此，自动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。该项目将建立基本的结构，并为汽车形式的现代理论和相关的兰兰兹计划做出实质性贡献。对于更广泛的影响，PI将继续培训研究生和博士后，向更广泛的社区提供有关他的研究的讲座，包括在各种环境中的公开讲座，主要的讲座和研究演讲，并为国际数学社区组织研究计划和Zoom-Seminars。 PI将继续对正方形的自动形式，L功能和Langlands功能性猜想的离散光谱进行研究。 PI将要研究的基本问题是自动形式的离散频谱的完善结构，在古典群体上，自动形态L功能的分析和算术特性，以及明确的Langlands通过自动形态整合的自动形式转换，尤其是使用Twist twist twist deScent twisted descc，可用于方形综合自动形式。一方面，PI打算根据内窥镜的存在研究精制的结构。另一方面，PI打算通过具有自动形核功能的积分转换来构建尖锐的自动形式的显式模块，以便可以通过自动形态积分转换来实现内窥镜转移。同时，PI还将发展本地理论，将局部领域组的谐波分析中的基本问题与局部兰兰兹猜想给出的算术数据有关。该项目的长期研究目标是了解自动形式理论中的一般局部全球形态原则，这反映了算术和数字理论的基本原理之一。该奖项反映了NSF的法定使命，并通过基金会的知识优点和广泛的影响来评估NSF的法定任务，并被认为是值得的。