Automorphic Representations and L-Functions

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。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、数论、数学物理、代数和分析，特别是通过称为自同构形式的函数。因此，现代自同构形式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，并对自同构形式的现代理论和相关的朗兰兹程序做出重大贡献。为了产生更广泛的影响，PI将继续培养研究生和博士后，向更广泛的社区发表他的研究报告，包括公开讲座、初级讲座和各种环境下的研究讲座，并为国际数学界组织研究项目和缩放研讨会。PI将继续他对平方可积自同构形式的离散谱、l函数和朗兰兹泛函猜想的研究。PI将研究的基本问题是经典群上自同构形式的离散谱的精细结构，自同构l函数的解析和算术性质，以及通过自同构积分变换，特别是使用扭曲自同构下降的平方可积自同构形式的显式Langlands泛函转移。一方面，PI打算在内窥镜存在的基础上研究精细结构。另一方面，PI打算通过自同态核函数的积分变换来构造倒刀自同态形式的显式模，从而通过自同态积分变换实现内窥镜变换。同时，PI还将发展局部理论，将局部场上群谐分析的基本问题与局部朗兰兹猜想给出的算术数据联系起来。本课题的长期研究目标是了解自同构形式理论中的一般局部-全局自同构原理，它反映了算术和数论的基本原理之一。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.