On the Automorphic Discrete Spectrum of Classical Groups: Constructions and Characterizations

该项目涉及数论的研究，数论是数学的一个分支，与许多其他学科有联系，包括数据安全和物理学。 自守形式是具有丰富对称性的函数，在现代数论中占有重要地位。在应用中，这些对称性可以作为理解我们宇宙中物体内在结构的指导方针。在数学中，这些对称性是许多不同学科的共同基础，包括几何、数论、数学物理、代数和分析;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。 该研究项目旨在研究自守形式的基本结构，并促进对自守形式的离散谱的理解，并可能应用于数论和算术的基本问题。 研究生参与了这个项目。这个研究项目调查了一系列关于平方可积自守形式的基本问题，以及在复群、真实的群和p-adic群的表示中相应的局部问题。 目标是理解自守形式的精细结构及其与调和分析和数论的联系。研究者打算构造尖点自守形式的显式模。此外，该项目旨在发展局部理论，将其与p-adic群的调和分析中的基本问题联系起来。长期目标是理解自守形式理论中的一般局部-整体-自守原理，它反映了算术和数论的基本原理之一。

This project concerns research in number theory, a branch of mathematics that has connections to many other subjects, including data security and physics. Automorphic forms, which play an important role in modern number theory, are functions with abundant symmetries. In applications, these symmetries serve as guidelines to understand the intrinsic structures of objects in our universe. In mathematics, these symmetries are common ground for many different subjects, including geometry, number theory, mathematical physics, algebra, and analysis; the modern theory of automorphic forms provides the organizing principle for further research in these areas. This research project aims to investigate basic structures of automorphic forms and to advance understanding of the discrete spectrum of automorphic forms, with potential applications to fundamental questions in number theory and arithmetic. Graduate students are involved in the project.This research project investigates a set of fundamental questions on square integrable automorphic forms, as well as the corresponding local problems in the representations of complex, real, and p-adic groups. The goal is to understand refined structures of automorphic forms and their connections to harmonic analysis and number theory. The investigator intends to construct explicit modules for cuspidal automorphic forms. In addition, the project aims to develop the local theory, relating it to basic problems in harmonic analysis of p-adic groups. The long-term goal 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.