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

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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.