Automorphic Representations and L-Functions

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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.