This project focuses on the modern theory of automorphic forms and the Langlands Program. Automorphic Forms are functions with abundant symmetries. These symmetries are the guidelines to understanding the intrinsic structures of objects in our universe. In Mathematics, these symmetries are common grounds for many different theories such as Geometry, Number Theory, Mathematical Physics, Algebra and Analysis. Hence the modern theory of automorphic forms, essentially the Langlands program, provides the organizing principle for further research in these areas. The research of the PI has a goal of establishing basic structures for automorphic forms. The PI will train graduate students and postodcs, and give lectures on his research to broader community, including public lectures, primary lectures and research talks in various occasions and conferences.
The PI, Dihua Jiang, 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 has been investigating 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. The theory of endoscopy, the existence of which was discovered by R. Langlands in 1980's and confirmed through the fundamental work of B.-C. Ngo and J. Arthur and others via the trace formula approach. 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 integral transforms, Moreover, the PI will develop the theory of twisted automorphic descents that can be used to prove substantially new cases of the global Gan-Gross-Prasad conjecture, establish new higher rank cases of non-vanishing of the central critical value of tensor product $L$-functions. Meanwhile, the PI also plans to develop the local theory, relating basic problems in harmonic analysis of groups over a local filed to the arithmetic data that are given by the local Langlands conjecture. The long term research goal of the PI is to understand the general local-global-automorphic principles in the theory of automorphic forms, which reflects one of the basic principles in the 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.