The PI, Dihua Jiang, has been working on some basic problems related to periods, L-functions and explicit Langlands functorial transfers for square-integrable automorphic forms. He investigates the basic structures of the discrete spectrum of automorphic forms and the related problems on the Langlands functoriality, and establishes explicit formulas for residues or special values of automorphic L-functions. In the local theory, his research attacks the local Langlands conjectures and related basic problems in harmonic analysis of p-adic groups. His 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 the arithmetic and number theory.

The PI, Dihua Jiang, is an expert in the modern theory of automorphic forms and the Langlands Program. Automorphic forms are functions with abundant symmetries. These symmetries are the guidelines to understand 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 Dihua Jiang establishes basic structures for automorphic forms and hence contributes essentially to the Langlands program.

Project Report

of the PI for NSF Grant DMS-1001672 for 2010--2013, extended to 2014. The goal of the research conducted by the PI is to understand the symmetric structures of certain generalized periodic functions defined on higher dimensional spaces with plenty symmetries. Such fundtions forms main objects in the modern theory of automorphic forms, whose classical theory has profound impact in the development of mathematics for more than 200 years. The Langlands program has been the organizing principle to reflect the deep arithmetic meaning of automorphic forms from their symmetric structures expressed in terms of harmonic analysis, group representation, and geometry. The research conducted by the PI has two aspects: one is to introduce more intrinsic invariants to characterize the behavior of the signal spectrum of the space of automorphic forms under certain basic Langlands functorial transfers, called the Endoscopy Transfers, for classical groups, and the other is to explicitly construct models which yield concrete realizations for any single spectrum of automorphic forms. Historically, such explicit realizations have great impact for applications of the highly transcendental automorphic functions to concrete problems in Number Theory, Arithmetic, Physics, and other disciplines. The PI published more than 15 research papers in highly regarded peer reviewed Math journals and gave presentation on his funding in various workshops and conferences, nation-wide and internationally. Th PI trained three graudate students and a few postdocs, who continued their careers in Mathematics. The Pi collaborated with experts in USA, China, Israel, Twiwan, and Singapore, has actively participated in organization of seminars, workshops and conferences, and has been the mentor for 10 undergraduate students to write their graduation senior thesis. The PI has been an editor for math journals, research proposal reviewers for US and some other countries.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Andrew Pollington
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University of Minnesota Twin Cities
United States
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