Automorphic Forms and the Langlands Program

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This award concerns research in number theory which studies integers, prime numbers, and solutions of a system of equations over integers or rational numbers following the long tradition from ancient Greeks. In the digital age, number theory has been essential in algorithms, cryptography, and data security. Modern mathematics has seen increasingly more interactions between number theory and other areas from a unifying perspective. A primary example is the Langlands program, comprising a vast web of conjectures and open-ended questions. Even partial solutions have led to striking consequences such as verification of Fermat's Last Theorem, the Sato-Tate conjecture, the Serre conjecture, and their generalizations.The PI's projects aim to broaden our understanding of the Langlands program and related problems in the following directions: (1) endoscopic classification for automorphic forms on classical groups, (2) a formula for the intersection cohomology of Shimura varieties with applications to the global Langlands reciprocity, (3) the non-generic part of cohomology of locally symmetric spaces, and (4) locality conjectures on the mod p Langlands correspondence. The output of research would stimulate further progress and new investigations. Graduate students will be supported on the grant to take part in these projects. The PI also plans outreach to local high schools which have large under-represented minority populations.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.