Applications of the Trace Formula to Shimura Varieties and the Langlands Program

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This research project concerns work in number theory, a branch of mathematics with applications to cryptography, coding theory, and cyber security. The Langlands program is a vast web of conjectures involving disparate areas of mathematics, with number theory playing a key central role. Fermat's Last Theorem followed from the verification of a small part of the Langlands program, and there are many other similar successes, often built upon fruitful interactions between objects encoding rich symmetries in different worlds: they are Shimura varieties (algebraic and complex geometry), automorphic forms (functional and harmonic analysis), and Galois theory. To deepen our understanding of these topics and especially the Langlands program, the principal investigator proposes to make new progress on the following problems: (1) new instances of the global Langlands correspondence for GSp(2n) and GSO(2n) (2) discrete Hecke orbit conjecture for Shimura varieties (3) local harmonic analysis questions for families of automorphic forms and (4) the p-adic Langlands program beyond GL(2,Qp). The output of research would stimulate further progress on these important problems and open up new research directions.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.