CAREER: Representations of p-adic groups and different incarnations of the Langlands Program

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Number theory is the study of the integers and objects built out of them. Groups are abstract mathematical objects which encode symmetries, for example the symmetries of a cube or crystal, and representation theory is the study of groups using matrices. Although number theory and representation theory are very different areas of mathematics, the Langlands correspondence predicts a fascinating connection between them. The ideas surrounding the Langlands correspondence are the driving force for many groundbreaking advances in mathematics including the famous proof of Fermat's Last Theorem, a conjecture that had withstood mathematicians' effort for almost 400 years. This project involves tackling a long-standing question concerning the representation theory side of the Langlands correspondence and studying the bridge to number theory. The PI will also organize workshops and activities to assist up-and-coming mathematicians in their career development and support underrepresented groups. A fundamental problem on the representation theory side of the local Langlands correspondence is the construction of all supercuspidal representations for all p-adic groups, which is the first main objective of this project. Solving this problem will involve tackling all the complications that arise in the non-tame case compared to the tame case. Based on these results, the PI will also advance various aspects of the Langlands program: the global Langlands program by constructing congruences between automorphic forms, the relative Langlands program by proving finite multiplicity results, and the explicit local Langlands correspondence.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.