Representations of p-adic Groups and the Local Langlands Correspondence

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This project is motivated by the Langlands program. The Langlands program is a far-reaching collection of conjectures that relates different areas of mathematics, including number theory and representation theory, and thereby provides a powerful tool to prove theorems and find explanations by transferring results between different areas. Recent work in Langlands program has lead to the resolution of various major conjectures. Number theory is one of the oldest areas in pure mathematics, which includes the study of integers and objects built out of them, such as integer solutions to equations. Apart from its fundamental role inside mathematics, number theory manifests important influences on our everyday life, for example, through cryptography. Representation theory, on the other hand, is the study of abstract objects, such as the symmetries of a cube (reflections and rotations that preserve the cube), using matrices and linear algebra. By realizing abstract objects as matrices, one can reduce abstract problems to questions in linear algebra, which are often easier to study. Representation theory has also applications outside of mathematics, for example, in physics. This project will enhance our understanding of the objects studied on the representation theory side and aims to achieve further progress towards an explicit construction of the local Langlands correspondence on the number theory side.One of the central open questions in the representation theory of p-adic groups is the construction of all smooth, complex, supercupsidal representations. In this project, the PI will provide a new construction of such representations and prove its exhaustiveness for all primes p. In parallel, the PI and her collaborators will explore another novel approach to obtain a better understanding of the structure of the representations of p-adic groups. Moreover, the PI intends to use the new construction of representations to gain new insights about the 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.