Geometric Methods in Representation Theory and the Langlands Program

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Representation theory is the study of symmetries using linear algebra. As such, it is deeply intertwined with a vast range of mathematical fields and has rich applications, to particle physics, physical chemistry, and computer vision for example. Since the dawn of the subject nearly a century ago, interactions between representation theory and algebraic geometry, which is the study of solutions of polynomial equations, have influenced the landscape of each area. More recently, in the past several decades, number theory has played an increasing important role in this world. The Langlands program is a far-reaching web of conjectures which predicts unexpected matchings between deep number-theoretic and representation-theoretic phenomena. This project includes opportunities for student research. This project is centered on geometric methods in representation theory within the framework of the Langlands program. The long-term objective is to construct a geometric theory of representations of p-adic groups. This will allow passage between rapid developments in the traditionally algebraic approach to representations of p-adic groups and rapid developments in geometric progress in the Langlands correspondence. Moreover, it will relate conjectural algebraic constructions of L-packets to deep geometric phenomena.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.