The PI proposes several research projects related to the local Langlands correspondence, and in particular to the modular representation theory of general linear groups over p-adic fields. One goal of these projects is to address the question of extending the local Langlands correspondence to a correspondence that works on the level of families of Galois representations. This builds on previous work of the PI and Matthew Emerton that has had applications to the Langlands program for global fields; in particular it plays a role in Emerton's proof of the Fontaine-Mazur conjecture and is likely to be important for generalizations of this result. A long term goal of the project is essentially a "local theory of congruences" for automorphic representations of general linear groups.
The proposed project has its roots in the Langlands program, a series of far-reaching conjectures that connect topics in harmonic analysis and representation theory to long-standing problems in number theory. The Langlands program is expect to shed light on questions in theoretical physics, and also has implications for the theory of elliptic curves. Advances in elliptic curves in particular could have considerable practical applications in cryptography and coding theory.
