Explicit Methods for the Local Langlands Correspondence

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This project is rooted in two ancient parts of mathematics: representation theory (the study of symmetry) and number theory (numerical solutions of equations). Though these appear to be two quite different areas of mathematics, the Local Langlands Correspondence (LLC) predicts surprising relations between them. Roughly speaking, it says that infinite-dimensional symmetries should correspond to equations whose solutions have a related collection of finite-dimensional symmetries. The goals of this project are first, to discover and explicitly verify new and interesting examples of the LLC, and second, to use the predictions of the LLC to make new discoveries in number theory and representation theory.This project is about interactions between representation theory and number theory, as predicted by the Local Langlands Correspondence for general reductive groups. The LLC is a body of predicted relations between the representation theory of local Galois groups to representation theory of reductive groups over a p-adic field. The goals in this project are to discover and understand new and interesting aspects of the LLC, via explicit methods. There are five proposed directions of inquiry: Lie-primitive representations of Galois groups into complex Lie groups; representations of p-adic groups whose parameters are Lie-primitive; extending known constructions of epipelagic representations to higher depth; inequalities for Swan conductors; and Jordan decomposition of depth zero representations and LLC.