Derived and perverse methods in the local Langlands correspondence

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The Langlands program is a series of far-reaching conjectures, dating back to the 1970s, that relate such widely separate branches of mathematics as number theory, representation theory, harmonic analysis, and mathematical physics. Central to the Langlands program is the local Langlands correspondence, which relates admissible representations of p-adic groups (objects from harmonic analysis) to Galois representations (objects from number theory). It has recently become clear that objects on both sides of this correspondence have a very rich geometric structure, and that the local Langlands correspondence often allows us to study the geometry on one side of the correspondence to deduce conclusions about the geometry on the other side.A natural question to ask is: to precisely what extent is the geometry of one side reflected in the geometry of the other? The most optmistic thing to ask for would be a description of the category of admissible representations a p-adic groups in terms of the corresponding Galois representations. Recent breakthroughs by many people suggest that this is not only plausible, but might be within reach in special cases.The goals of this research are: first, to obtain such descriptions in settings where it is feasible to do so, and second, to explore the consequences of such a description for various applications of the local Langlands correspondence in number theory.