The local Langlands correspondence in families

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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 in order to properly understand this correspondence, we should consider not just single examples of objects on both sides of the correspondence, but whole families of such objects, that vary as one varies certain parameters.This leads naturally to the following question: if one varies the objects on one side of the correspondence, how do the objects on the other side of the correspondence change? The proposed research seeks to develop a version of the local Langlands correspondence that addresses this question. Moreover, the answer to this questions should have extensive applications within the Langlands program, particularly to Emerton's "p-adic global Langlands correspondence". We seek to explore these applications and deduce new results in the Langlands program via these techniques.