I study the representation theory of reductive $p$-adic groups, although a large part of my research is really motivated by questions which would be classed as arithmetic geometry. The major problem in the field is to understand the (still largely conjectural) local Langlands correspondence, which naturally parametrizes Galois representations in terms of the representations of various connected, reductive $p$-adic groups.
The representation theory of $p$-adic groups admits some very explicit constructions in terms of the theory of types; one obvious question is whether these constructions have any interpretation in terms of Galois representations. This is the basic idea behind the "inertial Langlands correspondence". While it is easy to construct such a correspondence (assuming the existence of types), it is by no means obvious that the resulting correspondence is a useful, or even interesting one. In order to resolve this, one is forced to consider the "unicity of types", a technical property regarding the representation theory of $p$-adic groups. To date, my work has focused on establishing this property in some of the more accessible cases.
A useful consequence of the inertial Langlands correspondece is that it conveys essentially the same information as the local Langlands correspondence, but takes the form of a map between categories of representations of compact groups. This means that, using the inertial correspondence, it is actually feasible to consider compatbilities between the $\ell$-adic and $\ell$-modular local Langlands correspondences. Such thinking eventually leads one to the Breuil--Mézard conjecture for representations of $\mathbf{GL}_N$; one thing that my results should be able to do is establish a similar results for various classes of representations of other groups.
For a more technical account of my work, see my research statement. A list of publications and preprints may be found here.