I work on the Hitchin moduli space and wall crossing phenomena. Below are specific projects:
In joint work with Matei Ionita we are generalizing the non-abelianization construction of Gaiotto--Moore--Neitzke ffrom the case of GL(n)/SL(n) to the case of arbitrary reductibe algebraic groups, using cameral covers, and T-local systems on these to replace the use of 1d local systems in the GL(n)/SL(n) cases.
I am working on describing spectral networks and non-abelianization for Frobenius manifolds using the Stokes data of modifications of the extended structure connection. This should make rigorous some of the links between non-abelianization and asymptotics.
I am working on describing spectral data for a Higgs vector bundle associated to a Higgs G-bundle via some representation.