New perspectives on the Breuil-Mézard conjecture

I will discuss joint work with Matthew Emerton on geometric
approaches to the Breuil-Mézard conjecture, generalising a geometric
approach of Breuil and Mézard. I will discuss a proof of the geometric
version of the original conjecture, as well as work in progress on a
geometric version of the conjecture which does not make use of a fixed
residual representation.