Models for Galois deformation rings
An important input into modularity lifting theorems is an understanding of the geometry of Galois deformation rings, especially local deformation rings with p-adic Hodge theory conditions at $\ell = p$. Outside of a few cases (ordinary, Fontaine-Laffaille,...), the detailed structure of these deformation rings is very mysterious. I will introduce models which in generic situations conjecturally have the same singularities as those of a class of potentially crystalline deformation rings. This will ultimately have applications to the weight part of Serre's conjecture, the Breuil-M\'ezard conjecture, and modularity lifting in dimension $> 3$ where relatively little is known. This is joint work with Daniel Le, Bao V. Le Hung, and Stefano Morra.