|Workshop on Motives, Galois Representations and Cohomology Around the Langlands Program|
|Topic:||Models for Galois deformation rings|
|Affiliation:||University of Chicago|
|Date:||Thursday, November 9|
|Time/Room:||4:00pm - 5:00pm/S-101|
Abstract: 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-Mèzard 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.