Joint IAS/Princeton University Number Theory Seminar

I will begin with a brief introduction to the deformation theory of Galois representations and its role in modularity lifting. This will motivate the study of local deformation rings and more specifically flat deformation rings. I will then discuss Kisin's resolution of the flat deformation ring at $\ell = p$ and describe conceptually the importance of local models of Shimura varieties in analyzing its geometry. Finally, I will address how to generalize these results from $\mathrm{GL}_n$ to a general reductive group $G$. If time permits, I will describe briefly the role that recent advances in $p$-adic Hodge theory and local models of Shimura varieties play in this situation.