\(G\)-valued flat deformations and local models

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 ℓ=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 GLn 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.