Moduli of Galois representations and Serre's modularity conjecture

Serre's modularity conjecture (1973) was proven by Khare-Wintenberger building on work of Kisin in the mid-2000s. The methods and problems surrounding the conjecture have played an important role in the field modularity lifting and the p-adic Langlands program. One such development is the recent construction of moduli spaces of p-adic Galois representation by Emerton--Gee. I will explain the connection between Emerton--Gee's space and Serre's conjecture in the simplest case of two-dimensional representation of the Galois group of Q_p. I will then go on to describe how this picture helps us understand higher dimensional generalizations of Serre's conjecture based on joint work with Daniel Le, Bao V. Le Hung, and Stefano Morra.