The Taylor--Wiles method for reductive groups

The Langlands philosophy predicts a map from automorphic representations to representations of the Galois group $\Gamma$ of a global field. Automorphy lifting theorems can be used to go the other way: showing a representation of $\Gamma$ arises from an automorphic representation. Such theorems are proved via the Taylor--Wiles method; we discuss a generalisation allowing representations with small (residual) image. Time permitting, we conclude with applications to modularity of some elliptic curves, building upon the work of Boxer--Calegari--Gee--Pilloni, and to V. Lafforgue's construction of a global Langlands correspondence in the function field setting.