Rigidity of automorphic Galois representations over CM fields

The Taylor-Wiles method is a way of relating Galois representations with the cohomology of locally symmetric spaces. I will give an overview of this method and its generalisation due to Calegari-Geraghty. Besides automorphy lifting theorems, one of its applications is the rigidity of certain Galois representations. I will explain what this means and give a sketch of the proof.