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.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/158