Deformation Theory of Galois Representations and the Taylor--Wiles Method

In this chapter, we want to have an overview of the Taylor–Wiles patching method. For this purpose, at the first, we recall Mazur’s theory of deforming Galois representations and study both local and global deformation problems. Then, we go through the subject of Taylor-Wiles primes and examine the role that they play on the Galois side and the modular (automorphic) side. At the end, we arrive at the Taylor-Wiles patching method and use it to prove $R=\mathbb{T}$ in both minimal and non-minimal cases. Note that, in the Galois side, we will work with totally real number fields, but for the modular side, we will concentrate on $\mathbb{Q}$ to avoid difficulties of working with Hilbert modular forms.