Galois representations and reduction mod $p$

 For any prime number $p$, the process of  reduction mod $p$ of a $p$-adic Galois representation is well-defined up to semisimplification. Given a two-dimensional crystalline representation of the local Galois group Gal$(\bar{\mathbb{Q}_p} | \mathbb{Q}_p)$, the semi-simplified reduction can be computed  using the compatibility of the $p$-adic and mod $p$ Local Langlands Correspondences. We will explain the basic principle of this method  introduced by Breuil (2003) and Buzzard-Gee (2009), and then discuss what is known so far about the general behaviour of the reductions for varying weights and slopes. We give an explicit description of the reduction for the smallest positive integral slope 1, with families of examples of both reducible and irreducible reductions (joint work with E. Ghate and S. Rozensztajn). If time permits, I will talk about some recent work on the local constancy of the reduction process.