Geordie Williamson will offer the following one semester course starting on the 8th of March 2019, Friday 10:00--12:00, Carslaw room 830. Title: Langlands correspondence and Bezrukavnikov’s equivalence This course will be in two parts. 1) I will attempt to give a picture of what the Langlands correspondence is about, from an arithmetical point of view. I will start of with some basic questions (e.g. counting points of varieties over finite fields) and show how they lead to interesting L-functions which should have an automorphic incarnation. Explaining this will involve rather a lot of algebraic number theory, which I will try to go over briefly. I will then pass to the local case. I will review the structure theory of local fields and their Galois groups and state the main theorems of local class field theory. I will then explain what the local Langlands correspondence should be, and we will see that it boils down to class field theory if the GL_1 case. I will sketch a beautiful heuristic argument for the local Langlands correspondence for GL_2 when p is not 2, which can be turned into a proof which I wonât go into (so-called Jacquet-Langlands correspondence). 2) The second half of the course will focus on affine Hecke algebras and their categorifications. Roughly speaking this part of the course is about the local Langlands correspondence at the easiest âlayers of difficultyâ, which is still extremely rich. I will explain the two simplest instances of the local Langlands correspondence for a general G, namely the case of unramified representations (where the Langlands correspondence boils down to the Satake isomorphism), and the case of tamely ramified representations (so-called Deligne-Langlands conjecture). The Deligne-Langlands conjecture was proved by Kazhdan and Lusztig (and later Ginzburg). I will try to outline their proof. I will then explain how categorifying Kazhdan and Lusztigâs proof leads to a remarkable equivalence conceived by Bezrukavnikov in the late 90s but only recently written down. If time permits I will try to outline the proof of his theorem. This is an ambitious course in terms of scope, and, unless you have significant background, will require work and reading outside of the lectures. I will certainly not prove everything, however I will try to make everything as explicit as I can for GL_1 and GL_2. There is some chance it will spill over into second semester, depending on the interest of the participants and the stamina of the lecturer!