# Spring 2018: Seminar on the Langlands Program

This is a seminar class on the Langlands program understood in a broad sense, covering topics ranging from number theory to quantum field theory, including, but not limited to, the following:

class field theory

Artin L-functions and Hecke L-functions

Tate's thesis

representation theory of real and p-adic groups

local Langlands correspondence

modular forms and automorphic representations

L-functions of elliptic curves and modular forms

Fermat's last theorem and modularity theorem

motives and special values of L-functions

Langlands functoriality and trace formula

Ngo's fundamental lemma

Drinfeld's shtukas

V. Lafforgue's global function field Langlands

geometric Satake equivalence

non-categorical geometric Langlands

ramified geometric Langlands

local geometric Langlands

Betti geometric Langlands

Langlands duality and 2-dimensional conformal field theory

Langlands duality and 4-dimensional N=4 supersymmetric gauge theory

In the first few lectures, the instructor explains how some of these topics fit into a big picture. For the remainder of the term, each participant gives a few talks on topics chosen in discussion with the instructor. One aim is to learn how to give a good expository talk. The instructor will also give a series of talks on topics of interest for the audience.

Disclaimer: The instructor is not an expert on most of the suggested topics!

**Class notes **(none of us is an expert, so use at your own risk!)

1/23: Phil - Introduction

1/25, 1/30: Phil - Langlands Reciprocity

2/1, 2/6: Phil - Geometric Langlands

2/8, 2/13: Phil - Categorical Geometric Langlands and QFT

2/15: Andrew - Geometric Class Field Theory

2/20: Phil - Local-Global Compatibility of Geometric Langlands I

2/22: Matt - Tate's Thesis

2/27: Phil - Local-Global Compatibility of Geometric Langlands II

3/1: Elad - Local Langlands Correspondence I (Local Class Field Theory)

3/6: Alex - Intro to QFT: Path Integrals and Observables

3/8: Yau Wing - Global Class Field Theory I (Kronecker--Weber Theorem)

3/27, 3/29: Phil - Modular Forms and Modular Curves

4/3: Yau Wing - Global Class Field Theory II (Complex Multiplication)

4/5: Andrew - Geometric Satake I (Perverse Sheaves and Affine Grassmannians)

4/10: Matt - Heegner Points and the BSD Conjecture

4/12: Daping - Geometric Satake II (Statement and Examples)

4/17: Phil - Eicher--Shimura Relation

4/19: Alex - Geometric Satake III (QFT Perspective)

4/24: Elad - Local Langlands Correspondence II

4/26: Daping - Geometric Satake IV (Convolution Product)