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)