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Spring 2018: Seminar on the Langlands Program

This is a seminar class on the Langlands program understood in a broad sense, with topics selected from subjects 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 are experts, 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)