# Fields Academy Shared Graduate Course: Automorphic Forms and Representation Theory: an introduction to the Langlands Program

## Description

**Instructor:** James Arthur, University of Toronto

**Course Dates:** January 9 - April 10, 2023**Mid-Semester Break: **February 20 - 24, 2023**Lecture Times:** Monday, Wednesday, and Friday at 5:00 PM (with exceptions listed in the schedule below)**Office Hours: **TBA

**Registration Fee:** Free for all students

**Evaluation:** Assignments TBA

**Capacity Limit:** TBA

**Format: **Hybrid - In Person at Room 230, the Fields Institute | Remote via Zoom: https://zoom.us/j/99605921464

**Course Description**

Please view the complete course description here: https://www.math.toronto.edu/cms/graduate-program/current-students-grad/2022-2023-graduate-courses-descriptions-3/#MAT1197HS

Automorphic representations are at the heart of the conjectures that make up the Langlands program. The trace formula provides the most powerful technique for attacking them. It is the analogue for automorphic representations of the Plancherel formula for ordinary representations.

We shall begin with some of the basic properties of automorphic representations. We shall then discuss some of the Langlands conjectures, especially the Principle of Functoriality. The rest of the course will be devoted to an introduction to the trace formula, and if time permits, some of its applications.

**Prerequisites:**

Core courses in Analysis and Algebra or their equivalents, and the basic theory of Lie (and/or) Algebraic Groups. This course will be a natural successor to this year's course MAT1196 on representations of real groups, but I will try to keep other prerequsites to a minimum. And in particular, readers without a background in Lie Groups could always think of the general linear group of nonsingular (N x N) matrices (under multiplication) in place of an arbitrary Lie/algebraic group G.

**References:**

S. Gelbart, An elementary introduction to the Langlands program, Bulletin of the American Mathematical Society, Vol. 10 (1984), p. 177- 219.

J. Arthur, An introduction to the trace formula, in Harmonic Analysis, the Trace Formula, and Shimura Varieties, Clay Mathematics Proceedings, Vol. 4, 2005, p. 1- 263.

You can find all lecture recordings here: https://www.youtube.com/playlist?list=PLArBKNfJxuumETfbxHWz0OHr9iPDwWHMp

## Schedule

11:00 to 12:00 |
Lecture 01 | Automorphic Forms and Representation Theory: an introduction to the Langlands Program
James Arthur, University of Toronto Location:Bahen Centre |

11:00 to 12:00 |
Lecture 02 | Automorphic Forms and Representation Theory: an introduction to the Langlands Program
Location:Bahen Centre |

11:00 to 12:00 |
James Arthur, University of Toronto Location:Bahen Centre |

11:00 to 12:00 |
James Arthur, University of Toronto Location:Bahen Centre |

17:00 to 18:00 |
James Arthur, University of Toronto |

17:00 to 18:00 |
James Arthur, University of Toronto |

17:00 to 18:00 |
James Arthur, University of Toronto |

11:00 to 12:00 |
James Arthur, University of Toronto Location:Bahen Building, Room 6183 |

17:00 to 18:00 |
James Arthur, University of Toronto |

17:00 to 18:00 |
James Arthur, University of Toronto |

17:00 to 18:00 |
James Arthur, University of Toronto |

17:00 to 18:00 |
James Arthur, University of Toronto |

17:00 to 18:00 |
James Arthur, University of Toronto |

17:00 to 18:00 |
James Arthur, University of Toronto |

17:00 to 18:00 |
James Arthur, University of Toronto |

17:00 to 18:00 |
James Arthur, University of Toronto |

17:00 to 18:00 |
James Arthur, University of Toronto |

17:00 to 18:00 |
James Arthur, University of Toronto |

17:00 to 18:00 |
James Arthur, University of Toronto |

17:00 to 18:00 |
James Arthur, University of Toronto |

17:00 to 18:00 |
James Arthur, University of Toronto |

17:00 to 18:00 |
James Arthur, University of Toronto |

17:00 to 18:00 |
James Arthur, University of Toronto |

17:00 to 18:00 |
James Arthur, University of Toronto |

17:00 to 18:00 |
James Arthur, University of Toronto |

17:00 to 18:00 |
James Arthur, University of Toronto |

17:00 to 18:00 |
James Arthur, University of Toronto |

17:00 to 18:00 |
James Arthur, University of Toronto |

17:00 to 18:00 |
James Arthur, University of Toronto |

11:00 to 12:00 |
James Arthur, University of Toronto Location:Bahen Building, Room 6183 |

17:00 to 18:00 |
James Arthur, University of Toronto |

17:00 to 18:00 |
James Arthur, University of Toronto |

11:00 to 12:00 |
James Arthur, University of Toronto Location:Bahen Building, Room 6183 |

17:00 to 18:00 |
James Arthur, University of Toronto |

11:00 to 12:00 |
James Arthur, University of Toronto Location:Bahen Building, Room 6183 |

17:00 to 18:00 |
James Arthur, University of Toronto |