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