Trace Formulae

This is the website for the learning seminar about the trace formula organized by Héctor del Castillo and Andrei Seymour-Howell. In this seminar, we will cover the basics of the trace formula, aiming towards explicit trace formulae results in the co-compact case and explore ideas for extending the trace formula beyond this case.

The seminar will take place in Room 401 of the Education Convergence Building at Chonnam National University, Gwangju. The weekly schedule is Tuesday at 11:00 am. Topics that we will cover include:

Trace formula for finite groups
Basics on functional analysis: compact and trace-class operators
Basic on adèles and algebraic groups
L^2 space and cuspidal spectrum
(Relative) simple trace formula: traces and orbital integrals
Weyl's law
Trace formula beyond the compact case: SL(2)

The seminar will be based on the following references:

Lectures

Date	Speaker	Subject	Extra
18/11/2025	Héctor del Castillo	Induced representation and distributions. Topics around Sections 1 and 2 of D. Whitehouse.	Notes
25/11/2025	Héctor del Castillo	Kernels and Traces. Topics around Sections 1 and 2 of D. Whitehouse.	Notes
02/12/2025	Héctor del Castillo	Trace formula. Topics around Sections 1 and 2 of D. Whitehouse.	Notes
09/12/2025	Héctor del Castillo	Aplication: Frobenius Reciprocity. Topics around Sections 1 and 2 of D. Whitehouse.	Notes
16/12/2025	Héctor del Castillo	We introduce compact and trace-class operators. Section 3.1 of D. Whitehouse and Section 9.1, 16.1 and 16.2 of Getz and Hahn.	Notes
23/12/2025	Héctor del Castillo	We introduce Hilbert-Schmidt and trace-class operators. Section 3.1 of D. Whitehouse and Section 9.1, 16.1 and 16.2 of Getz and Hahn.	Notes
30/12/2025		Seminar Website.
13/01/2026	오규진 (Gyujin Oh)	We review basic on adèles: definitions and locally compact.
20/01/2026	이의관 (Euigwan Lee)	We review basic on adèle: Discreteness of base field and compact quotient.
24/02/2026	Ayan Maiti	We survey around strong approximation and finite volume results.	Notes
03/03/2026	Ayan Maiti	We survey about automorphic forms, taking special attention the GL(2) case.	Notes
10/03/2026	Ayan Maiti	Cuspidal forms and decomposition theorems.	Notes
17/03/2026	Ayan Maiti		Notes
24/03/2026	Ayan Maiti	Sketch of proof of the compact property of the regular operator in the cuspidal subspace.	Notes
31/03/2026	Ayan Maiti	Existence of cusp forms.	Notes

Learning seminar: Trace Formulae

Lectures