**I.- Location**

Math Department, BA6180, University of Toronto.

Tuesdays 4:00 – 6:00.

**II.- Lecture Notes**

The lectures are based in the notes here but sometimes we cover less than what is written in the notes.

- Lecture 1 Trace Class Operators
- Lecture 2 On Compact Quotient
- Lecture 3 On Compact Quotient Part 2
- Lecture 4 Trace Formula for Compact Quotient

**III.- Brief Comments**

*Meeting 1, September 12:*We covered the first notes completely. I do not know if we really need to use Lusin’s theorem in the proof of Duflo’s Theorem. In the proof I did, completing the details of Casselman, I did not use it. Hope I didn’t miss something!*Meeting 2, September 19:*We covered the first part of the second lecture. That is, we saw the proof of the discreteness of the spectrum. Notice that in my notes of that proof there is a little mistake on the proof, nothing serious, when proving that the H_i generate H. It get’s fixed by invoking its finite dimensionality.*Meeting 3, September 26:*We covered the rest of the notes of Lecture 2 and Lecture 3. We are ready to talk about the proof of the criteria on compactness now!*Meeting 4, October 3:*We concluded the complete set of notes 4. We are done with the compact quotient part of the seminar. We are still struggling with one of the equations of the deduction of the trace formula since evryone seems to use fubini but is kind of unclear why it should be applicable.

**IV.- Bibliography**

Here are the books from where I review and take material.

- Anthony W. Knapp
- Representations of Semisimple Lie Groups.
- Advanced Real Analysis.

- James Arthur
- An Introduction to the Trace Formula

- Andrew Knightly, Charles Li
- Traces of Hecke Operators

- Serge Lang
- Real and Functional Analysis

- Automorphic Forms, Representations and L-functions, Vol 1 and Vol 2.
- Anton Deitmar, Siegfried Echterhoff
- Principles of Harmonic Analysis

- I.M. Gelfand, M.I. Graev, I.I. Pyateskii-Shapiro
- Generalized Functions, Volume 6. Representation Theory and Automorphic Functions.

The articles that we have reviewed, at least partially, so far or that are related:

- For operators of trace class and Duflo’s Theorem:
- The proof of Duflo’s Theorem that we followed comes here: Compact Operators, Essays in Analysis, Bill Casselman in Essays in Analysis by Bill Casselman.
- Kernels of Trace Class Operators, Chris Brislawn.

- For the criteria of compactness:
*On the Compactness of Arithmetically Defined Homogeneous Spaces*, Mostow and Tamagawa.- For a proof of Mahler’s Lemma used in the previous article:
*Limits of Lattices in a Compactly Generated Group*by A.M.Macbeth and S. Swierczkowski

- For a proof of Mahler’s Lemma used in the previous article:
*Arithmetic Subgroups of Algebraic Groups,*Armand Borel and Harish-Chandra.

**V.- Links to other webpages.**