MMP Homepage

The aim of this course is to describe some basic constructions involved in cohomology of arithmetic groups. We will try to keep the material as elementary as possible and so not assume much more than first year graduate coursework in algebra and analysis. The more topology, differential geometry, Lie theory, or number theory you know the better, but I will try to introduce what we need from these areas as we go along. If you are interested in joining, please send me an email

Logistics

This course will be run hybrid as a PIMS network course. If you are at the University of Alberta, you should register for MATH 681, Lecture B1. If you are at one of the other schools signed onto the Western Dean's Agreement (WDA) , please register for MATH 681, Lecture 800. This will reduce the fees you need to pay to about $250 CAD. Please see the PIMS website for more information on registration (you need several signatures, so please plan accordingly as some of the deadlines imposed by various organizations are strict.)

Course description, Syllabus, and Grading

The course description can be downloaded here. It contains a list of useful references as well as a rough plan for this course.

The official syllabus for the course is contained here . The syllabus is just a pseudo-legal document, and the main useful thing it contains is the rubric for how grades are to be assigned. Roughly speaking, to get credit for the course you need to complete a few exercises. Some of these will take just a few minutes to do while others are more involved, so I leave it up to you to decide what you want or have time to do. I will maintain a list of exercises and solutions here. You should aim to submit 5 of these (of any difficulty) to me throughout the term.

Location and Zoom Link

The course will meet Tuesday and Thursday, 4-5:20pm Edmonton Time. If you are in Edmonton, we will meet at the University of Alberta campus in BS M-141 (Biological Sciences building). If you are attending online, please follow this zoom link , with the password which has been emailed to all participants. If the link does not work, the zoom meeting number is 954 7908 6002 .

Resources, Notes, Exercises

Handwritten notes from the lectures are here (watch out! there are many mistakes and errors...)>

Some typed notes from the course are posted here (a slightly edited version of the handwritten notes, still many errors! )

A list of exercises and solutions is contained in the typed notes (I will eventually separte them out as an individual section). Note: even after a solution has been posted, you can still turn it in to count towards your grade (just try not to look at it when writing your own solution).

Schedule

Here is the day-to-day schedule. You can click on the "Topic" for a brief summary of the lecture.

Date	Topic
Lecture 1, Sep. 3	We first define $H^{} (Γ, E)$ , the group cohomology, relate it to locally symmetric spaces, and then begin to describe the main case of interest, when $Γ$ is an arithmetic* group. We give examples of these and introduce the related notion of a lattice.
Sep. 5	CLASS CANCELLED
Lecture 2, Sep. 10	We recall some basic facts about haar measure and the Iwasawa decomposition. We are aiming to show that $\SL_{n} (Z)$ is a lattice in $\SL_{n} (R)$
Lecture 3, Sep. 12	We review the notion of Siegel sets for $\SL_{n} (R)$ and formulate the statement of reduction theory. Then we begin a discussion on some elementary Lie theory language for \( SL_n )
Lecture 4, Sep. 17	We introduce some more of the Lie industry terminology of roots, weights, etc. and then discuss the notion of highest weight reprentations.
Lecture 5, Sep. 19	We finish up the discussion of highest weight representations, introduce Chevalley forms, and then use these to construct Siegel sets using a minimization procedure.
Lecture 6, Sep. 24	We review the proof of existence of Siegel sets from last time, and use it to prove the fact that $S L_{n} (Z)$ is a lattice. We also formulate a useful compactness criterion due to Mahler.
Lecture 7, Sep. 26	We redo the discussion of Mahler, and show how it can be used to prove co-compactness of certain lattices. We then begin to discuss Lie groups from a differential geometric perspective .
Lecture 8, Oct 1	We use spectral sequences to relate the group cohomology with a sheaf cohomology on the locally symmetric space $K ∖ G / Γ$
Lecture 10, Oct 3	We explain the proof, due to Weil, of the de Rham theorem starting from a 'good' covering.
Lecture 11, Oct 8	We finish up the proof of the de Rham theorem and explain some 'twisted' variants. Then we return to the setting of $X / Γ$ and describe $H^{n} (Γ, M)$ in de Rham terms following Matsushima-Murakami.
Lecture 12, Oct 10	We explain the de Rham setup of Matsushima-Murakami and describe some further results on elementary Lie theory (Cartan decomposition at the Lie algebra level, Killin forms, etc.) which we need.
Lecture 13, Oct 15
Lecture 14, Oct 17
Lecture 15, Oct 22
Lecture 16, Oct 24
Lecture 17, Oct 29
Lecture 18, Oct 31
Lecture 19, Nov 5
Lecture 20, Nov 7
Lecture 21, Nov 19
Lecture 22, Nov 21
Lecture 23, Nov 26
Lecture 24, Nov 28
Lecture 25, Dec 3
Lecture 26, Dec 5

Date

Topic

Lecture 1, Sep. 3

We first define $H^{*} (Γ, E)$ , the group cohomology, relate it to locally symmetric spaces, and then begin to describe the main case of interest, when $Γ$ is an arithmetic group. We give examples of these and introduce the related notion of a lattice.

Sep. 5

CLASS CANCELLED

Lecture 2, Sep. 10

We recall some basic facts about haar measure and the Iwasawa decomposition. We are aiming to show that

\SL_{n} (Z)

is a lattice in

\SL_{n} (R)

Lecture 3, Sep. 12

We review the notion of Siegel sets for

\SL_{n} (R)

and formulate the statement of reduction theory. Then we begin a discussion on some elementary Lie theory language for \( SL_n )

Lecture 4, Sep. 17

We introduce some more of the Lie industry terminology of roots, weights, etc. and then discuss the notion of highest weight reprentations.

Lecture 5, Sep. 19

We finish up the discussion of highest weight representations, introduce Chevalley forms, and then use these to construct Siegel sets using a minimization procedure.

Lecture 6, Sep. 24

We review the proof of existence of Siegel sets from last time, and use it to prove the fact that

S L_{n} (Z)

is a lattice. We also formulate a useful compactness criterion due to Mahler.

Lecture 7, Sep. 26

We redo the discussion of Mahler, and show how it can be used to prove co-compactness of certain lattices. We then begin to discuss Lie groups from a differential geometric perspective .

Lecture 8, Oct 1

We use spectral sequences to relate the group cohomology with a sheaf cohomology on the locally symmetric space

K ∖ G / Γ

Lecture 10, Oct 3

We explain the proof, due to Weil, of the de Rham theorem starting from a 'good' covering.

Lecture 11, Oct 8

We finish up the proof of the de Rham theorem and explain some 'twisted' variants. Then we return to the setting of

X / Γ

and describe

H^{n} (Γ, M)

in de Rham terms following Matsushima-Murakami.

Lecture 12, Oct 10

We explain the de Rham setup of Matsushima-Murakami and describe some further results on elementary Lie theory (Cartan decomposition at the Lie algebra level, Killin forms, etc.) which we need.

Lecture 13, Oct 15

Lecture 14, Oct 17

Lecture 15, Oct 22

Lecture 16, Oct 24

Lecture 17, Oct 29

Lecture 18, Oct 31

Lecture 19, Nov 5

Lecture 20, Nov 7

Lecture 21, Nov 19

Lecture 22, Nov 21

Lecture 23, Nov 26

Lecture 24, Nov 28

Lecture 25, Dec 3

Lecture 26, Dec 5

Introduction to Cohomology of Arithmetic Groups

Fall 2024

Instructor: Manish M. Patnaik

contact: patnaik at ualberta dot ca

Logistics

Course description, Syllabus, and Grading

Location and Zoom Link

Resources, Notes, Exercises

Schedule