We will give some examples of arithmetic groups in \( SL_n(\mathbb{R})) \) and, in case \( n= 2 \) introduce some important aspects of the geometry of the upper half plane.

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

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.)

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.

I've compiled several useful resources which we will consult here.

Handwritten notes from the lectures are here.

Some typed notes from the course are posted here.

A list of exercises and solutions is posted here. * 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).

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 GSB 7-11. If you are attending online, please follow the zoom link, which has been emailed to all participants.

Occasionally, we will have invited guest speakers. The time and location will be indicated below.

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

Date | Topic | Notes | Remarks |
---|---|---|---|

Lecture 1, Sep. 3 | We will give some examples of arithmetic groups in \( SL_n(\mathbb{R})) \) and, in case \( n= 2 \) introduce some important aspects of the geometry of the upper half plane. |
additional remarks here | |

Lecture 2, Sep. 5 | More Lie theory | ||

Lecture 3, Sep. 10 | Fundamental domains and Siegel sets | ||

Lecture 4, Sep. 12 | Group, de Rham, and Lie algebra cohomology | ||

Lecture 5, Sep. 17 | Basic differential geometry of symmetric spaces | ||

Lecture 6, Sep. 19 | Vanishing of \( H^1(\Gamma, \mathfrak{g}) \) and Local Rigidity (following Weil, Garland) | ||

Lecture 7, Sep. 24 | Vanishing of \( H^1(\Gamma, \mathfrak{g}) \) and Local Rigidity (following Weil, Garland) | ||

Lecture 8, Sep. 26 | On \( H^1(\Gamma, \mathbb{C}) \) following Matsushima | ||

Lecture 9, Oct 1 | On \( H^1(\Gamma, \mathbb{C}) \) following Matsushima | ||

Lecture 10, Oct 5 | |||

Lecture 11, Oct 10 | |||

Lecture 12, Oct 15 | |||

Lecture 13, Oct 17 | |||

Lecture 14, Oct 22 | |||

Lecture 15, Oct 24 | |||

Lecture 16, Oct 29 | |||

Lecture 17, Oct 31 | |||

Lecture 18, Nov 5 | |||

Lecture 19, Nov 7 | |||

Lecture 20, Nov 19 | |||

Lecture 21, Nov 21 | |||

Lecture 22, Nov 26 | |||

Lecture 23, Nov 28 | |||

Lecture 24, Dec 3 | |||

Lecture 25, Dec 5 |