## Algant Master 2020: Lie groups and their lattices

Main goal: prove Margulis' superrigidity theorem.

Along the way we will discuss:
- Some generalities about locally compact groups, Lie groups, lattices
- Howe Moore theorem about unitary representations of Lie groups
- Borel density theorem, asserting that Lattices in semi-simple Lie groups are Zariski dense
- Some ergodic theorems
- Amenable groups

Times permitting we shall given an application to lattices in combinatorics: the construction Expander graphs.

**Lecture notes:** Here (update: May 11).

Exercise sheets:

Lc groups and lattices, Homogeneous spaces, Lie groups, ergodic theory,
Stationary measures, Linear and projective representations, Amenability.

### Updates:

- The
**final exam** will be held on Tuesday, May 26, 14:00-17:00.

Computation of the final grade for the course: Max(Final, 0.6 x final + 0.4 x Homework).
- The midterm exam is replaced by a graded Homework. Here is the Correction.
- Weakly reading (until May 15) :
**Chapter 6, sections 3.3, 3.4** (pages 50-54).
- Next meeting on Wednesday: We'll get back to the exercises on amenability. We will also discuss a fact used in Lemma 6.23.
- Last weak's Weakly reading (until May 8) : Chapter 6, sections 2, 3.1, 3.2.

Last update: 11/05/2020.