Syllabus for Math 8852 (Spring 2013).

Unitary representations and Kazhdan's property (T)

The course will start with a brief introduction to the theory of unitary representations of groups (about 5.5 weeks), after which we will turn to Kazhdan's property (T) and its applications (about 7.5 weeks). Below is a very tentative list of main topics I am going to discuss:

1.  Some generalities on unitary representations
2.  Unitary representations of compact groups
3.  Unitary representations of abelian groups
4.  Unitary representations of SL_2(R) and the Heisenberg group
5.  Kazhdan's property (T): definition(s) and basic properties
6.  Kazhdan's property (T) and affine isometric actions (property (FH))
7.  Relative property (T)
8.  Property (T) for SL_n(Z), n>=3
9.  Spectral criteria for property (T)
10.  Property (T) and amenability

The above list will likely be continuously adjusted throughout the semester depending on how fast we go and on the audience interests.

Prerequisites: I will try to make the course as self-contained as possible. The main prerequisites (in addition to first year graduate courses) are some exposure to representation theory of finite groups (at the level of Algebra-III) and basic theory of Hilbert spaces, including spectral measures and spectral theorems (at the level of Functional Analysis-I), but even these are not strict requirements if you are willing to do some extra reading on your own and possibly assume some facts for granted.

References: The main book for the course is `Kazhdan's Property (T)' by B. Bekka, P. de la Harpe and A. Valette, a preprint version of which is freely available here . Other references will be provided as we progress through the material.

Required work: There will be regular homework assignments (approximately every other week). Problems will not be due in written form; instead we will have a problem session (about 90 minutes long) after each assignment where solutions will be presented by students. If you are taking this course for a grade, you are expected to participate in those presentations on the regular basis, but all students sitting in the class are strongly encouraged to work on the assigned problems.