### Syllabus for Math 8851, Fall 2012. Expander graphs and linear groups

### TuTh 2-3:15 pm, Kerchof 128.

- Instructor: Mikhail Ershov
- Office: Kerchof 302
- e-mail: ershov
*at* virginia *dot* edu
- Office hours: by APPOINTMENT

The course will be somewhat non-standard both in terms
of
- content as it will include material from several
rather different areas and
- class format as a significant portion of the course
will consist of student presentations

The course will consist of three main parts:

A. Brief introduction to linear groups (5 weeks)

B. Introduction to expander graphs (6 weeks)

C. Linear groups and expander graphs (2 weeks)

The first two parts will be almost unrelated, while
the third part will use results and techniques from the first two.

In the first part we will discuss some classical results about
the structure theory of linear groups and develop necessary
background for the third part of the course. There is no specific text suggested for the
first part.

On the contrary, in the second part we will follow quite closely
the book *Expander Families and Cayley Graphs: A Beginner's Guide * by Krebs and Shaheen.
The plan is that material in the second part will be presented by students
taking (or sitting in) the course, with each student presenting one chapter
of the book in two consecutive classes, over the span of six weeks. For this reason
you should start reading the book at the very beginning of the fall term and make decisions
about what you would like to present by the middle of September. We will start discussing
expander graphs after the Fall break and probably cover chapters 1,3,4,5,7 and 8 of the book.

In the third part of the course I hope to talk about some recent work on expansion in linear groups.
A more precise plan for this part and suggested references will be announced later in the term.
#### Getting credit.

If you are officially registered for the course, in order to get credit you must do at least one of the following:
- (a) Make a presentation on expander graphs (2 lectures) during part B of the course (preferred)
or a presentation (1 lecture) on a topic related to the course material at the of the semester.
- (b) Submit solutions to some (reasonably many) problems formulated during the lectures

These are minimal requirements, and students are encouraged to do both (a) and (b). This applies
not only to students registered for the course, but also to those just attending.