Math 595: Expander Graphs in Number Theory

Fall 2015

MWF 3-3:50PM, 441 Altgeld Hall

Professor: Elena Fuchs
Office: 359 Altgeld Hall
Email: lenfuchs at illinois dot edu
Office hours: by appointment

This course will explore various aspects of expander graphs, with a view towards applications in number theory, specifically that of the Affine Sieve developed by Bourgain-Gamburd-Sarnak in 2009. We will give three definitions of expander graphs and show that they are equivalent, we will then explore the application of expander graphs in the Affine Sieve, and, finally, we will prove that certain concrete families of graphs (obtained as Cayley graphs connected to certain finitely generated subgroups of SL_2(Z)) are indeed expanders.

Textbook and Prerequisites:

We have no official textbook but a good reference for the expander graphs section is E. Kowalski's lecture notes on expander graphs. We will also be going into aspects of Bourgain-Gamburd-Sarnak's first paper on the affine sieve, as well as some papers on the number theory of Apollonian packings. Please see the list of useful links below for the most relevant literature.

The prerequisite for this course is a knowledge of linear algebra and the group theory part of a first semester graduate algebra course.


Your grade for the course is based on a project which you can do together with one other student (some topics are better suited for collaboration than others). The project will be to select one of the following topics and to present it in class. I have posted some suggested references, but you may supplement those that I have suggested with others if you wish. Topics are:

Detailed Plan:

The following is a rough outline of what we will be doing in lecture every day. It will be updated on a regular basis.
Some useful links, in alphabetical order:

Instructions in event of emergency