Seminar in Elementary Number Theory, Group
Theory, and Ramanujan Graphs
Spring 2025
About the seminar
This seminar is dedicated to the construction of expander graphs, which are sparse yet highly connected graphs with wide-ranging applications in computer science. Specifically, we will focus on Ramanujan graphs, which are optimal expanders. We will establish the expansion properties of classical and elegant constructions of Ramanujan graphs using elementary number theory and group theory, including some elements of representation theory, which we will develop from the ground up. We will mostly follow this book by Davidoff, Sarnak, and Valette, which shares its title with this seminar.
When & where
Thursdays 10:00-12:00. Location: TBD.
Tentative plan
As mentioned, we will mostly follow this book by Davidoff, Sarnak, and Valette, covering most of it. Some of the topics may be skipped if the number of students is lower than expected. The list of lectures is roughly as follows:
Lecture 1 - Introduction
Lecture 2 Ben (1.4) - The asymptotic behavior of the spectral expansion
Lecture 3 - Ido (2.1, 2.2) - Sums of two squares and Gaussian integers
Lecture 4 - Tzlil (2.3, 2.4, 2.5) - Quadratic reciprocity, sum of four squares, and quaternions
Lecture 5 - Omri (2.6) - The arithmetic of quaternions
Lecture 6 - Yuval (3.1, 3.2, 3.3 up to, not including, Theorem 3.3.4) - Some finite groups, simplicity, and structure of subgroups
Lecture 7 - Alon (3.3.3, Theorem 3.3.4 onwards)
Lecture 8,9 (3.4) - Representation theory of finite groups
Lecture 10 - Matan (3.5) - Degrees of representations of PSL_2(q)
Lecture 11 - Nadav (4,1, 4.2) - Construction of X^{p,q}
Lecture 12 (4.3) - Girth and connectedness
Lecture 13 (4.4) - Spectral estimates