Undergraduate Seminar (Fall 2017): Ramanujan Graphs and Number Theory

Organizer
Daniel Gulotta
Email
dgulotta .at. math.columbia.edu
Prerequisites
Linear algebra. It would also be helpful to know a bit of graph theory (the definiton of a graph), number theory (modular arithmetic), and group theory (the definition of a group, subgroup, homomorphism).
Location and time
Monday and Wednesday 4-5 in 528 Math
Office Hours
By appointment.

Resources

We will follow the book:

You may also find the following survey article helpful:

Requirements

Each student will give 3 lectures. Additional lectures are optional.

Schedule

Date Topic Speaker
Sep 11 Overview, adjacency matrices (1.1) Dan
Sep 11 Spectral gap (1.2 to 1.2.3a) Dan
Sep 18 Asymptotic behavior (1.2.3b-1.3) Dan
Sep 20 Asymptotic behavior (1.4 to 1.4.8 first step) Mevludin
Sep 25 Asymptotic behavior (1.4 from 1.4.8 second step) Nick
Sep 27 Large girth and large chromatic number (1.5-1.6) Dan
Oct 2 Sums of two squares (2.1-2.2.6) Mevludin
Oct 4 Sums of two squares (2.2 from 2.2.7) Yujin
Oct 9 Quadratic reciprocity (2.3-2.4.3) Nick
Oct 11 Sums of four squares, quaternions (2.4.4-2.5) Dan
Oct 16 Quaternions (2.6 to 2.6.5) Dan
Oct 18 Quaternions (2.6.6-2.6.11) Mevludin
Oct 23 Quaternions, some finite groups (text after 2.6.11-3.1 intro) Dan
Oct 25 Simplicity, structure of subgroups (3.1.1-3.3.6) Dan
Oct 30 Structure of subgroups (3.3.7-3.3.11) Dan
Nov 1 Representation theory (proof of 3.3.6-3.4.4) Dan
Nov 13 Representation theory (3.4.5-3.4.11) Yujin
Nov 15 Representation theory (3.4.12-3.4.18) Dan
Nov 20 Representation theory (3.4.19-end of 3.4) Nick
Nov 27 Degrees of representations of PSL2(q) (3.5) Yujin
Nov 29 Cayley graphs (4.1, 4.2.1) Mevludin
Dec 4 Girth and connectedness (4.2.2-4.3) Dan
Dec 6 Spectral estimates (4.4 to 4.4.4) Dan
Dec 11 Spectral estimates (4.4.4-end of 4.4), Jacquet-Langlands Dan