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:

G. Davidoff, P. Sarnak, A. Valette, Elementary Number Theory, Group Theory, and Ramanujan Graphs

You may also find the following survey article helpful:

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

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