Lecture notes

# Lectures 15-16: Gaussian processes and the Gaussian free field

Everything we covered on Gaussian processes can be found here.

# Lecture 14: Cover times

In this lecture, we studied hitting times, commute times, the effective resistance metric, and cover times of graphs. The book of Aldous and Fill has rather detailed discussions of all these notions. In particular, see Chapter 2, Section 6 for Matthews’ upper and lower bounds on cover times.

# Lectures 9-13

These lectures covered primarily the following four papers:

I will post comprehensive lecture notes soon. We only used the subspace enumeration algorithm from (2) and the bulk of our analysis used the Lovasz-Siminovits-style arguments of (3) (see also (4)).

# Lecture 7-8

We discussed combinatorial and spectral notions of graph expansion and sketched why a random d-regular graph is an expander for d large enough. Then I presented Margulis’ expander construction and the Gabber-Galil analysis. To end, we proved that expanders require logarithmic distortion to embed into any Euclidean space.

Here is a modified analysis of the Margulis expanders.

# Lecture 6

Here are the slides, and here’s a proof of the crossing number inequality.

From this paper, we know that for an n-vertex planar graph with maximum degree ${\Delta}$, we have for every k, the eigenvalue bound

${\lambda_k \leq O(\Delta k/n)}$

Open question: What about the eigenvalue gaps? I conjecture that the stronger bound ${\lambda_{k+1} - \lambda_k \leq O(\Delta/n)}$ holds for every k. I believe it is open even for k=2.