## Spherical codes and designs

Later this month, Hans Parshall will participate in a summer school on “Sphere Packings and Optimal Configurations.” In preparation for this event, Hans was assigned the task of writing lecture notes that summarize the main results of the following paper:

P. Delsarte, J. M. Goethals, J. J. Seidel,

Geometriae Dedicata 6 (1977) 363–388.

I found Hans’ notes to be particularly helpful, so I’m posting them here with his permission. I’ve lightly edited his notes for formatting and hyperlinks.

## Some news regarding the Paley graph

Let $\mathbb{F}_p$ denote the field with $p$ elements, and let $Q_p$ denote the multiplicative subgroup of quadratic residues. For each prime $p\equiv 1\bmod 4$, we consider the Paley graph $G_p$ with vertex set $\mathbb{F}_p$, where two vertices are adjacent whenever their difference resides in $Q_p$. For example, the following illustration from Wikipedia depicts $G_{13}$:

The purpose of this blog entry is to discuss recent observations regarding the Paley graph.

## MATH 8610: Mathematics of Data Science

This spring, I’m teaching a graduate-level special topics course called “Mathematics of Data Science” at the Ohio State University. This will be a research-oriented class, and in lecture, I plan to cover some of the important ideas from convex optimization, probability, dimensionality reduction, clustering, and sparsity.

The current draft consists of a chapter on convex optimization. I will update the above link periodically. Feel free to comment below.

UPDATE #1: Lightly edited Chapter 1 and added a chapter on probability.

UPDATE #2: Lightly edited Chapter 2 and added a section on PCA.

UPDATE #3: Added a section on random projection.

UPDATE #4: Lightly edited Chapter 3. The semester is over, so I don’t plan to update these notes again until I teach a complementary special topics course next year.

UPDATE #5: As mentioned above, I’m teaching a complementary installment of this class this semester. I fixed several typos throughout, and I added a new section on embeddings from pairwise data.

UPDATE #6: Added a section on the clique problem.

UPDATE #7: Added a section on the Lovasz number.

UPDATE #8: Added a section on planted clique.

UPDATE #9: Added sections on maximum cut and minimum normalized cut.

UPDATE #10: Added a section on k-means clustering.

## A neat application of the polynomial method

Two years ago, Boris Alexeev emailed me a problem:

Let $n \geq 2$.  Suppose you have $n^2$ distinct numbers in some field.  Is it necessarily possible to arrange the numbers into an $n\times n$ matrix of full rank?

Boris’s problem was originally inspired by a linear algebra exam problem at Princeton: Is it possible arrange four distinct prime numbers in a rank-deficient $2\times 2$ matrix? (The answer depends on whether you consider $-2$ to be prime.) Recently, Boris reminded me of his email, and I finally bothered to solve it. His hint: Apply the combinatorial nullstellensatz. The solve was rather satisfying, and if you’re reading this, I highly recommend that you stop reading here and enjoy the solve yourself.

## Notes on Zauner’s conjecture

Last week, I visited Joey Iverson at the University of Maryland, and we spent a lot of time working through different approaches to Zauner’s conjecture. In general, my relationship with this problem is very similar to Steve Flammia‘s description, as paraphrased by smerkel on Physics Stack Exchange:

[Flammia described] the SIC-POVM problem as a “heartbreaker” because every approach you take seems super promising but then inevitably fizzles out without really giving you a great insight as to why.

Case in point, Joey and I identified a promising approach involving ideas from our association schemes paper. We were fairly optimistic, and Joey even bet me \$5 that our approach would work. Needless to say, I now have this keepsake from Joey:

While our failure didn’t offer any great insights (as Flammia predicted), the experience forced me to review the literature on Zauner’s conjecture a lot more carefully. A few things caught my eye, and I’ll discuss them here. Throughout, SIC denotes “symmetric informationally complete line set” and WH denotes “the Weyl-Heisenberg group.”

## Introduction to the polynomial method (and other similar things)

The polynomial method has made some waves recently (see this and that, for instance), and last week, Boris Alexeev gave a very nice talk on various applications of this method. This post is loosely based on his talk. All errors are my own.

It’s hard to pin down what exactly the polynomial method is. It’s a technique in algebraic extremal combinatorics, where the goal is to provide bounds on the sizes of objects with certain properties. The main idea is to identify the desired cardinality with some complexity measure of an algebraic object (e.g., the dimension of a vector space, the degree of a polynomial, or the rank of a tensor), and then use algebraic techniques to estimate that complexity measure. If at some point you use polynomials, then you might say you applied the polynomial method.

What follows is a series of instances of this meta-method.

## The HRT Conjecture

A couple of weeks ago, I attended a workshop hosted by Darrin Speegle on the HRT Conjecture. First posed twenty years ago by Chris Heil, Jay Ramanathan, and Pankaj Topiwala in this paper, the conjecture states that every finite collection of time-frequency shifts of any nonzero function in $L^2(\mathbb{R})$ is linearly independent. In this post, I will discuss some of the key ideas behind some of the various attempts at chipping away at HRT, and I’ll describe what appear to be fundamental barriers to a complete solution. For more information, see these survey papers: one and two.

First, some notation: Denote the translation-by-$a$ and modulation-by-$b$ operators by

$(T_af)(x):=f(x-a),\qquad (M_bf)(x)=e^{2\pi i bx}f(x),$

respectively. Then the formal conjecture statement is as follows:

The HRT Conjecture. For every $0\neq f\in L^2(\mathbb{R})$ and every finite $\Lambda\subset\mathbb{R}^2$, the collection $\{M_bT_af\}_{(a,b)\in\Lambda}$ is linearly independent.

What follows are some of the popular methods for tackling HRT: