## Polymath16, twelfth thread: Year in review and future plans

This is the twelfth “research” thread of the Polymath16 project to make progress on the Hadwiger–Nelson problem, continuing this post. This project is a follow-up to Aubrey de Grey’s breakthrough result that the chromatic number of the plane is at least 5. Discussion of the project of a non-research nature should continue in the Polymath proposal page. We will summarize progress on the Polymath wiki page.

Activity on this project has slowed considerably, as we’ve gone 6 months without having to roll over to a new thread. As mentioned in the original thread, the deadline for this project is April 15, 2019, so we only have a couple of weeks remaining. Dömötör and Aubrey took the time to summarize the highlights of what we’ve accomplished in the last year (see below). While we don’t have a single killer result to publish, there are several branches of minor results that warrant publication. Feel free to comment on additional results that were missed in the summaries below, as well as possible venues for publication.

## Another simple solution to the Basel problem

Recall that $\zeta(2)=\sum_{k=1}^\infty 1/k^2$. Perhaps the shortest proof of $\zeta(2)=\pi^2/6$ applies (the sophisticated) Parseval’s identity of the Fourier series to $f(x)=x$. By contrast, the simple (but long) proof in this paper, which was recently popularized by the video below, uses basic ideas from Euclidean geometry.

The following argument interpolates between these two, resulting in a proof that is both simple and short.

## Packings in real projective spaces

There has been a lot of work recently on constructing line packings that achieve equality in either the Welch bound or the orthoplex bound. It has proven much harder to pack in regimes where these bounds are not tight. To help fill this void, about a month ago, I posted a new paper with Matt Fickus and John Jasper on the arXiv. We provide a few results to approach this case, which I outline below.

1. The minimal coherence of 6 unit vectors in $\mathbb{R}^4$ is 1/3.

The Welch bound is known to not be tight whenever $n$ lies strictly between $d+1$ and $d+\sqrt{2d+1/4}+1/2$ (see the next section for a proof sketch). As such, new techniques are required to prove optimality in this range. We leverage ideas from real algebraic geometry to show how to solve the case of $d+2$ vectors in $\mathbb{R}^d$ for all sufficiently small $d$. For example, our method provides a new proof of the optimality of 5 non-antipodal vertices of the icosahedron in $\mathbb{R}^3$, as well as the optimality of Sloane’s packing of 6 lines in $\mathbb{R}^4$.

## Conjectures from SampTA

Back in May, I attended this year’s SampTA at American University. I spoke in a special session on phase retrieval, and as luck would have it, Cynthia Vinzant spoke in the same session about her recent solution of the 4M-4 conjecture. As you might expect, I took a moment during my talk to present the award I promised for the solution:

Recall that Cynthia (and coauthors) first proved part (a) of the conjecture, and then recently disproved part (b). During her talk, she also provided a refinement of part (b). Before stating the conjecture, recall that injectivity of the mapping $x\bmod\mathbb{T}\mapsto |Ax|^2$ is a property of the column space $\mathrm{im}(A)$.

## Alexander Grothendieck

Readers of this blog are probably already aware that Alexander Grothendieck died on Thursday. He is widely regarded as one of the most influential mathematicians in the twentieth century. Since his is not my field of study, I felt that now was a good time to learn a little about why he is so well regarded — I took the day to read a couple of articles from 10 years ago that provide an overview of his life, research, personality, and philosophies. I highly recommend the read: here and here.

## My research explained, using only the 1000 most common English words

[Inspired by ScottAfonso and Joel, using the Up-Goer Five Text Editor, which in turn was inspired by this xkcd. I actually only use 261 distinct words.]

Let’s say you have a picture, a piece of music, or a movie that you want to store on your computer. You can do this without taking up your entire hard drive, but why? Because there’s a way to look at each of these things so that they appear very simple: Imagine someone is making a movie of you reading this. You’re just sitting there. Maybe a fly is flying around the room, but not much is changing. That means each moment of the movie looks a lot like the one right before, and this makes it very easy to store the entire movie on a computer.

The fact that pictures and such are so simple allows you to do other cool stuff. Let’s say you find your favorite movie in the back of a second-hand store, but when you watch it at home, different marks pop up every now and then. Since movies are so simple, you can use a computer to fill in what you can’t see, and make it good as new.

## Boris’s Law of Blogging

Before I started this blog, I got a helpful blog-writing tip from Boris Alexeev. We’ve been friends since I started studying at Princeton, so I try to live up to his advice:

Boris’s Law of Blogging
The time between any two consecutive blog posts should be between 1 and 2 weeks.

In fact, his advice has saved this blog from dying a couple of times, as the periodic deadlines have kept me motivated to write; also, the one-week lower bound has prevented me from writing too much and burning out. Afonso and Joel have taken Boris’s Law a step further by devising a two-player version of the upper bound: You must post within one week of your opponent’s most recent blog post. Overall, I highly recommend following Boris’s Law if you start a blog.

Last week, my PhD was conferred; at the ceremony, I saw Aretha Franklin get an honorary PhD in music! Tomorrow morning, I’ll move to Dayton, Ohio to become an Assistant Professor of Mathematics at the Air Force Institute of Technology. As you can imagine, the past couple of weeks have been rather hectic, so I haven’t been able to write a technical blog post, but I still want to abide by the Law.

I’ll post something more serious in the next couple of weeks. In the meantime, enjoy this short discussion of “crocodile matrices.” 🙂