# The Baruch Distinguished Mathematics Lecture Series

I am happy to announce the beginning of the Baruch Distinguished Mathematics Lecture Series. In this series we will bring established mathematicians to give talks to a general mathematical audience.

Our first Distinguished Lecture, by Bjorn Poonen, will be “Undecidability in Number Theory”. Click here for the full details. The talk is open to everyone, and includes refreshments. After the talk we will also go to lunch with the speaker.

# Incidences in a Recent Work of Walsh

Recently, Miguel Walsh posted a very interesting paper on arXiv. The main purpose of the paper is to study various properties of polynomials and varieties. These properties are related to incidence problems – some originally arose from studying incidences. Walsh also presents new incidence bounds as applications of his results. In this post I’ll briefly discuss the incidence aspects of Walsh’s paper. Since this is a rather specialized post for incidence researchers, it assumes familiarity with the problems and notation.

Miguel Walsh

It seems to me that Walsh’s paper makes three main contributions to the study of incidence problems. The first two contributions are explicitly stated in the paper:

• The paper derives a new polynomial partitioning theorem for points on a variety, with a good dependency on the degree of the variety. This was the missing step for removing an extra $\varepsilon$ in the exponent from most of the existing incidence bounds in ${\mathbb R}^d$. For example, it should remove the extra $\varepsilon$ in the results of several of my papers, such as this one and that one.
• As an application of his results, Walsh introduces an incidence problem that I did not see before. In this problem we study incidences between points and varieties in ${\mathbb R}^d$, but the degrees of the varieties are not constant. That is, the degrees of the varieties are allowed to depend on the number of points and on the number of varieties. It would be interesting to see how this variant develops. For example, what applications in other subfields it might have.

The two above incidence results are definitely interesting, but I suspect that the paper would have a more significant effect on the incidence community in a different way. The paper makes significant progress in our understanding of polynomials and varieties, which directly affect the algebraic methods used to derive incidence bounds. This improved understanding could potentially be what we need to address some of the main problems that we have been trying to solve for a while. Me and other incidences researchers will definitely spend time reading the paper carefully, and think how it might fit with these problems. Walsh also mentions a follow-up paper, and I wonder if he has something similar in mind. I am planning to soon write a post about the current main open problems in incidence theory, and will then mention more clearly what I am referring to.

It might also be worth discussing the lower bounds part of the paper. The paper contains matching lower bounds for some of its incidence results. However, these lower bounds seem to apply specifically to the new variant of incidences with varieties of non-constant degrees. The “standard” problem of incidences with bounded-degree hypersurfaces in ${\mathbb R}^d$ remains open. In particular, we have matching lower bounds only in some cases. For an up-to-date discussion of the known lower bounds, see for example the relevant part in the introduction of my recent work with Thao Do.

# An Uncitable Result?

My colleague Pablo Soberon just showed me an unusually problematic result to cite, and I wanted to share this weird story. If you have other weird citation stories, do tell!

Yes, this is a second silly post in a row. Lately I’m not finding the time to write more serious ones. And the silly stories need to be documented somewhere…

This story begins with a Japanese anime show called The Melancholy of Haruhi Suzumiya. I don’t know much about this show, but apparently there are several different orders in which one can watch the episodes. This led a fan of the show to ask for the minimum number of episodes one needs to watch, so that you saw all the episodes in every possible order. In other words, the minimum sequence of the numbers from $\{1,\ldots,n\}$ that contains every possible permutation of the numbers. (if I understand correctly, each order has to appear consecutively, with no additional numbers in between). This was on 4chan in 2011. A solution was then offered by an anonymous user, and this disappeared among the other weird anime discussions around the web.

It turns out that some people have been studying the above question as a serious math problem, prior to the show and not aware of it. The sequence containing all of the possible permutations is referred to as a superpermutation. See for example here and here. One paper about this was even published in the journal “Discrete Mathematics” in 2013.

Now the people coming from the mathematical angle discovered the original 4chan discussion, and in it the solution to the problem. So can they cite this result? It is by an anonymous person and appeared on an anime fans website. If this is not complicated enough, the relevant website no longer exists. Instead, the original discussion was discovered on a site that archives old online discussions. And it’s unclear how stable this archive site is. Luckily, this is not my problem!

# Mathematical Energy: Etymology

This might be the silliest post I’ve written so far (yes – worse than “Was Disney trying to kill mathematicians?”). I urge you to stop reading now unless (i) you are quite familiar with the mathematical notion of energy (e.g., additive energy), and (ii) you have a horrible sense of humor.

The term energy was coined by Tao and Vu. I like “energy” as a name for this object, but I never had a good answer when asked why this is how it’s called. That is, until a referee report provided me with an answer. And this wonderful referee probably didn’t even know it.

In a recent paper, Cosmin Pohoata and I used “color energy”. We have a graph $G=(V,E)$ with colored edges. Denote the color of an edge $(v_1,v_2)$ as $\chi(v_1,v_2)$. The color energy of $G$ is

$E(G) = \left|\left\{(v_1,v_2,v_3,v_4)\in V^4 :\ \chi(v_1,v_2) = \chi(v_3,v_4) \right\}\right|.$

The referee complained about our notation for the multiplicity of a color $c$ (the number of edge of color $c$), and asked to change it to $m_c$. After this change of notation, the energy is defined by the standard formula

$E = \sum m_c^2.$

# NYC Discrete Geometry: Introductory Meeting

I am excited to announce the official start of our new Discrete Geometry group! This event will also be the first meeting of the NYC Geometry Seminar in the CUNY Graduate Center (midtown Manhattan). It will take place at 2pm of Friday August 31st. If you are in the NYC area and interested in Discrete Geometry and related topics – come join us!

The introductory meeting would not consist of the standard seminar presentation. Instead, the purpose of the event is to meet people who are interested in Discrete Geometry, Computational Geometry, and so on (with people coming from NYU, Princeton, various CUNYs, etc). Participants will introduce themselves to the audience and mention the main topics that interest them. You are welcome to either introduce yourself, or just to come listen and have some pastries. If you introduce yourself, you are encouraged to briefly state a major open problem you wish you could solve.

For the exact location, see the event page. Personally, I plan to have more technical math discussions with some participants before and after this meeting.

# Linear Algebra Riddle

I’d like to tell you about a nice riddle, which I heard from Bob Krueger (one of our current REU participants, who already has four papers on arXiv!!). The riddle requires very basic linear algebra and is in the spirit of the previous post.

Riddle. A library has $n$ books and $n+1$ subscribers. Each subscriber read at least one book from the library. Prove that there must exist two disjoint sets of subscribers who read exactly the same books (that is, the union of the books read by the subscribers in each set is the same).

Hint: Very basic linear algebra. Try the first thing that comes to mind.

# Discrete Geometry Classic: Two-distance Sets

This post presents another simple and elegant argument in Discrete Geometry. This classic is based on the so-called “Linear Algebra method”.

Warm-up problem. What is the maximum size of a set ${\cal P} \subset {\mathbb R}^d$ such that the distance between every two points of the set is 1?

By taking the vertices of a $d$-dimensional simplex with side length 1 in ${\mathbb R}^d$, we obtain $d+1$ points that span only the distance 1. It is not difficult to show that a set of $d+2$ points in ${\mathbb R}^d$ cannot span a single distance.

The two-distance sets problem. A point set $\cal P$ is a two-distance set if there exist $r,s\in {\mathbb R}$ such that the distance between every pair of points of $\cal P$ is either $r$ or $s$. What is the maximum size of a two-distance set in ${\mathbb R}^d$?

There is a simple trick for constructing a two-distance set of size $\binom{d}{2}$ in ${\mathbb R}^d$. You might like to spend a minute or two thinking about it. Here is a picture of Euler to prevent you from seeing the answer before you wish to.

Consider the set of all points in ${\mathbb R}^d$ that have two coordinates with value 1 and the other $d-2$ coordinates with value 0. There are $\binom{d}{2}$ such points, and the distance between every pair of those is either 1 or 2.

Larman, Rogers, and Seidel showed that the above example is not far from being tight.

Theorem. Every two-distance set in ${\mathbb R}^d$ has size at most $\binom{d}{2}+3d+2$.

Proof. Let ${\cal P} = \{p_1,p_2,\ldots, p_m\}$ be a two-distance set in ${\mathbb R}^d$, and denote the two distances as $r,s\in {\mathbb R}$. Given points $a,b\in {\mathbb R}^d$, we denote the distance between them as $D(a,b)$. We refer to a point in ${\mathbb R}^d$ as $x=(x_1,\ldots,x_d)$. For $1\le j \le m$, define the polynomial $f_j\in {\mathbb R}[x_1,\ldots,x_d]$ as

$f_j(x) = \left(D(x,p_j)^2 - r^2\right) \cdot \left(D(x,p_j)^2 - s^2\right).$

That is, $f_j$ is a polynomial of degree four in $x_1,\ldots,x_d$ that vanishes on points at distance $r$ or $s$ from $p_j$. Every such polynomial is a linear combination of the polynomials

$\left(\sum_{j=1}^d x_j^2\right)^2,\qquad x_k \sum_{j=1}^d x_j^2,\qquad x_k,\qquad x_\ell x_k,\qquad 1,$

for every $1\le \ell, k\le d$. Since every $f_j$ is a linear combination of $t = \binom{d}{2}+3d+2$ polynomials, we can represent $f_j$ as a vector in ${\mathbb R}^t$. In particular, the vector $V = (v_1,\ldots,v_t)$ corresponds to the polynomial

$v_1 \cdot \left(\sum_{j=1}^d x_j^2\right)^2 + v_2\cdot x_1 \sum_{j=1}^d x_j^2 + v_3\cdot x_2 \sum_{j=1}^d x_j^2 + \cdots + v_{t} \cdot 1.$

For $1\le j \le m$, let $V_j$ denote the vector corresponding to $f_j$. Consider $\alpha_1,\ldots,\alpha_m\in {\mathbb R}$ such that $\sum_{j=1}^m \alpha_j V_j = 0$. Let $g(x) = \sum_{j=1}^m \alpha_j f_j(x)$. By definition, the polynomial $g(x)$ is identically zero. For some fixed $p_k\in {\cal P}$, note that $f_j(p_k)=0$ for every $j\neq k$ and that $f_j(p_j) = r^2s^2$. Combining the above implies

$g(p_k) = \sum_{j=1}^m \alpha_j f_j(p_k) = \alpha_kr^2s^2.$

Since $g(x)=0$, we have that $\alpha_k=0$. That is, the only solution $\sum_{j=1}^m \alpha_j V_j = 0$ is $\alpha_1=\cdots = \alpha_m = 0$. This implies that the vectors $V_1,\ldots,V_m$ are linearly independent. Since these vectors are in ${\mathbb R}^t$, we conclude that $m \le t = \binom{d}{2}+3d+2$. $\Box$

One can improve the above bound by showing that the polynomials $f_j$ are contained in a smaller vector space. This is left as an exercise of your googling capabilities.