Orthogonal groups of four-dimensional real quadratic spaces

I’m taking a class on quadratic forms, orthogonal groups, and modular forms this semester, and the class is formatted such that students present extended solutions to exercises in pairs on a rotating basis. Well, this week it is my turn to present a description of four dimensional real quadratic spaces over ${\bf R}$, along with my comrade Jad Hamdan. Neither Jad nor I had had any exposure to Lie groups before this week, so we worked out the following with a great deal of guidance from the instructor of the course, Prof. Henri Darmon. I’m writing this blog post to organise and collect my thoughts before the actual presentation. I’ll go ahead and assume a level of mathematical background equal to my own before the class started, namely, an undergraduate understanding of group theory, linear algebra, and topology but no experience with differential geometry or topological groups.

The lattice of arithmetic progressions

Jad Hamdan and I have uploaded our paper “The lattice of arithmetic progressions” to the arXiv. In it we study the partially ordered set $L_n$ of all subsets of $[n] = \{1,2,\ldots,n\}$ that are arithmetic progressions, including the empty set and trivial progressions of length $1$ and $2$. This poset is a lattice, but for $n\geq 4$, it is not graded. We derive formulas and recurrences regarding the numbers $p_{nk}$ of arithmetic progressions in $[n]$ of length $k$ as well as the number $b_{nk}$ of chains in $L_n$ of length $k+2$ that contain both $\emptyset$ and $[n]$. Let $\mu_n$ denote the Möbius function of the lattice; we give three short, independent proofs of the fact that for $n\geq 2$, $\mu_n(L_n) = \mu(n-1)$, where $\mu$ is the classical (number-theoretic) Möbius function. We finish off by computing the homology groups of the order complex $\Delta_n$ of $L_n$.

Update. (07 Sep 2021) We have added a second version of the paper. Our good friend Jonah Saks has joined the cause, helping us to strengthen the topological results in the second half of the paper. In particular, we are now able to show that $\Delta_n$ has the homotopy type of a sphere or a point.

The combinatorial Nullstellensatz

I haven’t done a proper math blog post for awhile, and I’ve been meaning to learn more about the polynomial method, so I thought I’d sit down and get to grips with the “combinatorial Nullstellensatz” (see Alon (1999)). This was prompted when I realised that there were a couple of exercises in of Knuth’s books (namely Volume 2 and Volume 4, Fascicle 5) that concern the Nullstellensatz. (I’m working off a preliminary draft of Mathematical Preliminaries Redux (MPR), which was posted to Knuth’s website before the book came out; it is no longer there because the book is now out! I do not know if the exercises (and numbering) in my draft correspond to the exercises in the actual book.) I must confess to having seen the proof of the Nullstellensatz before in Tao and Vu’s Additive Combinatorics which I read (read: skimmed) about three months ago, but I have forgotten it and so this post will see me trying to reconstruct it, with some hints from Knuth.

Tonogenesis in Dene languages

I’m taking a seminar class this semester on the Na-Dene languages as part of my linguistics minor, and this week it is my turn to give a presentation on a paper in the field. I am covering a 1999 paper by Jeff Leer called “Tonogenesis in Athabaskan”, and I thought it’d be good preparation to collect my thoughts in a blog post while reading the paper. This is a bit of a departure from the mathematical content I usually deliver, but should be in the original spirit of the blog. Most of my early posts involve me grappling and experimenting with new ideas, and I’m sure there will be a lot of that in this post, since most of the linguistic concepts found in this paper are completely new to me. Of course, you might get more out of this post if you have read/are reading the paper as well, but I intend to make this a self-contained summary of the paper for a somewhat more general audience.

Arithmetic subsequences in permutations

Rosie Zhao and I have just uploaded our paper “Arithmetic subsequences in a random ordering of an additive set” to the arXiv. In its simplest form, the problem we consider is the following. Given an ordering of the numbers $1$ through $n$, what is the length of the longest arithmetic subsequence that is embedded in this permutation? For example, in the ordering $(2,7,1,6,3,4,5)$, the longest arithmetic subsequence is $(2,3,4,5)$, which has length $4$. If we store the permutation in an array, the problem of finding the length of the longest arithmetic subsequence is a popular programming interview question that can be solved efficiently using dynamic programming (it is a medium-difficulty problem on LeetCode). But if we take the ordering to be random, with each of the $n!$ possibilities occurring with equal probability, then the length $L_n$ of the longest arithmetic subsequence becomes a random variable. Our paper studies the asymptotic behaviour of $L_n$ as $n$ gets large.

Update. (28 Sep 2021) Our paper was accepted to Integers: Electronic Journal of Combinatorial Number Theory! There were some small errors in the original lemmas, which have been corrected in both the published and arXiv versions as well as in the blog post below.