## One year of writing and submitting short stories

In early July of 2021, I was on a holiday with my parents in Canmore. This was the first stop in a long journey during which I planned to split off from my parents in Revelstoke, meet my university friends in Kelowna, and then drive across Canada back to Montréal, so I had brought a decent stack of books with me. In that stack was a short story collection by Margaret Atwood called Dancing Girls. The stories themselves were pretty good (honestly I don’t remember the details of most of them), but it was the first page that intrigued me. It looks like this (TeX reproduction because I don’t have access to a good digital camera right now):

## Alternating-sum statistics for certain sets of integers

Jonah Saks and I have uploaded our paper “Alternating-sum statistics for certain sets of integers” to the arXiv. We show that if ${\cal F}$ is a set family in our class, then a certain alternating-sum statistic is constant. This constant equals $-1$ in the case where ${\cal F}$ is the set of all finite primitive sets. Towards the end of the paper, we generalise the notion of primitive sets to $s$-multiple sets and show that if $s\ge 2$, then the alternating-sum statistic we study is not constant, but as $n$ increases it equals $(-1)^s {n-2\choose s-1}$.

## 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.