## Hot off the presses!

So let’s talk a little more in detail about the guts of the previous paper I just discussed. Just as a recap, the idea of that paper came from the fact that in MacMahon’s paper there were 8 variations discussed, and I had previously shown that two of them satisfied Nice PropertiesTM. I was curious about the rest of them, and so I went digging.

To understand what this paper is about, we need to talk a little bit more about theta functions which were mentioned in one sense in the first post in this series. However, in this case we are looking them a little differently. In a sense, all of the work in this paper comes from obtaining a better understanding of the ins-and-outs and the quirks of theta functions[1].

We’ve probably all burned our hands when touching a hot piece of metal; a pan, an oven rack, whatever. If only there were some way to predict how hot your fire poker would be…

It turns out that mathematicians long(ish) ago came up with a way to describe how heat flows in materials. They creatively called the resulting equation the “Heat Equation”; if your figure out the solution to it, then you can describe how heat propagates in whatever piece of metal you’re holding. (Of course, heat isn’t too too interesting as it propagates; it basically does what you think it does, and disperses evenly throughout whatever it’s in. If you start with an uneven distribution of heat, you end up with something smoother. But that’s not important right now)

Figure 1: Sort of like this! Bumpy to the left, smooth to the right.
These are the ravages of time, but the opposite way that people face them.

So what does this have to do with theta functions? Well, the theta function that was described earlier in relation to counting how many dots are are a fixed distance from a starting point:

1x0 + 2x1 + 0x2 + 0x32x4 + 0x5 + 0x6 +0x7 + 0x8 +  2x9 + …

Figure 2: This one

is in fact a solution to the heat equation[2].

This is one of those functions that shows up in many places in mathematics, two of which we see here. On the one hand, we have a function that describes the propagation of heat. On the other hand, we have something that is related to counting points in a lattice—magically, these are in fact the same thing. Math!

The upshot of this sort of connection is that you can use ideas from one domain to help study the other domain. So in particular, the fact that the theta function describes the propagation of heat has some strong implications for the coefficients that describe the counting of lattice points.

### The hard part

So the easy part is the what I just described, where we are able to connect the counting functions of stacks of boxes:

Figure 3: It’s so minimalistically Scandinavian…

to theta functions. This is done much in the same way that I learned from George Andrews (something that is, once you learn it, a pretty standard application of the so-called Jacobi Triple Product, a subject that I intend to write about in the future).

The idea is to take the parts that count boxes, realize you can write them as a big product that is secretly a collection of modified theta functions. After all of this, you end up with the following beautiful expression:

$\displaystyle \sum_{k=0}^\infty (-1)^k A_{S, n, k}(q) x^{2k} = \prod_{j=1}^s\frac{\vartheta_{\alpha(n, \ell_j)}(q^n, -\zeta)}{\vartheta_{\alpha(n, \ell_j)}(q^n)}$

Figure 4: Ta-da!

Well, trust me that it’s beautiful.

But now we want to move on to understanding this elusive property, modularity. The parts are all there; theta functions tend to beget modular forms, but these are a little twisted so it’s not quite so clear. While trying to understand if and how these functions were modular—and determining how that is the case in the end—I ended up finding out something far more interesting.

In the end, I was studying a whole bunch of functions (the 8 families that MacMahon studied, as well as some other ones that come up pretty naturally) which happen to have my nice property, modularity. While interesting, this is a pretty broad property. However, it turned out that there is a special way in which these are modular—namely, all of the building blocks are the same! It’s sort of like if you saw a bunch of different sculptures from far away, and then when you got closer you realized that they were made out of Lego—that they all consisted of the same pieces used over and again to produce something magnificent.

Amazingly, this all follows from paying careful attention to the formula shown above in Figure 4. Even more amazing, this technique applies to all of the box-counting functions that MacMahon investigated, as well as the infinitely many more suggested by his work. Unlike my first paper which used special techniques, the ideas in this paper worked far more broadly.

In some sense, this paper is my favourite of all that I have written. It started with a natural question that had arisen in my work, and it bounced around in my head for a few years and through a few separate institutions. When I finally came to tackling it, not only did it answer in the affirmative the question I’d been asking, but it told an even richer tale.

However, as I mentioned in the previous post, there is still one outstanding part of this puzzle. All of the above is lovely and true—but where is the geometry? In fact, the nicer answer discovered in this paper teases me even more, since the fact that the building blocks are all the same suggests to me some deeper underlying regularity. But unfortunately, that is a question for another time, and another (as yet unwritten) paper.

[1] Not the least of which is how mathematicians write them. A normal Greek lowercase letter theta, as anyone clearly knows, is written

$\theta$

However, mathematicians also use a variant cursive font (which we pronounce—I kid you not—“vartheta”) which is written as

$\vartheta$

The more you know! Basically, mathematicians care far more than most anyone about fonts. We’re a bit weird.

[2] Well, sort of. There is a tweak that you have to put in place to account for variation over space and time.