Math papers are an odd beast. They out of necessity contain a lot of technical details and can appear quite abstruse. They are filled with jargon, formulæ, and other bits that can easily scare the non-specialist reader.
At the same time, there is a lot of beauty, elegance, and interesting ideas floating about. I hope, over the next few blog posts, to go over some of my published papers and to express their content in a way that will appeal to the interested non-specialist reader (Hi John!).
A happy accident
My first paper came out almost accidentally. I was working on a part of my thesis, mucking around with some computations that I was trying to understand, when I happened across a curious connection between my thesis problem and some mathematical work studied by Percy MacMahon in 1921. I couldn’t explain it, and so it was recommended that I appeal to George Andrews who was far more experienced in this subject. After a quick “I find your question extremely interesting” response from him, a few days later we had put together a short little paper which proved to be invaluable in my thesis.
So let’s go over this paper. It’s not a particularly long one, and it has a couple important ideas that crop up all throughout my work, which we will introduce and discuss here. The overview of the paper is that we prove some surprising connections between some mathematical work of the wonderfully bearded Pafnuty Chebyshev and some other work of Percy MacMahon (whose moustache is pretty spectacular as well). This ends up having some pleasant consequences for my thesis, which will be discussed at a later point.
But what does that all mean? Well, to begin with,
Let’s talk about Generating Functions
Let’s start by assuming you have a collection of numbers that you find interesting. We could start with 1, 2, 3, 5, 7, 11, say. You can start with you favourite set of numbers (I recommend looking around on the Online Encyclopedia of Integer Sequences, for example), your list can be as long—even infinite!—as you like.
Briefly speaking, what we are interested in is a way to talk about all of these numbers together at once, while respecting their order—which, after all, is part of their structure. We want to be able to then manipulate the whole list in ways that illuminate and provide information about the nature of the underlying numbers, or that provide information about whatever our source of these numbers was.
Let’s look at an example. The Fibonacci numbers are the following:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
That is, we start with the numbers 0 and 1, and then the next number is the sum of the previous two numbers. So for example, we have that 5 + 8 = 13, 5 and 8 being the two numbers before 13.
[There is a fun aside about Fibonacci numbers: If you take the ratio of successive Fibonacci numbers, say 3/2, 5/3, …, 144/89, …, then these get closer and closer to the “golden ratio”, which is approximately 1.619. By pure coincidence, the ratio of a mile to a kilometre is very close to 1.609, which is quite close to 1.619. So if you want to convert from miles to kilometres, you can just choose the next Fibonacci number to get a good approximation. For example, since the next Fibonacci number to 8 is 13, it follows that 8 miles is pretty close to 13 kilometres. In practice, you only need to memorize the first few Fibonacci numbers to get a pretty good guess for this.]
So what is the “generating function” for the Fibonacci numbers? This is given by
0x0 + 1x1 + 1x2 + 2x3 + 3x4 + 5x5 + 8x6 + 13x7 + 21x8 + …
where you can see that we’ve put the Fibonacci numbers (in red) as the numbers in front of powers of a variable x. To use an analogy due to Herbert Wilf, we can think of a generating function as hanging the Fibonacci numbers on a clothesline.
Figure 1: A clothesline
Anyhow, you could ask and answer a lot of questions about these numbers using this clothesline very quickly. For example, you can show that the 7-th Fibonacci number is the closest integer to
Figure 2: Math!
(Try it out! It’s kind of cool!) This of course allows you to compute other Fibonacci numbers without having to compute all of the ones before it.
The moral of the story is that generating functions are tools that we can use in mathematics to help us understand different quantities by studying their generating functions—and hence better understand whatever the numbers that make them up are.
What about Modular Forms?
So let’s look at a few particular examples of generating functions. Consider a regular lattice of points in a line, a plane, or any higher dimensional space as in the following image.
Figure 3: Colourful telephone poles
This is a one-dimensional lattice; you can think of it like the integers along a real line, or evenly spaced telephone poles along the road. I’ve marked one special one, which I call the “origin”, in red. What we want to do is count the number of points a fixed distance from this origin. For technical reasons we will see shortly, we count them not by their distance d, but by the square of the distance, d2. For example, the blue dots are of distance 2 from the red dot, but the squared distance is 22 = 4.
We will now write down a generating function so that the coefficient in front of xd2 is the number of points whose squared distance from the origin is d2. For the case above, we obtain the expression
1x0 + 2x1 + 0x2 + 0x3 + 2x4 + 0x5 + 0x6 +0x7 + 0x8 + 2x9 + …
since there is exactly one point whose distance is zero from the red dot (the red dot itself), and then two points whose distance is one, two points whose distance is two (so the distance squared is four), two points whose distance is three (and hence distance squared is nine), and so forth. There are of course no points whose distance squared is 2, or 3, and so on. This expression is called the theta function of the lattice.
Our next example is the following lattice.
Figure 4: Dotty dotty dots
This new example is a two-dimensional square lattice. The colours are chosen so that red is again the origin, and then each other colour represents all the points a fixed distance from the (red) origin. If we count them up, we end up with
1x0 + 4x1 + 4x2 + 4x4 + 8x5 + 4x8 + …
As we discussed above, the powers of x are the squared distance, and the coefficient is the count. So there are 8 orange dots, all of whose squared distance is 5 from the red dot (Remember your Pythagorean theorem—a2 + b2 = c2!).
Figure 5: 1 + 4 = 5, I promise
In particular, this is why we use the squared distance instead of the distance—this will always be a whole number.
Having written these two generating functions down, we get to our point. If you start with a lattice (as always, there are a few technical conditions that I’m eliding over), then when you build a theta function as we did above, you will always get what we call a modular form. In particular, both
1x0 + 2x1 + 2x4 + 2x9 + 2x16 + 2x25 + …
1x0 + 4x1 + 4x2 + 4x4 + 8x5 + 4x8 + …
are modular forms.
So what are modular forms? Well, I’m the first to admit that I’m not an expert in their study. To me, they are simply a particular kind of generating functions that interest other mathematicians. More glibly, they are generating functions that have coefficients that are likely to show up on the OEIS. Of course, these theta functions are not the only things that are modular forms; many other generating functions—although a vanishingly small proportion of all generating functions—are modular (The theta functions are however the only ones for which we have any real sense as to what their modularity means are these theta functions; everything else is a mysterious and lovely accident).
In the end, the study of modular forms is a rich and classical subject, that unites many seemingly disparate areas of mathematics. It ties together geometry and number theory (as above), and possibly even also analysis (studying the generating function itself) and even of late, some branches of theoretical physics. They were, in fact, used in 1994 by Sir Andrew Wiles to prove the famous Fermat’s Last Theorem.
A = B
A surprising amount of mathematics comes down to statements of the form A = B. While this seems not so interesting in some cases (1 + 1 = 2), in its most exciting versions we are showing that two seemingly different objects or quantities are equal to each other, or possibly that there are two ways of writing the same thing.
We now get to the main point of our paper. At its heart, it’s an A = B statement, namely that there are two ways of writing a certain generating function (of Chebyshev polynomials, the aforementioned work of Pafnuty Chebyshev). What do we mean by that in this case? Well, look at the following image.
Figure 6: Up Up Down Down Left Right…
If we look at this generating function from one side (so to speak), we get a generating function for Chebyshev polynomials. If we look at it from the other side, we get one for some functions studied by MacMahon. This in and of itself is a nice and non-obvious statement, and was not known by MacMahon. But the real point of the paper was to examine certain consequences of this fact.
The Chebyshev polynomials are given by the rows in the diagram above, and MacMahon’s functions are given by the columns. Due to the fact that they describe the same collection of data—together with what we know about Chebyshev’s polynomials—we can deduce that the columns are modular forms, a fact which was also not known by MacMahon!
To re-iterate, we find a novel way of writing a generating function for Chebyshev polynomials, which we then leverage to show that another type of function is a modular form.
What are those functions, you might ask? They themselves are generating functions, specifically those that count a certain geometric type of object. However, as they are the subject of two other papers that I have written, I’ll discuss them in more detail in a later blog post.