The foreign language of mathematics
When my husband and I first began courting, I asked to read a copy of his dissertation because I thought it would be romantic. The blue leather-bound book was fairly thin but the title, “On Bilinear Maps of Order Bounded Variation,” was making my brain hurt before I even opened it. When I did peek into the pages, I discovered symbols I’d never encountered before. This man was bilingual, I realized. He spoke a language called mathematics that I only thought I understood.
About six years later, seven months ago now, I met another mathematician here at Northeastern, assistant professor Ben Webster. At the time he was about six months into receiving a prestigious Career Award from the National Science Foundation. With $400,000 over the course of five years, Webster would be exploring “representation theory of symplectic singularities.” I read the grant summary before our meeting, just as I did again a few minutes ago, before writing this post. But it was written in that other language that I simply cannot speak. No problem, I told myself, I’ll just get Professor Webster to break it down for me when I get to his office.
But when I asked him the main goal of his research he recalled a clip from a Nova documentary about proving Fermat’s last theorem in which some of the most brilliant mathematicians in the world attempt to explain something called modular functions:
Though modular functions have nothing to do with Webster’s research, he said, “I feel like the first three guys in that montage.” It’s hard to put into plain English the abstract work he’s devoted his life to. Instead of laughing at me though, like the second guy in the clip, Webster kindly pulled one bit of his work out of the mix, the part that he said is most amenable to cocktail party conversation, and tried to explain that.
He gave me an hour-long course in knot theory, after which I felt vaguely more educated in this other language. But it was really more like spending an hour learning how to pronounce the word “mañana” in Spanish and walking away still saying “banana.” When I got back to my office I think I sat in front of the computer staring at a blank screen for a while and then decided to do something else on my list. I wrote “Webster” in my planner under “To Write” and moved on. For the next five weeks, I copied Webster’s name to my next “To Write” list. I was having a hard time figuring out how to write about something that I didn’t have the vocabulary for. I took a break from copying the name because I knew it wasn’t happening, but, determined, I started copying it again a few weeks later. It has now been repeated 13 times in my planner: “To Write: Webster.”
And so, here we are.
I’ve decided that in order to write this post I’m just going to have to get over the fact that I don’t speak the language and tell you what Webster told me, because even though I can’t speak the language, I can understand that it is beautiful and, well, cool.
Early in our conversation, he said that “math is really about looking at something and saying, ‘well, what sort of structure here is important and what things can I just forget?’” That is, mathematicians only care about the most fundamental elements of a problem. If you’re looking at the mathematics of something being shot out of a cannon, he said, it really doesn’t matter what that something is. Only its trajectory, and the force with which it is launched, for example. “So when I say knot theory,” Webster continued, “I mean thinking about closed loops of string, or closed loops of anything — that’s one of the things you’d want to forget. The important thing is what shape does it have? How is it tangled up on itself?”
Knots. Yep. That’s what we spent the first hour of our two hour conversation talking about. I never knew there was so much math in the pesky tangles that hurt my finger nails to resolve but it turns out knots have a long and rich history. In the 19th century, Lord William Thomson Kelvin got the idea that the different elements might be tiny strings tied up in knots, and that the properties of the elements were somehow dependent on the properties of knots. While this turned out not to be Lord Kelvin’s most on-the-mark insight, it did get people thinking about knots in a more systematic way. The first question they asked themselves, Webster told me, was how to distinguish between different knots. What does that mean exactly? Let’s say you have a rubber band:
Now say you take your rubber band and twist it:
It looks different, but it’s still the same unknot. You can easily see how you’d untwist it and get back to the first picture. This is easy to see because the unknot is so simple. But say you have a really complicated knot, like this one:
If you had some extra twists and turns (not interlacements though, to be clear), it might easily be confused for an entirely different knot. That was the case with one particular knot, which was listed twice on the proverbial Table of Knots for ninety years before someone was able to prove that they were actually the same exact thing.
Prove. An important word in the mathematical vocabulary. Mathematicians aren’t satisfied that something is true until they have proof. But it’s very difficult to prove that two complicated knots are not the same, because you can never be certain that there isn’t a really weird maneuver that you just didn’t see. “It’s a very hard concept to realize you can’t check infinitely many things by just doing them one by one,” said Webster.
And this is where his own work fits into the puzzle. “You have two knots,” said Webster, “and let’s say you’re pretty convinced that they’re different. But you need to check. So what do you do? Well, you look for some way of extracting information from the knot that you actually can compare.”
As it happens, there’s a lot of information you can extract from a knot, you just need the right invariant to find it. An invariant is something that doesn’t vary in an individual knot, no matter how masked it is behind unnecessary twists and turns. One example is the unknotting number, or the minimum number of crossings that must be reversed in order to get back to the unknot I mentioned above. If the unknotting number is seven then you will never be able to unknot it with fewer that seven maneuvers. You could also perhaps do it with eight, nine, or 100 maneuvers, but they are all unnecessary. Only the seven matter — that never varies.
How do you find the unknotting number, or any other invariant of a knot? You send it to a factory.
Okay, not really, but that’s the analogy Webster used for something that he really couldn’t explain to me using our common language. I gather that it’s basically a collection of rules about knots, and when you put a knot into the factory it’s sorted based on its own properties. I know this doesn’t help you much, that’s because I don’t understand it myself. But this is what Webster does when it comes to knots. He creates new factories for outputting new interesting things about knots.
But, as I mentioned earlier, this is only the cocktail-party-friendly bit of his work. And it turns out it’s not even the biggest part. “If you went and looked at my grant proposal you’d see ‘section one: some other stuff; section two: some stuff about knots; section three: a bunch of other stuff’,” he said. “The hard work is going on somewhere else and then you say, ‘oh, actually, this has some interesting consequence where you can do this thing with knots.’”
The reason he talks about knots he says, is because he can draw pictures of them. When it comes to the other stuff, he said, “what can I say?” …kind of like the first few guys in the Nova clip.
Okay, so…what’s the point of it all? That was my final question to professor Webster after this two hour conversation in “math-lish.” Maybe it’s an unfair question, but he had a beautiful answer. In addition to the implications in bacterial genetics, where the organisms’ genomes are examples of actual biological knots, and in quantum mechanics, where the path of particles in a two-dimensional space have similar properties as knots, Webster said there is also some very significant value in exploring knots simply for the sake of the exploration:
“Basic science exactly means that you don’t do it with a particular application in mind. I think at this point basic science has an exceptionally good track record of turning out to be useful even though you didn’t know beforehand how it was going to be. And somehow I think this is a really important point and one that’s hard to penetrate, that you can’t just make progress in science by waiting until you need this stuff and saying, ‘okay, we’ll figure it out when we get there.’”
Einstein’s theory of general relativity is necessary for GPS technology, webster said. Curved geometry, worked out hundreds of years before there was any idea of its value, makes it possible for airplanes to travel around the globe. Work done 300 years ago makes it possible for credit cards to travel around the Internet safely.
So even if most of us can’t speak the language, there’s no doubt that mathematics — the most basic research out there — is itself the point.