The foreign language of mathematics

Photo via Thinkstock.

Photo via Thinkstock.

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:

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NOTE: While this doesn’t quite fit the Eng­lish def­i­n­i­tion of a knot (“an inter­lace­ment of the parts of one or more flex­ible bodies forming a lump or knob (as for fas­tening or tying together),” according to Merriam-​​Webster), we’re in the land of Math­e­matics here, where you’ll recall that knots are simply closed loops. In this lan­guage this non-​​interlacing closed loop is called the “unknot.”

 

Now say you take your rubber band and twist it:

104777935It 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:

10_147If 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.