“Topology is the study of whether there are holes in a space,” says mathematician Dennis Sullivan of the City University of New York Graduate Center. He just won one of the most prestigious awards in mathematics, the Abel Prize, which is handed out annually by the Norwegian Academy of Science and Letters on behalf of the country’s Ministry of Education and Research. Sullivan was recognized for his work on topology, which investigates how you can bend, stretch or twist a shape without changing its basic nature. “The cliché is that a doughnut and a coffee cup are the same to a topologist, because there’s a hole in the doughnut, and there’s also a hole in the coffee cup,” he says.
The prize was announced on March 23. Sullivan spoke to Scientific American about his recent win, his career trajectory and how geometry and string theory intersect.
[An edited transcript of the interview follows.]
Were you always a “math person”?
I was sort of a bad student in high school, in the sense that I didn’t study. I think it’s because I was actually nearsighted. I didn’t know you were supposed to be able to see the blackboard. So when I told my teacher that I was applying to Rice University, which is a very good school right in Houston, Tex., where I lived, she said, “You shouldn’t apply there. You’re not a good enough student.”
How did you fall in love with math?
I went to Rice as a chemical engineer. We took a lot of science. I didn’t know there were such things as mathematicians. And then, in my second-year math course, the professor, [the late] Guy Johnson, was showing us a wonderful theorem in math that went way beyond the intellectual quality of things like equations and calculations, which is what I thought math was. And I was just very struck by it.
Once you understand that deeper real mathematics, it’s sort of amazingly clear what things mean. There’s no ambiguity, whereas almost everything else is much woolier and murkier, compared with mathematics. There’s some great comfort in that. That’s what attracted me.
What does that deeper level look like?
Mathematics is built on two fundamental concepts: counting and space. We live in a three-dimensional space, and then physical processes such as wind, the movement of currents, weather, fluids, everything that happens in science takes place in that space. So a lot of mathematics, such as solid geometry, was just built up from studying space. But it’s a deep thing; it’s not just something you learn like reading a cookbook. And it’s very beautiful.
How does topology work?
Albert Einstein has his famous theory describing how the force of gravity is a property of what’s called the “curvature” of space. As mass and energy move around, the space curves and changes. But its topological properties don’t change, even though its geometry does. So imagine you have a flat surface, and something bubbles up or some wind blows across it. It’ll change the shape of something like that, but it’ll still be the same topological surface.
Over the course of your career, you have taken a multidisciplinary approach to mathematics. From your experience, what is the relationship between math and fields of science such as physics?
Most mathematicians—myself included, even though I’ve been trying for decades—don’t really understand physics at all. In math, if you make a statement, it’s either unknown or it’s known to be true or false. We have this way of understanding where we have definitions of the concepts we’re using. And physicists have different criteria. They have physical experiments, and when they want to understand how to predict what will happen in the future, they’ll describe what they observe. So it’s a different game. But it’s amazing because the two games are very closely related.
You helped develop the field of string topology, which combines quantum mechanics and classical physics with topology. How did that start?
Well, it’s not unreasonable to ask, “How are two things that I don’t know about related?” I often do that. And I try to find connections between fields that I don’t understand. About 20 years ago my collaborator (who’s now my wife, Moira Chas) and I found a theory called string topology.
The physicists had been talking about string theory, which was kind of a revolution from the previous paradigm they’d had, which was that everything is made up of little points, such as points of mass or points of electrons or whatever. And when they come close, they push each other apart or attract each other, depending on the situation. But there were some really serious problems when physicists tried to make that picture include all of the forces of nature, such as, say, gravity.
But then they found that if they just imagined that the little points were actually tiny strings, like tiny little rubber bands or vibrating wires, these strings might hit each other or come together and touch at a point, and you would get something similar. And when they did the calculations for that theory, all the terms that had infinities before didn’t have infinities anymore. So this is the best candidate for being able to unite all the basic forces of nature into one physical theory that hopefully makes sense.
Purely topologically, without any physics—those operations are basic algebraic operations. So string topology was just a very simplified mathematical version of what they were talking about, which we could actually prove mathematically and then discuss.
Did you and your wife meet through mathematics?
Yes. She was also a mathematician, and I was introduced to her in front of the elevator going up to the math department by another mathematician. She likes to joke that on our first date, instead of getting a box of chocolates, she got a question about curves on a surface.
How did it feel to receive the Abel Prize?
It’s gratifying because I work on math all the time. Mathematicians don’t get the Nobel Prize, and this is something that’s supposed to be analogous to that in mathematics. It’s about, you know, the human effect of people thinking that you’re good at something. I mean, everyone wants to be taken seriously, right?
And also, I think it’ll make the graduate students listen more carefully to what I say.
Are you working on anything right now?
In fact, I am working on something. [More than] 250 years ago [Swiss mathematician Leonhard] Euler came up with this equation that uses calculus for how fluids move, such as air or water, a flowing river or ocean. There’s a whole set of equations, and it’s a very good mathematical theory. But then, in the early 1990s, I found out that, in three dimensions, we don’t know that these beautiful equations actually have solutions. So I started working on that. I wondered, “Why is it so hard? I want to try to study this question by not thinking of it in terms of the calculus equations, which involve this infinite assumption at the beginning. I want to produce a finite algorithm that will give me finite predictions about finite aspects to this very complicated, infinite situation.”
Anyway, it’s nice to get the Abel Prize so I don’t have to solve this!