*This article was amended as it included minor factual errors and typos.
“The word mathematics has a bad rep,” Dennis Sullivan tells me, minutes after we’d been running across a hotel hallway in Oslo, and an hour before he had to go and receive the Abel Prize. It was a crazy few days for Sullivan. Between festivities, lectures, meet-ups with young mathematicians, and a meeting with the Norwegian King, he managed to squeeze in an interview.
There’s no Nobel Prize in mathematics. The Abel Prize is probably the closest thing to it (there’s also the Fields Medal, which is awarded every four years to mathematicians under the age of 40), but Sullivan didn’t seem too flustered by this. He says he hopes his life won’t change that much after receiving the prize.
“This [prize] is random in a way, there are many good people they could give it to. I think they go around different fields [of mathematics] and try to be democratic.
“I have a good life and now it’s just that I get more attention from people. Maybe some of my more reluctant graduate students will take me more seriously now, but I hope they should do what they have to do and not be influenced by this.”
Sullivan loves simplicity — he says so himself. Every time he grapples with a problem, he tries to break it into simpler and simpler bits. In fact, all of math can be broken down into one of two things: numbers and space. It’s a concept so simple that even toddlers have some experience with it.
“If you think about all of mathematics, everything can be traced back to something related to space or something related to counting numbers. It’s very basic. Toddlers for instance aren’t really playing — they’re working. They’re figuring out how things work. First, they do space and then they do numbers. They’re starting — they’re little mathematicians.”
But as we grow up, we stop thinking of mathematics as a means to explore the universe — and it becomes something else.
“A lot of students, even bright ones, think they’re bad at math. ‘I’m very bad at math’, everyone says that but it’s ridiculous. When they were 5 or 6 years old they were fine, they were exploring space and being curious. People who stay mathematicians are just the ones that got past the difficulties and are still interested in new discoveries because it is fascinating.”
At some point, mathematics becomes a chore, a school thing, instead of something that’s related to understanding and personal growth. It becomes something for people who are good at school, and that’s just counterproductive, Sullivan says.
“The association of mathematics with things like young people that are very good at taking tests and very good in competitions is not exactly a good connection. People who are really interested and curious, they don’t have to be clever or fast, they just have to realistically analyze what they’re trying to learn and understand.”
Oftentimes, it’s not the “best students” who do well in mathematics. In fact, Sullivan recalls one instance from his undergrad years in which there were plenty of other good students, but they couldn’t solve a particular problem that had been handed to them.
“I just kept working on it for some reason, it was a very strange problem, and after two weeks I got it – and I wasn’t the best student in class, there were several really good students. It was then that I realized that if you really want to understand something, you have to keep plugging at it and you’ll eventually understand it, and that’s basically it. So then you have to find simple things that you can work with so you’re not totally confused by complexity and information.”
It’s even worse if you happen to be a part of one of the underrepresented groups in mathematics — like a woman for example. Sullivan knows a thing or two about this, as his wife, Moira Chas, is also a well-established mathematician.
“She’s a mathematician too and she thinks it’s unfair… well, she knows it’s unfair [that there aren’t more women in mathematics],” Sullivan mentions. The underrepresentation happens at all levels in mathematics. At the highest level, just one woman has been awarded the Abel Prize: Karen Uhenbeck. In the Fields Medal, a very recent ceremony celebrated only the second female laureate since the prize was instituted in 1936 (the Abel Prize started in 2002). But this is just the tip of the iceberg.
“Women are underrepresented at all levels of mathematics,” Sullivan mentions. My daughter was terrific in math when she was 6. Then, she went to school and she said ‘boys are better at math’. They had three tables in class based on how good they were at math and she was at the middle table, but I knew that she understood everything probably as good as the teacher. This is the old mentality, that girls aren’t supposed to run that fast, they’re not supposed to do well in math… It’s slowly changing but just officially.”
If we want to bring more people into mathematics, a good first step would be to help them see the simplicity in mathematics. Sullivan is very fond of the idea of simple things in mathematics and believes that understanding is more important than memorizing theorems and proofs. So teaching mathematics, therefore, should focus on understanding, and breaking everything down into its simplest components.
But it’s an uphill battle, particularly as maths has such a “bad rep” — a small branding change could help.
“Look, I think you should change the name,” he says while laughing. “Call it “The Logic of Space and Number” because those are the defining concepts, and they’re more attractive. Yeah, that’s my new name for mathematics: the Logic of Space and Number.”
Sullivan’s own work has forged space and numbers together. For over half a century, he has been a leading figure in algebraic topology, one of the fields of mathematics which, to an average person, will seem anything but simple. Topology is concerned with the properties of geometric objects that are preserved under continuous deformations. If you can take a geometric object and stretch, twist, crumple, or bend it to form another object — they’re the same, from a topological standpoint. You can pull a circle until it becomes a square, but you can’t can’t deform the surface called a sphere to the surface of a donut. So in this regard, a circle is the same as a square but a sphere is not the same as a donut. Algebraic topology takes tools from abstract algebra (think numbers) to study topology.
Sullivan dove into the field of topology head on. It was an exciting and challenging field, and it still is though it has matured somewhat, precisely thanks to the work of the likes of Sullivan. A self-described lone wolf, who constantly enjoys new challenges, Sullivan jumped from one field to another. Even now, he seems less interested in talking about his past, award-winning work, and more interested in what he’s working on now.
“I kind of like it when things are vague because you can go and make it simple and clear,” he says. “If you weren’t here, I’d be writing some little thoughts on the next page,” he says, taking out a small notebook.
In addition to algebraic topology, Sullivan has also worked in geometric topology and dynamical systems; his recent interest lies closer to the latter. The draw to this field also stems from Sullivan’s own background: he started out as a chemical engineering major, and then switched to mathematics.
“I’m trying to work on something that’s about space and understanding the model for fluid motion. In three dimensions, the equations don’t have a good theory and so we’re not sure if it’s the right way to think about things.”
The crux of the problem is numbers: really really small numbers. Mathematicians work with infinitely small numbers. In calculus, they use these infinitesimals to study the rate of change. But the problem is that infinitely small things don’t exist in nature. There’s a limit to how small things can get in terms of mass and size. So then, using things like calculus for this purpose is not exact — it’s an approximation.
In most fields of science, this approximation is good enough. But we’re already starting to see its limits. Calculus doesn’t really work well on the quantum scale, and people like Sullivan are looking for better alternatives in other fields, like fluid motion. So there needs to be some sort of discrete approach to the problem as well — it can include calculus, but it needs to have some discrete, finite element.
“Decimals can go on forever, and sometimes they don’t repeat. Like Pi. So how do you multiply numbers like this? You have all these digits and you don’t know what they are — so on computers you have to approximate and you have to cut the number somewhere and then you multiply these approximate numbers. Sometimes you round off or you have some rule. But on the other hand, every textbook uses this sort of law, and it’s just not true, it’s an idealization.”
“So when you take an idealization of fluid motion, there are a lot of possible errors. So we imagine infinitely small quantities, but in real life that’s not true because physics doesn’t make sense below thirty-three zeroes,” Sullivan says, alluding to the Planck units.
It’s still a work in progress, but it’s not the only thing that interests Sullivan. He seems dismayed by the amount of disinformation in the world. But mathematics could help in one way. Everything is clear in mathematics, and that could be used as a tool to fuel critical thinking and prime people against disinformation.
“We need more critical thinking,” Sullivan says. “People have beliefs based on intuition and emotion and partial information, and there’s a huge amount of misinformation. You can’t decide something’s true just because you want it to be true. So critical thinking would be an obvious application of math… the logic of space and number. The method is just critical thinking, simple and clear.”
“That’s one of the main features [in mathematics]: all the statements have clear meaning and the once they have a precise meaning that everyone agrees to,” he adds.
Staying interested is not just a work thing for Sullivan — it’s a way of life. In fact, it’s the thing he deems most important; it’s the key to a happy life, and it’s also the key to success, in mathematics and elsewhere.
“People should follow their interests. The real advice is to try to find something you’re interested in.” Oh, and one thing about math specifically is I don’t think it’s that important to be some kind of a math genius in a cliché sense of the world. If you’re interested and you keep trying to understand it, you’ll understand it. That brings great pleasure so you’ll want to do it again. So try to understand, don’t try to learn a lot, try to understand.”
Sullivan stayed in the hotel room, discussing math, earthquakes, and plenty of other things that weren’t included here, until the very last minute he could. “I only need one minute to get to the car,” he tells me, right before zooming downstairs and off to the ceremony.
At the Abel Prize ceremony, Sullivan made good on his promise and tried to convince people to refer to The Logic and Space and Number. He gave a simple and charming speech, much of which he previewed in the hotel room. During the speech, I couldn’t stop thinking how he would take a second to think before responding to every question — except for one.
“Do you still enjoy mathematics?” — I had asked him.
“Yeah, of course,” he said without skipping a beat.
