# The Logic of Space and Number: Dennis Sullivan’s quest for simplicity

# BLOG: Heidelberg Laureate Forum

You’re always a bit nervous when talking to a laureate, but as I was running with Dennis Sullivan in the hallway of a hotel in Oslo, my worries quickly vanished.

In about two hours’ time, Sullivan was about to be awarded the Abel Prize, one of the highest distinctions in mathematics.

“It’s the most I’ve run in ten years,” he tells me, but you can tell. We’re sprinting and dashing, and the hotel is surprisingly large. Eventually, we reach the room where the interview is supposed to take place. A car is about to pick the laureate up in 60 minutes. I ask if we should stop the interview after 50 minutes, so he has time to prepare.

“What do you mean? I only need one minute to get downstairs and I’m ready. We’ll stop after 59 minutes.”

Sullivan’s love for mathematics comes across clearly in everything he does. “Whoever is serious about math will get time with Dennis. I may not get time as his wife,” says Moira Chas, Sullivan’s wife, herself an accomplished mathematician at Stony Brook University.

Much of Sullivan’s acclaimed work comes from the field of topology, a field which is hard to comprehend, even for the initiated. Topology, which is concerned with the properties of a geometric object that are preserved under continuous deformations, is sometimes called “rubber-sheet geometry”. In topology, two objects are considered equivalent if they can be continuously deformed into one through twisting, stretching, and shrinking, without tearing apart or gluing their parts. You can distort a circle into a square, but you can never make it into a donut that has a hole at its center. Among his many contributions to topology, Sullivan (along with Moira Chas) started the field of string topology, which studies the homology of free loop spaces (the homology being a way of associating a sequence of algebraic objects with other mathematical objects such as topological spaces). Sullivan also worked on classifying high-dimensional manifolds with surgery theory.

Dennis Sullivan (Credit: Peter Badge)

But Sullivan’s definition of topology is discouragingly simple:

“Topology is the structure of certain properties of space,” he says, giggling to himself. In fact, his description of mathematics is also very simplistic in nature.

“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,” he says. “In fact, I like to talk about the logic of space and the logic of numbers, that’s my new name for mathematics: the Logic of Space and Number.”

Sullivan is in a constant search for simplicity, and that’s what he wants other people to think about too. Math is not really about being the smartest in the room, he says, nor about being able to calculate much or quickly. In fact, thinking about calculating fast is “sort of silly,” he says. Mathematics is about understanding things; and if you want to understand things, the best way to go at it is to break things into simple components and then work on them until you understand them better.

“When I was a math major,” Sullivan recalls, “we had a homework assignment that had one problem on it that no one could do so I just kept working on it for some reason. It was a very strange problem, and after two weeks I got it. But 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.”

He also thinks about redefining math, especially for the public eye. Math doesn’t need to be overly complex, it can be broken down into simple things. But math has an undeservedly “bad rep,” the laureate notes:

“The word mathematics has a bad rep. ‘I’m very bad at math.’ Everyone says that but that’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.”

“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.”

A lot of people say “I’m bad at math”, Sullivan says, or “math is my worst subject” – and it all boils down to how we approach things. Everything in math must be well defined, he says, before promptly delving into a definition of what a pillow is and using the pillows in the room as examples. Mathematicians sometimes seem to be slightly detached from world practicalities, but this doesn’t seem to be the case for Sullivan.

Changing the way people see math is anything but simple. But it’s worth a shot.

“Look, I think you should change the name,” Sullivan says while laughing. “Call it The Logic of Space and Number because those are the defining concepts, and they’re more attractive. I met the King [of Norway] today, he told me ‘I wish someone had told me that.'”

I couldn’t help but ask Sullivan about women in mathematics. Out of 25 Abel Prize laureates, there has only been one woman: Karen Uhlenbeck. This may be changing somewhat, but change is slow and women are still underrepresented.

“They’re underrepresented at all levels of mathematics,” Sullivan prompts. He recalls how his daughter was terrific at math when she was around 6 years old. “But then she went to school and she said ‘boys are better than girls in mathematics.’ They had 3 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.”

The problem is all the more significant for Sullivan because of Moira Chas:

“My wife is a mathematician too and she thinks it’s unfair… in fact, she knows it’s unfair.”

“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.”

I pushed for a conclusion: “Unofficially, it’s still problematic?”, I asked. “Exactly,” Sullivan said.

The discussion winds back and forth, ranging from earthquakes and vagueness to how we need more critical thinking to fight disinformation – and how mathematics could help with that. He was right, every additional minute is well spent and serves a purpose. I couldn’t help but ask a question to which I already knew the answer. I also couldn’t help but feel that despite a lifetime of spectacular contributions, Dennis Sullivan still has a lot to give.

“Do you still enjoy mathematics?” I asked.

“Yeah, of course,” Sullivan responded.

“Are you working on anything right now?”

“If you weren’t here, I’d be writing some little thoughts on [pulls notebook] the next page.”

Dennis Sullivan was awarded the Abel Prize for “his groundbreaking contributions to topology in its broadest sense, and in particular its algebraic, geometric and dynamical aspects.”

The title “The logic of space and numbers” already shows that it is also about didactic questions, because pure mathematicians do not care about which category their field of work belongs to, even if there are efforts to standardize mathematics.

Didactically, the question arises as to how young people should be introduced to mathematical questions. Today, everyone learns to deal with numbers and geometric figures. It seems to me that “The logic of space and numbers” offers a plan that leads from school geometry to thinking in mathematical spaces and that leads from the numbers of school mathematics to the higher number concepts of today’s mathematicians.

An important aspect of mathematical work is also covered in this article: mathematics requires hard mental work. Even for a genius. I think many who fail in mathematics fail due to a lack of effort.

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