*This article has been corrected for minor factual errors and typos.
“words Math Dennis Sullivan told me minutes after he was running through the corridors of his Oslo hotel, an hour before he was due to receive the Abel Prize. It’s been a crazy few days for Sullivan. Between festivals, lectures, meetings with young mathematicians, and meetings with the King of Norway, he managed to get an interview.
There is no Nobel Prize in mathematics. The Abel Prize is probably the closest thing to it (he also has the Fields Medal, which he awards every four years to mathematicians under 40), but Mr. Sullivan didn’t seem too upset by this. He says he hopes his life doesn’t change much after his win.
“this [prize] It’s random in a way, but there are plenty of good people out there who can give it away.I think you are traveling in many different fields. [of mathematics] And try to be democratic.
“I’m living a good life, but now I’m getting more attention from people. Some of the reluctant graduate students may take me more seriously now.” But I hope they do what they have to do and don’t let this affect them.”
Sullivan loves simplicity. He says so himself. Every time he tackles a problem he tries to break it down into simpler ones. In fact, all mathematics can be broken down into two parts: numbers and spaces. It’s such a simple concept that even toddlers have some experience with it.
“When you think about all mathematics, everything can be traced back to something related to space, or to something related to counting numbers. They’re working instead of being in. They understand how things work.First they do spaces, then they do numbers.They’re starting — they’re little mathematicians. is.”
But as we grow older, we stop thinking of mathematics as a way to explore the universe and mathematics becomes something else.
“Many students, even good ones, think they are bad at math. At five or six, they were feisty, space-exploring, and curious. Those who remain mathematicians are those who have overcome difficulties, and new discoveries are fascinating, so I am still interested. “
At some point, mathematics becomes a chore, a school thing, rather than something related to comprehension and personal growth. According to Sullivan, it doesn’t matter to those who do well in school, and it’s counterproductive.
“The association between mathematics and things like young people who are very good at taking tests and who are very good at competitions is not always a good relationship. You don’t have to be fast, you just have to be realistic about what you want to learn and what you’re trying to understand.”
In many cases, it’s not the “good students” who are good at math. In fact, Sullivan recalls one instance during his undergraduate years when there were many other brilliant students, but he was unable to solve the particular problem given to them.
“I kept working on it for some reason. It was a very strange problem. Two weeks later I got it. I wasn’t the best student in my class. I realized that if I wanted to understand, I had to keep understanding it, and eventually I had to understand it, so to avoid being completely overwhelmed by complexity and information, I wanted to find something simple that I could work with. I need to find it.”
If you’re one of the underrepresented groups in mathematics, like women for example, it’s even worse. Sullivan knows a little bit about this, as his wife, Moira Chas, is also an established mathematician.
“She’s also a mathematician and thinks it’s unfair… well, she knows it’s unfair. [that there aren’t more women in mathematics]’ said Sullivan. Underrepresentation occurs at all levels of mathematics. Karen Uchenbeck is the only woman to have won an Abel Prize at the highest level. The Fields Medal only celebrated her second female laureate in the most recent ceremony since the award was instituted in 1936 (the Abel Prize began her in 2002). But this is just the tip of the iceberg.
“Women are underrepresented at all levels of mathematics,” said Sullivan. My daughter was 6 when she was very good at math. Then she went to school and she said, “Boys are good at math.” There were three tables in the math class, and she was in the middle table. It’s an old-fashioned way of thinking that girls don’t have to run that fast and they don’t do well in math…it’s slowly changing, but only officially. “
If you want to get more people into mathematics, a good first step is to help them understand the simplicity of mathematics. We believe that understanding is more important than memorizing proofs and proofs. Therefore, teaching mathematics should focus on understanding and breaking everything down into its simplest components.
But this is an uphill battle. Mathematics, in particular, has such a “bad reputation” that a small change in branding might solve it.
“Look, I think we should change the name,” he said with a laugh. . Well, that’s my new name for mathematics: the logic of space and numbers.
Sullivan’s own work forged space and number together. For more than half a century, he has been a leading authority on algebraic topology, one of his areas of mathematics that seems unsimplistic to the average person. Topology is concerned with the properties of geometric objects that are retained under continuous deformation. If a geometric body can be stretched, twisted, crumpled, or bent to form another body, they are the same from a topological point of view. You can pull a circle until it becomes a square, but you cannot transform a surface called a sphere into a donut surface. 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 (thinking numbers) to study topology.
Sullivan dived headlong into the field of topology. It’s an exciting and challenging field that has matured to some extent thanks to the work of people like Sullivan, and still does. Jumped into another field. 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 like things to be vague because it can be straightforward,” he says. “If you weren’t here, I’d write some thoughts on the next page,” he said, pulling out a small notebook.
In addition to algebraic topology, Sullivan has also worked on geometric topology and dynamical systems. His recent interest is closer to the latter. His fascination with the field also stems from Sullivan’s own background. He majored in chemical engineering and then switched to mathematics.
“I’m working on something about space and trying to understand a model of fluid motion. In three dimensions, I don’t know if it’s the right way to think of things because there’s no good theory for the equations.”
The heart of the matter is numbers. A really small number. Mathematicians deal with infinitely small numbers. Calculus uses these fractions to examine rates of change. But the problem is that infinitely small things do not exist in nature. There are limits to what can be made small in terms of mass and size. So using something like calculus for this purpose isn’t exact — it’s an approximation.
This approximation is sufficient for most scientific disciplines. But the limits are already starting to appear. Calculus doesn’t really work well on the quantum scale and people like Sullivan are looking for better alternatives in other areas such as fluid motion. . This can include calculus, but requires discrete finite elements.
“A decimal number can go on forever, and it may never repeat. Like a pie. So how do you multiply numbers like this? We have all these numbers, but what are they?” We don’t know, so the computer has to approximate it, rounding the numbers down somewhere, and then multiplying these approximate numbers, possibly with rounding or some sort of rule. But on the other hand, all textbooks use laws of this kind, which are idealized rather than true.”
“So idealizing fluid motion can lead to a lot of errors. So we imagine infinitely small quantities, but physics makes no sense below 33 zero, so in practice It’s not for ,” Sullivan says, alluding to the Planck unit.
It’s still a work in progress, but it’s not the only thing Sullivan is interested in. He seems disappointed by the amount of disinformation in the world. But mathematics is useful in some ways. Everything is clear in mathematics, and it promotes critical thinking and can be used as a tool to protect people from disinformation.
“We need to think more critically,” Sullivan says. “People base their beliefs on intuition, emotion, and partial information, and there is an enormous amount of misinformation. So critical thinking is clearly an application of mathematics, the logic of space and numbers.The method is critical thinking, plain and simple.”
“This is one of our main features. [in mathematics]: Every statement has a clear meaning, a precise meaning that everyone agrees on once,” he adds.
Staying interested isn’t just a job for Sullivan, it’s a way of life. In fact, it’s what he considers most important. It’s the key to a happy life, and it’s also the key to success in math and other areas.
“People should follow their own interests. The real advice is to try to find something that interests you.” I don’t think being a genius is all that important. You will want to do it again because it brings so much joy. So try to understand, don’t try to learn much, try to understand. “
Sullivan stayed in his hotel room, discussing math, earthquakes, and many other things not included here until the last minute. “It only takes a minute to get to the car,” he told me, just before zooming downstairs and heading to the ceremony.
At the Abel Awards ceremony, Sullivan lived up to his promise and tried to persuade people to mention logic and space and numbers. He gave a simple and engaging speech, much of which he previewed in his hotel room. During his speech, I couldn’t help but think he would take a second to think before answering every question — except one.
“Do you still enjoy math?” — I asked him.
“Yes, of course,” he said without skipping a second.