Connecting Different Realms with Representation Theory

Two completely disparate fields will show the same type of mathematical structure, hinting at a connection. Yi Sun of the Simons Society of Fellows uses representation theory to explore the mystery of these deep symmetries

Yi Sun

Is math invented or discovered? Mathematicians tend to be divided in their answers to this longstanding philosophical mystery. Yi Sun of Columbia University admits that he’s spent time on both sides. “I used to think it was definitely invented,” he says. “But there’s this strange phenomenon in mathematics where the same type of structures — like a sequence of five highly specific numbers — will appear in two unrelated fields. A priori, you have a lot of trouble giving any type of explanation for these coincidences — but it doesn’t feel like mathematicians invented them.”

Mysterious symmetries are at the core of Sun’s work in representation theory, a field of mathematics concerned with identifying deep structures in abstract algebra. “It was originally used by physicists in the 1800s,” Sun says, “but what I find very interesting is that the situations it addresses also show in up statistics, machine learning and other probabilistic settings.” Although the technical details may defy common intuition, Sun speaks about his theoretical work with a disarming groundedness — not to mention an occasional Tetris analogy. An edited version of my interview with him follows.

What is representation theory, and what drew your interest to it?

Roughly speaking, representation theory is the algebraic study of symmetry. An object like a sphere is rotationally symmetric, and representation theory gives you a way to derive some quantitative consequences of that symmetry.

It’s a small part of the standard introductory abstract algebra class in college; many of my friends and I who took this class thought representation theory wasn’t even a real field of study. But later, almost randomly, I took a class from my future Ph.D. adviser, which was only about representation theory. And he really showed me that it applies to many different areas and situations in math. It seems to isolate some set of fundamental principles that can relate to a lot of different things.

Can you describe one of these other domains that representation theory describes?

A lot of the problems I work on now have some motivation from physics. For example, in statistical physics, you often want to study an interface between two surfaces. A simple way to imagine this is playing a game of Tetris: if you look at your top line of blocks, that’s some random interface. And depending on the rules you’re playing by, that interface is going to have different statistical properties: you might expect it to be very jagged, or maybe very smooth.

People have studied how this interface fluctuates and how it evolves — not for the rules of Tetris, but for some set of rules that is closer to the natural world. And they found various statistical properties that happen to align almost exactly with certain properties of random matrices, which are probabilistic objects which I study using techniques from representation theory.

What is the difference between random matrices and statistical physics?

A random matrix is like a grid of numbers, and each of those numbers is chosen in some random way. The point of random matrix theory is to say that actually, despite the randomness, there’s still something very close to deterministic and structured about this matrix. The quantities of the random matrix that have this deterministic property are called eigenvalues.

Statistical physics is a branch of physics focused on deducing large-scale properties of systems from an understanding of their small-scale behavior.  As an example, it would seek to go from an understanding of how individual gas molecules bounce off each other to an understanding of the pressure the gas exerts.  This pressure would be an average of the impacts of many individual small gas molecules.

The definitions of these two fields really have no intersection. If you just told someone about each field in isolation, you would never know that they might be related, but some techniques based on representation theory seem effective in describing them both. It’s actually kind of a mystery why this is the case.

Many people think of mathematics as rigid and literal, rather than something with “mystery” in it. How did you come to view math this way?

Both my parents are chemists, and so they always emphasized math and science in my education. But I really never thought of math as a special thing at all until middle school, when I started doing a lot of math competitions. Most people think of math as just solving equations, or even at a more basic level, just multiplying some numbers. But in these competitions, they give you problems where it’s not obvious how to solve the problem. I’d always been really good at elementary school math. But when I began doing these competitions, for the first time I saw that there are a lot of problems that I had to think a lot more about. It really showed me that math is something much deeper.

Some theoretical mathematicians treat their work almost as a creative art form, where others approach it more as a science. Where do you stand?

I’m definitely more towards the latter. What I like to do is try to find new behavior and then try to explain it. I don’t have that much understanding of how to do applied work in mathematics, but it seems that you must be very adapted to your problem domain. You understand those phenomena very well, but you perhaps don’t have the time to look at several domains that really aren’t that similar, which is what I like: looking at each of these domains from a high level. And what’s interesting for me is to try to see whether there’s some theoretical phenomenon that really underlies all of them. And, you know, that doesn’t always exist. But when it does, it’s pretty nice.

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