The Geometry of Minimal Surfaces: An Interview with Antoine Song

Born and raised in France, Antoine Song completed his undergraduate and master's degrees at École normale supérieure in Paris before coming to the United States to do doctoral work at Princeton University. After earning his PhD, Song completed a three-year postdoctoral fellowship at UC Berkeley and then came to Caltech as an assistant professor of mathematics in 2022.

Song specializes in differential geometry, the study of the shape of objects or spaces using analysis and differential equations. He is on a quest to deeply understand minimal surfaces, geometrical shapes that minimize total surface area that are the lowest energy surfaces in their local space. We recently sat down with Song to learn more about his research and his first year as a professor at Caltech.

What is a minimal surface?

It's a generic term to describe a whole subset of objects that are mathematically interesting. They have a special shape, a specific geometry. If you think about all the possible surfaces you could visualize in three dimensions, most of them do not have a beautiful or meaningful shape. But minimal surfaces—those that minimize energy—are optimal in some sense.

Can you give an example of a minimal surface?

One of the simplest examples is a soap film. If you dip a wire frame in soap water, when you take it out you have a soap film that has a very particular shape. It will not be bumpy, for example. I mean, you can imagine all sorts of shapes that have the same boundaries as the soap film, shapes that can exist within the same wire frame. But the shape made by the soap film uses the minimum amount of energy. It is at a resting state. And in that sense, it is an optimal shape.

Another example that's very simple to understand is if you imagine a bunch of points in a plane: if you link all these points with the least amount of string possible, this creates a minimal surface. This surface has the most connections possible with the least amount of energy.

What have you learned about minimal surfaces?

Just 10 years ago, it was not clear how many minimal surfaces there might be. At first it was thought that although they abound in the familiar three-dimensional space of our everyday life, they might otherwise be quite rare. But the research I did for my PhD thesis showed that there are infinitely many minimal surfaces. It's not always obvious to pinpoint them, but it's a fascinating class of objects.

Are there applications for your work on minimal surfaces, or is this concept still solely within the realm of pure mathematics?

Honestly, my direct motivation in studying the geometry of minimal surfaces is not any potential applications of the work. I sometimes say that maybe in a hundred years there will be such an application, but that's not my goal. Fundamentally, of course, there's no real distinction between pure and applied math. And personally I'm interested in exploring more applied mathematics over the long term.

What helps you most when you're trying to think through a difficult problem? Where are you when you get your best ideas?

Walking helps a lot. The Caltech campus is great for that. Sometimes, though, it can get you into trouble, if you're so deep in your thoughts that you don't recognize people who are trying to say hello to you. When I was doing my graduate work at Princeton, I used to walk around in the library a lot, not looking at the books or anything, just moving to help myself think. One day the librarian came to me and asked me if I was all right. She was worried about me.

Do you ever wonder if an advance in mathematics will put the foundations of all the simpler math the rest of us learn in school into question?

No, no. Mathematics is not like physics, where a new theory could either destroy or completely revolutionize what came before. Just by the nature of mathematics, I think this is less possible. We like to present the history of math as some kind of linear progress, but it's more subtle than that. It is true that the theorems of the ancient Greeks are as fresh now as when they were first proved. They're correct and very useful. But the idea of a linear progression is complicated in the sense that there is no predetermined direction in mathematics. We may be working in areas that seem important to us, but one day people may realize that these are not the right questions to ask or the right kind of theorems to prove. They may be completely correct, but they may be abandoned if they are not considered beautiful, rich, or meaningful enough. So, mathematicians may set this work aside and move in other directions.

What courses are you teaching?

Last year I taught a course about minimal surfaces—my own specialization. This year, I'll be offering a class on Riemannian geometry, which is used in physics to describe general relativity.

What do you like about Caltech so far?

The size works really well for me. Caltech is able to offer things that bigger math departments cannot. This quarter, for example, I'm hosting a seminar called Frontiers of Mathematics. The goal is to bring in someone each week, whether a faculty member of a postdoc, to give a talk about their research. In this way, undergrads can learn about the wide range of problems mathematicians are working on now and also about how a seminar works. Many universities aren't able to offer something like this to their undergrads.