Beyond Squares and Triangles: Deciphering the Mysteries of Complex Shapes

Two-dimensional shapes are easy to understand. But what about abstract, imaginary shapes — ones that come in four, five or six dimensions? Mathematician Aleksander Doan is up to the challenge. A member of the Simons Society of Fellows, Doan works to understand the geometry of manifolds. In mathematics, manifolds are intriguing because they appear to be two things at once: Any two manifolds of the same dimension look the same at close range, but when you zoom out, you can see that they are in fact distinct from each other. This property of manifolds makes their study formidable and inspiring.

Doan is adept both at describing what manifolds are in everyday language and at using some of the most advanced tools in mathematics to uncover their secrets. He completed a bachelor’s degree at the University of Warsaw and a doctorate at Stony Brook University. He is now a Junior Fellow of the Simons Society of Fellows, working at Columbia University, and a Junior Fellow of Trinity College at the University of Cambridge in the United Kingdom.

Doan and I recently discussed how his work could catalyze new insights in physics and geometry.  Our conversation has been edited for clarity.

One of the fundamental concepts you study is known as a manifold. What is a manifold?

Manifolds include simple shapes like a line, which have just one dimension, or two-dimensional shapes like a sphere. In math a “dimension” refers to how many coordinates you need to use to specify a point on an object. A line is one-dimensional because you only need one coordinate to locate anything on that line. You need two coordinates to specify any point on a sphere, such as the latitude and longitude.

There are also manifolds of higher dimensions. Some of these manifolds appear naturally in various problems of engineering and physics, but most are not physical objects; they exist only in mathematical reality.

Because the human eye can only see in three dimensions, many manifolds are invisible to us. In some ways these higher-dimensional manifolds are just like lines and surfaces, and in other ways they are completely different. The basic property of manifolds is that all manifolds of a given dimension look the same at small scales, but at large scales they can be quite distinct.

Let me give you an example. Imagine that you’re looking at a sphere. If you zoom in and look at it very closely, it looks like a flat plane — just like the planes you may have studied in high school geometry. But of course spheres are not flat planes, which is something we can only tell at larger scales, when we zoom out and see the entire structure. The Earth is a sphere, after all, and is certainly not flat. It took people thousands of years to understand this, though, because the Earth looks flat to anyone standing on it.

Mathematicians are interested in studying manifolds of any dimension, from one to infinity. I study a particular area of mathematics called topology. Topologists study properties of manifolds that remain the same when we stretch, twist, bend or modify them in any other way that does not destroy the integrity of the initial shape. For example, you can’t add a hole to a manifold that does not already have one without destroying its integrity. I am hopeful that this focus will help us learn how to distinguish manifolds from each other.

Would you say that it’s difficult to tell manifolds apart?

Well actually, sometimes it’s easy. Let’s go back to the sphere. Imagine you’re looking at the surface of that sphere as well as the surface of a doughnut, which mathematicians call a torus.

Right away you can see that the torus has a hole through the middle, while the sphere does not.  They are topologically distinct because you cannot transform one into the other without creating — or filling in — that hole.

You can make this distinction because the sphere and torus are easy to visualize. However, oftentimes manifolds are not presented to us visually but with a written description, which can be quite involved and difficult to imagine. With higher numbers of dimensions — more variables, essentially — it becomes impossible to visualize manifolds and much harder to tell them apart. Eventually you need mathematical tools to distinguish manifolds from each other.

Why is it important to be able to distinguish manifolds?

That’s a good question, and in fact often before embarking on any problem mathematicians ask themselves, “What would be the point of solving this problem?”

I’m interested in this challenge because any difficulties in distinguishing or classifying manifolds reflect gaps in our knowledge of geometry.

By trying to answer these questions, we are forced to develop new tools, which often turn out to be useful for solving other problems of mathematics and physics. For example, many of the tools of differential geometry, a branch of geometry that uses calculus to study manifolds, turned out to be crucial to Einstein’s theory of relativity, and in recent times also in quantum field theory and string theory. Each of these mathematical advances offered critical insight for theoretical physicists, and I think many more advances lie ahead.

I try to bear in mind that many elementary facts about shapes or numbers which seem obvious to us —such as the Pythagorean theorem or the existence of irrational numbers — were groundbreaking discoveries in ancient Greece. Over time, scientists and engineers incorporated this knowledge into everyday practice so deeply that it changed how we live, but that took time. By that measure, topology is indeed quite abstract — it involves the study of shapes that typically don’t exist in real life — but I believe that in the long term its results can be just as consequential.

What topological tools and concepts do you use to study manifolds?

Topologists have developed many such tools, but I am particularly interested in using gauge fields to study manifolds. In mathematics, a field is a way for us to associate a mathematical quantity — which can be a number but doesn’t have to be — to every point on the manifold.

The simplest example is a field that associates a number to every point, like how meteorologists measure the temperature or atmospheric pressure at different points on the Earth’s surface.

A more complicated example is the electromagnetic field that permeates our space. At every point in space, this field has not only a size but also a direction, making it a more complicated mathematical object than a single number. The dynamics of this electromagnetic field are governed by what are known as Maxwell’s equations. Similarly, in physics any fundamental force of nature is carried by another type of field — gauge fields — that, just like the electromagnetic fields, are described by a complicated system of equations. These equations provide a bridge between mathematics and physical reality.

What is remarkable is that gauge fields, which were long the domain of particle physics, can also help us solve various problems in pure mathematics, including distinguishing manifolds.

It’s an amazing and surprising fact that some manifolds, depending on their geometry, can have many gauge fields, while others have none at all. We can use this information to differentiate manifolds from each other. This discovery was a major breakthrough in topology in the 1980s, which led to spectacular advances in understanding manifolds of three and four dimensions.

How have you worked with gauge fields?

During my doctoral work at Stony Brook University, I was fortunate to work with Simon Donaldson, who pioneered the use of gauge fields in four-dimensional topology.

He also proposed that it might be possible to generalize these methods to study manifolds of higher dimensions, and I focused on that problem.

For example, consider a class of six-dimensional manifolds known as Calabi-Yau manifolds. These manifolds are especially important in string theory and in another mathematical field called algebraic geometry. My collaborator Thomas Walpuski, at the Humboldt University in Berlin, and I study problems related to gauge fields on such manifolds.

People typically assume that gauge fields are smooth, meaning that the field does not jump abruptly as you move from one point on the manifold to another. In the manifolds we studied, we were quite surprised to discover an abundance of gauge fields that were not smooth.

There’s still much that we don’t understand about such non-smooth gauge fields, so there’s a lot of work left to do.

That’s exciting! And it brought you to Columbia University, where you’re currently completing a postdoctoral appointment with Mohammed Abouzaid. What’s your focus today?

I’m still studying Calabi-Yau manifolds, but from a different angle. As I just described, one way to probe their geometry is to analyze gauge fields. Another is to look at a special class of two-dimensional surfaces, called holomorphic surfaces, that sit inside Calabi-Yau manifolds. These surfaces are special because they have minimal volume, very much like soap bubbles. Like gauge fields, they tell us a lot about the geometry of the manifold inside which they live. There is an interesting connection between gauge fields and these surfaces: As we change or deform the manifold, some gauge fields can become very strong in a small region near such a surface, and very weak everywhere else. This is a fascinating picture that’s not so well understood, which makes it an exciting research project for my postdoc at Columbia. I hope it spurs all sorts of new discoveries, by myself and others.