We humans perceive the world in three-dimensional space. So how can one study a geometric object that might exist in, say, 25 dimensions?
Tarik Aougab works with a set of mathematical techniques that help him to do just that. One of the keys to his work, says the new Tamarkin assistant professor of mathematics, is symmetry.
“If you’re interested in geometric objects that live in 25-dimensional space, you can’t really look at it and you don’t really know much about it,” Aougab said. “But maybe you know something about its collection of symmetries. That’s really useful because you can reconstruct geometric facts about this high-dimensional object by studying its symmetries.”
Take a simple, two-dimensional example: a circle. You can rotate a circle through any angle, and its shape doesn’t change. Each transformation in which an object’s shape doesn’t change is called a symmetry. If you know about an object’s set of symmetries, you can infer things about the object’s geometry without ever seeing it.
“Suppose I had a mysterious two-dimensional shape in my pocket,” Aougab said. “I’m not going to tell you what it is. I’m only going to tell you that it’s an object that is really symmetric — so symmetric that you can rotate it through any angle and it doesn’t change shape. Then it wouldn’t be hard to guess that the object is actually a circle. This tells you that the symmetries actually encode the identity of the shape.”
The technique isn’t necessary to understand circles. All we need to do to understand the geometry of a circle is look at it. Where the technique comes in handy is for objects that we can’t simply look at. One of Aougab's specific interests is in an object of many dimensions known as Teichmüller space.
“What I do is I study a much lower-dimensional object that has the same set of symmetries as this Teichmüller space does,” he said. “The geometry of this low-dimensional object is basically a graph — a network with edges connected by points. The graph is engineered so that it has the same symmetries of the Teichmüller space, and it tells me about the geometry of these very high-dimensional objects.
So why study objects that we can’t really look at?
“You study such high-dimensional shapes because they’re encoding information about lower-dimensional shapes that we would encounter in applications and in theoretical physics and in many other areas,” Aougab said.
Aougab comes to Brown from Yale University, where he received his Ph.D. earlier this year. But he’s no stranger to the Brown community or the Providence area. For the last four years, he’s been coming here to help with a program at ICERM, Brown’s National Science Foundation math institute. The program trains young mathematicians to do research.
“Undergraduates from around the country and also from Brown come here and work on research-level math problems,” he said. “We help to advise them and guide them through that research process. So I’m familiar with Brown and Providence because of that.”
Aougab’s first year at Brown will be dedicated to his research, which is supported by a grant from the National Science Foundation. The grant gives people a chance to go to a university in order to work with a particular researcher.
“I applied for a grant to come to Brown and work with Jeff Brock,” Aougab said. Brock is chair of the Department of Mathematics and has pioneered research in hyperbolic geometry and Teichmüller space.
“My work is highly influenced and inspired by his work. That’s one reason I wanted to come here. It gives me the chance to work closely with him.”