PopSci's Fourth Annual Brilliant 10

Maryam Mirzakhani

Princeton University

She finds new ways to describe arcane geometric objects.

One of Maryam Mirzakhani's favorite movies is Dogville, a stark look at Depression-era America. "There are no walls and no sets. You have to fill a lot in for yourself," she says. Mirzakhani's taste in films reflects the open-ended nature of her research, which involves pinning down the characteristics of unusual geometric forms. "There are times when I feel like I'm in a big forest and don't know where I'm going," she says. "But then somehow I come to the top of a hill and can see everything more clearly. When that happens, it's really exciting."

Mirzakhani, 28, grew up in Iran. After winning the International Math Olympiad twice in high school, she attended the Sharif University of Technology in Tehran. In 1999, at Harvard University, she tackled a problem that had stymied many a mathematician: calculating the volumes of moduli spaces of curves-geometric objects whose points each represent a different hyperbolic surface. Some hyperbolic surfaces are oddly shaped, like doughnuts or amoebas. Mathematicians had been trying to calculate the volume of all possible variants of these forms. Mirzakhani found a new way, using a strategy that involved drawing a series of loops on the surface of the shapes and calculating their lengths.

Few practical applications now exist for Mirzakhani's research, but if the universe turns out to be governed by hyperbolic geometry, her work would help define its precise shape and volume. "Maryam is great at finding new connections," says James Carlson, president of the Clay Mathematics Institute. "She can rapidly move from a simple example to a complete proof of a deep and comprehensive theory."

-Elizabeth Svoboda