Maryam Mirzakhani is the first woman to win the Fields Medal due to her contributions in understanding of the symmetry of curved surfaces.
Mirzakhani and Eskin decided to tackle one of the largest open problems in their field. It concerned the range of behaviors of a ball that is bouncing around a billiard table shaped like any polygon, provided the angles are a rational number of degrees. Billiards provides some of the simplest examples of dynamical systems — systems that evolve over time according to a given set of rules — but the behavior of the ball has proven unexpectedly hard to pin down.
We've seen simple examples of these questions, and are familiar with using the "reflection property" to help us imagine the path of the ball. Specifically, instead of imagining the ball bouncing away, we let the ball continue in it's path and instead reflect the entire billiard table. In this way, the numerous reflected billiard tables allow us to easily consider the path of the ball.
Andrew extends this question further in Bouncing Off of the sides of a Triangle Forever, and posed a question where the "billiard table" is a triangle.
In David's set Complete and Utter Chaos, he demonstrates how this simple idea is related to chaos theory.
Note: Maryam was also the first female contestant fielded by Iran at the IMO!