http://www.simonsfoundation.org/quanta/20140812-a-tenacious-...
Mirzakhani became fascinated with hyperbolic surfaces — doughnut-shaped surfaces with two or more holes that have a non-standard geometry which, roughly speaking, gives each point on the surface a saddle shape. Hyperbolic doughnuts can’t be constructed in ordinary space; they exist in an abstract sense, in which distances and angles are measured according to a particular set of equations. An imaginary creature living on a surface governed by such equations would experience each point as a saddle point.
It turns out that each many-holed doughnut can be given a hyperbolic structure in infinitely many ways — with fat doughnut rings, narrow ones, or any combination of the two. In the century and a half since such hyperbolic surfaces were discovered, they have become some of the central objects in geometry, with connections to many branches of mathematics and even physics.