The metric in this case is not the one that we “See” when we live on the surface of any doughnut, but the something-like-a-square idea shows how we can define it. If the topology of a surface is “Complicated enough”, the surface admits so-called hyperbolic metrics. Like the Pac-Man metric on the doughnut, these metrics cannot be visualized as distances measured on the outer layer of a solid. A remarkable property of these hyperbolic metrics is that in every deformation class, only one closed curve has the shortest possible length in the class. Part of Maryam’s work involved counting these geodesic curves on surfaces with a hyperbolic metric. One of the questions Maryam answered is about the growth in the number of geodesics with no crossings.
A bit more than a decade ago when the mathematical world started hearing about Maryam Mirzakhani, it was hard not to mispronounce her then-unfamiliar name.