Maryam Mirzakhani has become the first woman Fields medalist for drawing deep connections between topology, geometry and dynamical systems.
"But when Mirzakhani started graduate school, some of the simplest questions about such surfaces were unanswered. One concerned straight lines, or “geodesics,” on a hyperbolic surface. Even a curved surface can have a notion of a “straight” line segment: it’s simply the shortest path between two points. On a hyperbolic surface, some geodesics are infinitely long, like straight lines in the plane, but others close up into a loop, like the great circles on a sphere.
The number of closed geodesics of a given length on a hyperbolic surface grows exponentially as the length of the geodesics grows. Most of these geodesics cut across themselves many times before closing up smoothly, but a tiny proportion of them, called “simple” geodesics, never intersect themselves. Simple geodesics are “the key object to unlocking the structure and geometry of the whole surface,” Farb said.
Yet mathematicians couldn’t pin down just how many simple closed geodesics of a given length a hyperbolic surface can have. Among closed geodesic loops, the simple ones are “miracles that [effectively] happen zero percent of the time,” Farb said. For that reason, counting them accurately is incredibly difficult: “If you have a little bit of an error, you’ve missed it,” he said."
