It is a classical problem to count the number of closed geodesics of bounded length L on hyperbolic surfaces and study how this number grows with L. Huber proved in the 50’s that this number grows asymptotically as e^L/L and this result has since been generalized by many people in many directions for example by considering other surfaces and manifolds, resulting in various exponential growth rates. However, in her thesis Maryam Mirzakhani studied the subset of geodesics that are simple (i.e. no self intersections) which behave very differently, and proved that on a hyperbolic surface S the number of simple closed geodesics of length L grow asymptotically like a constant times L^{6g-6}, where g is the genus of S. In fact, she proved something stronger: she counted the curves inside a mapping class group orbit of a any simple geodesic. Some years later, using very different methods, she generalized this result to hold also for closed geodesics with self-intersections. 

 

In this mini-course I will give an idea of how to prove Mirzakhani's results. In fact, we will give a different proof from the original ones, using simpler methods, which allows us to treat both cases at once. We will translate the problem of counting geodesics into a statement about convergence of certain measures (which is also what Mirzakhani did for the case of simple geodesics). Along the way we will give a brief introduction to measured laminations, geodesic currents, train tracks, and the Thurston measure. We will end by discuss some applications to Mirzakhani’s counting theorems to studying random multicurves.