In this course we will give an introduction to geodesic currents and length functions on them with a view towards Mirzakhani’s curve counting theorems. In her thesis Mirzakhani obtained the asymptotic growth of the number of primitive simple closed geodesics of length bounded by L on a hyperbolic surface and showed that it is a constant times $L^{6g-6}$, where g is the genus of the surface. In fact, she obtained this growth for each topological type of simple geodesics (i.e. those in a fixed mapping class group orbit of a simple geodesic) and later extended it, through very different methods, to also hold for orbits of non-simple geodesics. We will discuss how rephrasing the counting question to one about measures on the space of currents gives a unified proof of the two theorems which is more combinatorial in nature and has the key feature that it gives flexibility in the choice of length used to measure the geodesics/curves.
De Maciej Bocheński
De Neza Zager Korenjak
