Counting curves on surfaces à la Mirzakhani.

Resumen: I will give an overview, aimed at non-experts, of my work centering around Mirzakhani's curve counting theorem. It's a classical result that the number of closed curves (geodesics) on a hyperbolic surface of length bounded by some L is an exponential function in L. Mirzakhani proved that if we only look at those closed curves that have no self-intersections, then the function is polynomial in L. The degree of the polynomial only depend on the topology (genus) of the surface, but the coefficient gives other interesting information and for example can be used to describe what a "typical" simple curve looks like. I will explain these results and mention some of  mine that generalizes, or are inspired by, Mirzakhani's work.