Long curves on hyperbolic surfaces and the geometry of moduli space

In her thesis, Maryam Mirzakhani proved that for any closed hyperbolic surface, the number of simple closed geodesics of length at most L grows like a polynomial in L. Thisbreakthrough theorem and the subsequent explosion of related results use techniques and draw inspiration from Teichmüller theory, symplectic geometry, surface topology, and homogeneous dynamics. Starting from first concepts, in this talk I’ll survey some of these connections as well as discuss answers to the following qualitative questions: What do hyperbolic surfaces with long curves look like? And what do long curves look like on a hyperbolic surface?