Random hyperbolic surfaces via random flat surfaces

What does it mean to pick a ``random’’ hyperbolic surface, and how does one even go about ``picking’’ one? Mirzakhani gave an inductive answer to this question by gluing together smaller random surfaces along long curves; this is equivalent to studying the equidistribution of certain sets inside the moduli space of hyperbolic surfaces. In this talk I’ll describe a new method for building random hyperbolic surfaces by building random {\em flat} ones, and a template for translating theorems from the flat world to the hyperbolic one. As time permits, we will also discuss the application of this technique to Mirzakhani’s ``twist torus conjecture.’’ This is joint work (in progress) with James Farre.