I will give a formula for the Masur-Veech volume of the moduli space of quadratic differentials in terms of psi-classes (in the spirit of Mirzakhani's formula for the Weil-Peterson volume of the moduli space of hyperbolic surfaces). I will also show that Mirzakhani's frequencies of simple closed hyperbolic geodesics of different combinatorial types coincide with the frequencies of the corresponding square-tiled surfaces. I will conclude with a (mostly conjectural) description of the geometry of a "random" square-tiled surface of large genus and of a "random" multicurve on a topological surface of large genus.
The talk is based on joint work in progress with V. Delecroix, E. Goujard and P. Zograf. It is aimed at a broad audience, so I will try to include all necessary background.