Multicurve count, Masur–Veech volumes and topological recursion

Based on the work of Mirzakhani, I will explain how the count of multicurves on bordered surfaces can be solved by geometric and topological recursion techniques, both in the hyperbolic and in the combinatorial setting. I will also explain how the asymptotics of such counts compute the Masur–Veech volumes of the moduli space of quadratic differentials. The talk is based on joint works with J.E. Andersen, G. Borot, S. Charbonnier, V. Delecroix, D. Lewański and C. Wheeler.