Title: Topological recursion as a mean to quantise spectral curves

Abstract: For some decades, deep connections have been forming among enumerative geometry, complex geometry, intersection theory and integrability. Topological recursion is a universal procedure that helps building these connections. It associates to some initial data called spectral curve, consisting of a Riemann surface and some extra data, a doubly indexed family of differentials on the curve, which often encode some enumerative geometric information, such as volumes of moduli spaces, matrix model correlation functions and intersection numbers. After an introduction to topological recursion and its relation to different topics, I will focus on the integrability side of the story. The quantum curve conjecture claims that one can associate a differential equation to a spectral curve, whose solution can be reconstructed by the topological recursion applied to the original spectral curve. I will present this problem in some simple cases and comment on some of the technicalities that arise when proving the conjecture for algebraic spectral curves of arbitrary rank, like having to consider non-perturbative corrections. The last part will be based on joint work with B. Eynard, N. Orantin and O. Marchal.