Computing Gromov-Witten invariants by topological recursion

Following an inductive procedure developed in the setup of random matrices, it is possible to solve many problems of enumerative geometry with a unique formula called topological recursion. In particular, this procedure allows to compute higher genus Gromov-Witten invariants of some manifolds in terms of a mirror partner. In this talk, I will present this procedure and show that, in some particular cases, it is equivalent to a quantization formalism developed by Givental for expressing the potential of a semi-simple cohomological field theory in terms of KdV tau functions. This correspondence explains the universality of this procedure for computing Gromov-Witten invariants.