Topological recursion, enumerative geometry and quantization of quadratic Lagrangian
Together with Bertrand Eynard, we developed a formalism allowing to compute the asymptotic expansion of a large class of matrix integrals when the matrix is very large. We later generalized this inductive formalism under the name topological recursion as an efficient way to solve many different problems related to enumerative geometry such as the computation of Gromov-Witten invariants of different manifolds, of the Weil-Petersson volume of moduli spaces of Riemann surfaces, of intersection numbers, Hurwitz numbers or sums over (plane-)partitions. It is also believed to generalize the volume conjecture by computing many knot invariants.
Even if a full general and geometric understanding of this success is still missing, I will present in this talk a brief introduction of the original formalism and its applications as well as a new point of view recently pointed out by Kontsevich and Soibelman presenting it as a way to quantize quadratic Lagrangians by providing the full WKB expansion of the associated wave function.