Key words : integrable systems, algebraic and enumerative geometry, combinatorics, topological recursion, resurgence theory, random matrices, string theory, statistical physics, maps.

Quantum systems are often defined 'perturbatively', from a classical system, as an asymptotic expansion series, whose coefficients can be defined either from a differential equation (e.g. Schroedinger), deformations relations, combinatorial expression, recursion,... One goal will be to show, at least in examples, that all definitions lead to a common universal recursion known as 'topological recursion'. And moreover that deformation relations satisfy an integrable system. These asymptotic series are divergent, and a resummation method is needed. We shall use the 'resurgence' method. The goal of the project is to study and prove, in examples or in general, some relations between topological recursion, integrable systems and resurgence. An integrable system was initially defined as a dynamical system with enough conserved quantities to make it « solvable ». It was rephrased as a set of Poisson-commuting Hamiltonians, and as the existence of a « Tau-function » whose differential is generated by the commuting Hamiltonians. In physics, the Tau function is the partition function. The Tau function is characterized by some relations satisfied by its differential, and in particular a nonlinear equation called « Hirota equation ».