Nouvelles relations pour les corrélateurs et les nombres d'intersection

Correlation functions are central objects in Physics and Mathematics. We define them in the context of integrable systems using determinantal formulas and explore their properties and applications on intersection theory in the moduli space of curves. Moreover, we present the relation between topological recursion, integrable systems and matrix models. This thesis comprises two articles and code implementation. The first article presents a method for deriving differential equations and recurrence relations for correlation functions in integrable systems and random matrices, allowing for the construction of coefficients that are rational functions. The second article introduces a new formula for intersection numbers of psi classes over the moduli space of curves, solving a conjecture from previous work. Python implementation for computing the following integrals over the moduli space of curves: simple and triple Hodge integrals with or without theta classes. We begin by introducing random matrix theory from a historical perspective.