Correlation functions of random matrices are related both to the theory of topological recursion and to free probability theory. Whereas topological recursion deals mainly with only one random matrix in all orders, the main thread of free probability is the multivariate situation in leading, planar order. Thus free probability can be seen as a kind of multivariate extension of the planar sector of topological recursion. The goal of this project is to give precise meaning to this observation and develop consequences both for topological recursion and for free probability theory. In particular, we aim at a better understanding of the problem of symplectic invariance of topological recursion.