Maps are discretization of a given surface obtained by glueing a finite collection of polygonal faces along pair of edges. In this definition, the faces in the map can be quite singular, due to self-glueings. Although geometrically puzzling, this convention dates back to Tutte and simplifies the enumeration. It is also natural from the point of view of matrix model techniques, in which maps arise as Feynman diagrams.

Maps with n ordinary boundaries are just maps with n distinguished marked faces. I introduce a notion of maps with simple boundaries -- this roughly means that the edges along the boundaries are not allowed to touch each other so as to form a "singular" face. I will explain how planar maps with ordinary boundaries can be retrieved bijectively from maps with simple boundaries, and the functional relations between their generating series determining one in terms of the other. This provides a map interpretation of the notion of free cumulants of Voiculescu and second order free cumulants of Speicher and Collins.

For arbitrary topologies, the enumeration of maps with ordinary boundary is governed by the topological recursion of Eynard and Orantin. I will present a conjecture to enumerate maps with simple boundaries in terms of the topological recursion which would give a combinatorial interpretation of the "symplectic invariance property" of the topological recursion.

This is work in progress with Elba Garcia-Failde.