Abstract
Combinatorial maps are discrete surfaces obtained as gluings of polygons. The first enumerative results about them were obtained by Tutte in the 1960's. During the 1980-90's, they were intensively studied in theoretical physics (under various names: planar diagrams, fatgraphs, ribbon-graphs...) due to their links with two-dimensional quantum gravity and matrix models. Finally, since the 2000's, new combinatorial and probabilistic approaches led to important developments.
After giving an overview of this long history, I will mention some results, obtained in collaboration with Emmanuel Guitter and Grégory Miermont, on the enumeration of maps with geodesic boundaries. They suggest an analogy with hyperbolic geometry, already observed in other contexts, such as topological recursion. Our hope is to arrive to a "bijective" understand of this analogy.
