Bijective counting of one-face maps on surfaces

A one-face map is a graph embedded in a compact surface, in such a way that its complement is a topological disk. Dually, it can be viewed as a polygon of even size, in which edges have been pasted pairwise to create a surface. These objects have very nice enumerative properties, discovered years ago by Lehman, Walsh, Harer and Zagier, but until very recently their combinatorial interpretation remained mysterious. I will present a bijection that enables us to understand the structure of these objects better, and obtain all enumerative results very easily (in particular the product formula counting one-face maps of given genus, involving Catalan numbers). I will also present a recent extension made jointly with Olivier Bernardi (MIT) to non-orientable surfaces.