Abstract:

The generating series for maps (embeddings of graphs) in surfaces (orientable and non-orientable) can be expressed both in terms of symmetric functions and in terms of matrix integrals through combinatorial arguments. These series occur, for example, in mathematical physics (the early model of two-dimensional quantum gravity) and in determining the Euler characteristic for the moduli space of curves. Moreover, each of these instances leads to a conjecture about combinatorial properties of maps. One of them suggests a bijection between 4-regular maps and all maps, which holds for all genera. The other suggests the existence of a new invariant of maps with a conjectural interpretation for the moduli spaces of complex and real curves. In this talk, I shall concentrate upon the algebraic combinatorics of this topic.