Counting Graphs on Surfaces
The problem of map enumeration concerns counting, up to homeomorphism, connected spatial graphs, with a specified number j of vertices, that can be embedded in a compact surface of genus g in such a way that its complement yields a cellular decomposition of the surface. As such this problem lies at the cross-roads of combinatorial studies in low dimensional topology with graph theory. The determination of explicit formulae for map counts, in terms of closed classical combinatorial functions of g and j, has been a decades-long problem with motivations stemming from random combinatorial structures as well as statistical physics related to random matrix theory. In joint work with Joceline Lega and Brandon Tippings, we have recently obtained a complete solution to this problem in a broad range of cases. This talk will describe those results as well as what underlies them, which brings together a range of ideas from dynamical systems theory, asymptotic analysis and analytical combinatorics.