Colloquium

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 combinatorics and statistical physics. In joint work with Nick Ercolani and Brandon Tippings, we have recently obtained a complete solution to this problem in the case of graphs of even valence. This talk will chronicle our approach, which brings together a range of ideas from dynamical systems theory, asymptotic analysis, analytical combinatorics, and transfer matrix analysis.