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.
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