The interplay between graph theory and surface topology has become a classic topic in both combinatorics and geometric topology. This talk will be about surprising links via moduli theory, which is originally about measuring distances between surface shapes.

From curve counting, to combinatorics and hyperbolic geometry, the talk aims to show how graphs can help us understand moduli spaces and how moduli spaces can help us understand graphs. This will include recent progress on generalizing the so-called crossing lemma, which helps quantify how far from being planar a graph can be in terms of its number of edges and vertices.