Lecture III: Positive connections and generalizations.
An n-web is an n-valent bipartite ribbon graph. (Planar) webs arise in representation theory: they are combinatorial devices used to understand invariants in tensor products of irreducible representations. Webs on surfaces can likewise be used to parameterize the character- or representation-variety of the surface.
Intriguingly, webs on surfaces also arise in the n-dimer model of statistical mechanics. In these lectures we discuss webs and their connection with the probability and statistical mechanics of the dimer model, giving a vast generalization of the classical Kasteleyn theorem which counts dimer covers (perfect matchings) of planar graphs.
These lectures are based on joint work with Dan Douglas, Nicholas Ovenhouse, Haolin Shi, Haihan Wu.