Monodromies as tête-à-tête graphs

In 2009, A'Campo introduced the notion of tête-à-tête graphs: these are metric ribbon graphs that satisfy a special property that codifies the topology and a periodic mapping class of a surface with boundary. His original motivation was to model monodromies of isolated plane curves in a combinatorial way.  In this talk we discuss the basics of tête-à-tête graphs and introduce the notion of mixed tête-à-tête graphs: a generalization that allows one to model some pseudo-periodic mapping classes. We will finish by stating a realization theorem that characterizes mixed tête-à-tête automorphisms as the monodromies of isolated holomorphic map germs defined on isolated complex surface singularities. If time permits, we will sketch some of the proofs and discuss natural questions that arise from this investigation.