In recent years an interesting connection has been established between some moduli spaces of algebro-geometric objects (e.g. algebraic stable curves) and some modulispaces of polyhedral objects (e.g. tropical curves).In loose words, this connection expresses the Berkovich skeleton of a given algebrogeometric moduli space as the moduli space of the skeleta of the objects parametrizedby the given space; it has been proved to hold in two important cases: the modulispace of stable curves and the moduli space of admissible covers. Partial results areknown in other cases.This connection relies on the study of the boundary of the algebro-geometric moduli spaces and on its recursive, combinatorial properties, some of which have beenlong known and are now viewed from a new perspective.
Caporaso, L. (2018). Recursive combinatorial aspects of compactified moduli spaces. In Proceedings of the International Congress of Mathematicians, ICM 2018 (pp.653-670). World Scientific Publishing Co Pte Ltd.