Connectedness of moduli spaces: From plane curves to rational maps

In this talk we describe some ways that fundamental groups and other topological notions have been applied to determine connectedness of moduli spaces. In the 1930s Zariski studied the question of whether the moduli space of plane curves of fixed degree with a certain set of prescribed singularities is connected. He famously showed that plane sextic curves with six cusps and no other singularities is disconnected: the two components corresponding to whether the cusps or not the cusps are in general position or lie on a conic. To prove this, Zariski developed and used tools around fundamental groups and branched coverings, showing that purely topological data can be used to answer geometric questions: the deformability of six cuspidal sextics one to the other. Recently, Milnor asked an analogous question in the realm of rational maps: Is the moduli space of quadratic rational maps of a fixed degree and a prescribed orbit portrait for one of the critical points connected? This problem is still wide open. In this talk we will describe some partial results with a particular focus on the use of topological techniques a la Zariski in this context.