Given a finite-sheeted, possibly branched covering space between surfaces, it's natural to ask how the mapping class group of the covering surface relates to the mapping class group of the base surface. In this talk, we will take a journey through this question for surfaces with boundary. It will feature appearances from the fundamental groupoid, the Birman-Hilden theorem, the Burau representation and new embeddings of the braid group in mapping class groups.

This is joint work with Alan McLeay. Speaker(s): Tyrone Ghaswala (University of Waterloo)