Classical mapping class groups, i.e. for surfaces of finite type, are well-studied but they are not particularly interesting from the point of view of topological groups as they are discrete.
When we turn our attention to surfaces of infinite type, the situation changes drastically: In particular, the mapping class groups are now "big" (here: uncountable) and we can define an interesting (here: non-discrete) topology on them. In particular, big mapping class groups are Polish groups and we can ask many new questions such as on automatic continuity or their (large-scale) geometry.
In this talk, I will give an introduction to surfaces of infinite type and big mapping class groups and then focus on the question of topological behaviour of conjugacy classes. The second part is based on joint work with Jesús Hernández Hernández, Michael Hrušák, Israel Morales, Manuel Sedano, and Ferrán Valdez, and will feature tools from model theory in the proofs.