The mapping class group, $\mathrm{Map}(S)$, of a surface $S$, is the group of all isotopy classes of homeomorphisms of $S$ to itself. A mapping class group is a topological group with the quotient topology inherited from the quotient map of $\mathrm{Homeo}(S)$ with the compact-open topology.

For surfaces of finite type, $\mathrm{Map}(S)$ is countable and discrete. Surprisingly, the topology of $\mathrm{Map}(S)$ is more interesting if $S$ is an infinite-type surface; it is uncountable, topologically perfect, totally disconnected, and more importantly, has the structure of a Polish group. In recent literature, this last class of groups is called "big mapping class groups.''

In this talk, I will give a brief introduction to big mapping class groups and explain our results on the topological structure of conjugacy classes. This was a joint work with Jesús Hernández Hernández, Michael Hrušák, Manuel Sedano, and Ferrán Valdez.