Explore countable subgroups of homeomorphism groups in this 50-minute lecture from the Workshop on Orderable Groups. Delve into problems concerning finitely generated and countable subgroups with specific properties, focusing on regularity in dimension one and examining aspects of joint work with S. Kim on critical regularity. Learn about a program to generalize one-dimensional results to higher dimensions, and discover recent findings on the model theory of homeomorphism groups of compact manifolds. Gain insights into potential future generalizations of these ideas, presented by Thomas Koberda at the Centre de recherches mathématiques (CRM).