Countable Subgroups of Homeomorphism Groups: Dimension One and Beyond - Lecture 1

Explore the intricacies of countable subgroups within homeomorphism groups in this comprehensive lecture from the Orderable Groups conference. Delve into problems concerning finitely generated and countable subgroups of homeomorphism groups with specific properties, focusing on regularity in dimension one. Examine aspects of joint work with S. Kim on critical regularity and discover the outline of a program to extend results from dimension one to higher dimensions. Gain insights into recent findings on the model theory of homeomorphism groups of compact manifolds, developed in collaboration with S. Kim and J. de la Nuez Gonzalez. Consider potential future generalizations of these concepts and their implications for the field of mathematics.