Abstract:

 In this talk, we will  begin by tracing the origins of discrete groups and their early explorations  in different contexts, highlight key milestones and the motivations behind the study of discrete groups, including their intricate interplay among algebra, topology and geometry.  Motived by Atiyah-Singer index theory, Alain Connes and Paul Baum formulated a collection of  remarkable  conjectures in the 1980s.  These  conjectures  provide  a bridge between the algebraic properties of discrete groups and the geometric and analytic properties of spaces associated with these groups.    We will explain  the fundamental ideas of assembly maps, K-theory, and operator algebras that lie at the heart of the conjectures.  Towards the conclusion of the talk, we will briefly discuss some recent work in collaboration with Paulo Carrillo Rouse and Hang Wang, focusing on the  details/well-definedness of Baum-Connes assemble maps and some applications of these new details (if time permits).