Title: Variants of the Atiyah conjecture and their relation to K-theory
Abstract: L^2-Betti numbers serve as an equivariant analogue of Betti numbers, allowing one to assign meaningful numerical invariants to arbitrary G-spaces for discrete groups G. Compared to their classical counterparts, L^2-Betti numbers enjoy the curious feature of taking arbitrary non-negative real numbers as values, no longer just integers. A famous question first asked by Atiyah is whether these numbers are rational at least for universal coverings of finite CW-complexes. The original question has since been answered in the negative, but a number of "Atiyah conjectures" have emerged in its stead and are still open. In this talk, I will discuss the full range of these conjectures and their relation to the questions about the K-theory of group rings and von Neumann regular rings.