Product systems represent powerful contemporary tools in the study of mathematical structures. A major success in the theory came from Katsura (2007), who provided a complete description of the gauge-invariant ideals of many important C*-algebras arising from product systems over Z+. This result recaptures existing results from the literature, illustrating the versatility of product system theory. The question now becomes whether or not Katsura's result can be bolstered to product systems over semigroups other than Z+ and, if so, what applications do we obtain? An answer has been elusive, owing to the more pathological nature of product systems over general semigroups. However, recent strides by Dor-On and Kakariadis (2018) supply a more tractable subclass of product systems that still includes the important cases of C*-dynamics, row-finite higher-rank graphs, and regular product systems. 

In this talk we will build a parametrisation of the gauge-invariant ideals, starting from first principles and gradually increasing in complexity. We will pay particular attention to the higher-rank subtleties that are not witnessed in Katsura's theorem, and comment on the applications.
 

A group is called W*-superrigid if its group von Neumann algebra completely remembers the original group. In this talk, I will present a recent joint work with Stefaan Vaes in which we generalize W*-superrigidity for groups in two directions. Firstly, we find a class of groups for which W*-superrigidity holds in the presence of a twist by an arbitrary 2-cocycle: the twisted group von Neumann algebra completely remembers both the original group and the 2-cocycle. Secondly, for the same class of groups, the superrigidity also holds up to virtual isomorphism.