ABSTRACT: In December I described the relationship of generalized forms of amenability to generalized versions of the Novikov Conjecture, and concluded by explaining how the theory of expander graphs yielded examples of spaces which were not 'generalized amenable' and therefore had the potential to be counterexamples to generalized Novikov conjectures.
In this talk I will explain the results obtained since that time by Higson, Lafforgue, and Skandalis which have led to a family of counterexamples to the Coarse Baum-Connes Conjecture, the Groupoid BC conjecture, and the Foliation BC conjecture. I will conclude by asking what the prognosis is for the Novikov Conjecture itself.