The idea of the direct splitting method for the Baum-Connes conjecture (BC) is that in order to split the assembly map for a group G, it is enough to construct an operator (or perhaps a family of operators) on a G-Hilbert space satisfying certain geometric properties. A proper Kasparov cycle is just a pair of a G-Hilbert space and an operator with such geometric properties. I will introduce this notion of proper Kasparov cycles with examples and non-examples and explain the idea of the direct splitting method. In the last part of this talk, I will describe how an asymptotic version of this idea can be used to explain the proof of the Higson-Kasparov theorem (BC for a-T-menable groups) in a new light.

Because of the Corona crisis, the lectures will be given as online lectures via zoom (or other video conference software). Please contact us by sending a message to our secretary Elke Enning, if you are interested to participate, so that we can send you an invitation for the lectures.