Room 326

The formulation of the Baum-Connes conjecture

Ruben Martos

University of Paris 7

The Baum-Connes conjecture, formulated in 1982 by Paul Baum and Alain Connes,
is one of the most active topic of research in noncommutative geometry and despite of the
relevant developments after his formulation, the conjecture is still unsolved.
If it is satisfied, then it will allow to prove other well known conjectures of different
areas of mathematics like in geometry (Novikov conjecture) or in analysis (Kadison-
Kaplansky conjecture).
In this talk, we are going to introduce in a simple way the tools used for being able to
formulate the Baum-Connes conjecture in a proper way. More precisely, we’ll define the
classifying space of proper actions of a group G, the G-quivariant K-homology with compact
supports and we’ll show an explicit way of constructing the assembly map. Finally, we’ll do
an overview of the current situation of the conjecture.