The BaumConnes Conjecture 
Formulations 
The BaumConnes conjecture proposes a means of calculating Ktheory for the reduced C*algebra of a locally compact group using a combination of group homology and the representation theory of compact subgroups. The conjecture was first set forth in 1982 by Baum and Connes; the current formulation was given in 1993 by Baum, Connes and Higson. The conjecture originates in work of Kasparov and Mishchenko on the Novikov higher signature conjecture, ideas of Connes in foliation theory, and Baum's geometric description of Khomology theory. The validity of the conjecture has implications in geometry and topology, most notably the Novikov conjecture and the GromovLawsonRosenberg conjecture on positive scalar curvature. In addition there are close connections to issues in harmonic analysis, for instance the problem of finding explicit realizations of discrete series representations and the problem of inducing supercuspidal representations of padic groups from compact open subgroups. Indeed a very striking feature of the conjecture is its generality, and the breadth of mathematics with which it makes contact. For this reason, the conjecture has drawn a great deal of attention from a wide variety of sources. The links at the left give further information about aspects of the conjecture. 

Surveys  
Classifying Spaces  
Special Cases  
Injectivity  
Coarse Geometry  
Topology  
Representation Theory  
Generalizations  