Abstract: In the area of mathematics which is now called non-commutative geometry, the usual way to study a singular object is to associate to it an operator algebra which reflects, in a sense, the main features of this object. For example, Poincaré duality is well known for smooth manifolds, but how can one formulate and prove Poincaré duality, say, for arbitrary simplicial complexes?
In this talk I plan to explain how operator algebras are used for this. These ideas find application in the study of K-theory of group C*-algebras of discrete groups. The conjectures which are related with this are the conjectures of Novikov and Baum-Connes.