Course  Materials  

Equivariant KKtheory and noncommutative index theory
In noncommutative geometry, a C*algebra is regarded as an algebra of continuous functions on a certain 'noncommutative space'. There are two natural topological invariants of such spaces. On the one hand, Ktheory of C*algebras extends the AtiyahHirzebruch Ktheory of topological spaces. On the other hand, Khomology arises from the study of analysis on the noncommutative space. These two theories, which are dual to each other, find a natural unification in the KKtheory of Kasparov, which arises from a deep understanding of the AtiyahSinger index theory. The first part of this course will provide the necessary background, the definition and the fundamental properties of KKtheory. In particular, we shall cover the basic properties of Fredholm modules, and their characters, which are given by natural cyclic cocycles first defined by Alain Connes. Hilbert modules provide natural extension of Hilbert spaces and are a key technical notion required for the definition of KKtheory. A very important property of KKtheory, which makes it very useful in applications, is the existence of a compositiontype product, which we shall discuss in some detail. The second part of the course will introduce an equivariant version of KKtheory which will be very useful in the study of properties of group actions on noncommutative spaces. The third part of the course will acquaint the student with the BaumConnes conjecture. This hypothesis, which has generated intense interest and some beautiful results over the past two decades, serves as a very natural view point on the interaction between topologydifferential geometry and the Ktheory of C*algebras. Prerequisites for this course include basic functional analysis (operators on Hilbert space), C*algebras and elementary topology and differential geometry. Course Summary:

Part I
Jacek Brodzki
Notes by P. Witkowski


Download  Last update  
20.05.2007 

Table of contents  


Part II
Paul F. Baum
Notes by P. Witkowski


Download  Last update  
05.07.2007 

Table of contents  

Exam  Exam questions  

The exam was on 12th June 2007. It consisted of the written part (five exercises) and oral part. In the oral part each student had to answer two questions: easy one and difficult one (chosen from the two difficult questions). Two students (on the graduate level) passed the exam. 
Exam, written and oral 