2007/08, 8th semester

Course Materials
The Baum-Connes conjecture, localisation of categories and quantum groups

The Baum-Connes conjecture has been an outstanding focal point of noncommutative geometry for over twenty years. The first part of this lecture course will show how to apply new ideas from homological algebra and homotopy theory to understand the conjecture more conceptually and to widen its scope towards locally compact quantum groups. The second part of the course will be devoted to applications of the Baum-Connes conjecture and its relations to classical topology.

The following topics will be covered:

  1. Categorical aspects of Kasparov's KK-theory
  2. Duality in Kasparov theory
  3. Homological algebra in triangulated categories
  4. The Baum-Connes assembly map for locally compact groups
  5. Dirac-dual-Dirac method
  6. The Baum-Connes conjecture for discrete quantum groups that are duals of compact groups
  7. Towards the Baum-Connes conjecture for locally compact quantum groups
  8. Corollaries of the Baum-Connes conjecture
  9. Baum-Douglas model for K-homology
  10. Equivariant-bivariant Chern character
  11. Geometric KK-theory

Prerequisites: Basic knowledge of K-theory of C*-algebras and elementary category theory.

Part I
Ralf Meyer
Notes by P. Witkowski
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Table of contents
  1. Noncommutative algebraic topology
    1. What is noncommutative (algebraic) topology?
    2. Kasparov KK-theory
    3. Equivariant theory
    4. Quantum groups
    5. Some applications of the universal property
  2. Kasparov theory as a triangulated category
    1. Additional structure on Kasparov theory
    2. Puppe sequences
    3. The rst axioms of a triangulated categories
    4. Cartesian squares and colimits
    5. Versions of the octahedral axiom
    6. Localisation of triangulated categories
    7. Complementary subcategories and localisation
    8. Homological algebra in triangulated categories
    9. From homological ideals to complementary pairs of subcategories
    10. Localisation of functors
    11. The Baum{Connes conjecture
    12. Towards an analogue of the Baum{Connes conjecture for quantum groups
Part II
Paul F. Baum
Notes by P. Witkowski
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Table of contents
  1. The Baum-Connes conjecture
    1. Proper actions
    2. Universal G-space for proper actions
    3. The Baum-Connes conjecture
    4. The conjecture with coeffcients
    5. Assembly map
    6. Reduced crossed product algebra
Designed by: Pawel Witkowski