Bivariant homology theories for noncommutative spaces.

Thanks to the Gelfand-Naimark theorem a noncommutative (pointed compact) space is simply a noncommutative C*-algebra (at least according to one viewpoint). Well-known bivariant homology theories for C*-algebras include KK-theory and local cyclic homology. Such bivariant homology theories have striking applications to certain problems in topology and algebra. I am going to present a generalized setup for studying such bivariant homology theories. This generalized setup is very desirable from the perspective of homotopy theory and new homology theories for noncommutative spaces. Keeping in mind the audience of the seminar I am going to try to keep the presentation non-technical.