Alladi Ramakrishnan Hall

Homotopy and homology of non-commutative spaces

Snigdhayan Mahanta

University of Muenster

In noncommutative geometry (NCG) one typically treats unital C-algebras
as noncommutative compact spaces via Gelfand{Naimark duality. In various applications of
NCG to problems in geometry or topology it is customary to rst reformulate these problems
in terms of certain (co)homology theories for noncommutative spaces. The celebrated Baum{
Connes conjecture, that reduces the Novikov conjecture to an assertion in bivariant K-theory,
is a prime example of this principle. However, the category of C-algebras is well-known to
be decient from the viewpoint of homotopy theory or index theory. In this talk I am going
to rst survey certain (co)homology theories for noncommutative spaces, then present my
proposed solution to the aforementioned problem, and nally (time permitting) discuss some
applications. I will try to keep it non-technical and accessible to a wide range of audience.