#### 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 decient 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.

Done