The Localization Theorem is a very convenient tool to compute the equivariant cohomology of $G$-spaces. It provides an isomorphism between the localized equivariant cohomology of a $G$-space (with respect to $S$) and the localized equivariant cohomology of its $G$-fixed points. We then have a complete description of the equivariant cohomology of $X$ up to $S$-torsion. In this talk, we review some properties regarding Localization (in algebra), then explain the statement of the Localization Theorem and finally give some examples of its applications.

Meeting ID: 997 4840 9440 Passcode: 911104

Spectral triples (A,H,D) can be interpreted as unbounded representatives for classes in KK-theory, specifically in KK(A,C). It therefore seems natural to investigate if, and how, constructions from KK-theory are reflected back in noncommutative geometry. In this talk we will look at a specific case of this; given a submersion pi:M->B there is a class, pi!, in KK(C(M), C(B)) such that there is a Kasparov product [M] = pi! x [B]. In this talk we will cover an article by W. van Suijlekom and J. Kaad on how this Kasparov product works at the level of spectral triples and correspondences. It turns out that the factorization is exact, up to a curvature term.