Noncommutative geometry encompasses a wide range of approaches to noncommutative gauge theory, from strong connections on quantum principal bundles to the spectral action principle. We use recent advances in unbounded KK-theory to provide the missing link: a practicable theory of noncommutative Riemannian principal bundles with compact connected Lie structure group. As time permits, we will look at the main classes of examples, both commutative and noncommutative, as well as the outlook for one (probable) non-example, the q-Hopf fibration.

This is joint work with Bram Mesland.