Bundles in noncommutative geometry

Noncommutative geometry is a generalisation of differential geometry that allows for an inclusion of quantum effects, such as those expected from quantum gravity. The aim of this research is to study noncommutative analogs of principal bundles, called Hopf-Galois extensions, within different approaches to noncommutative geometry. One the one hand, we will develop the synthetic geometry of noncommutative principal bundles and connections internal to braided monoidal categories, such as the representation categories of quasi-triangular Hopf algebras. Such scenarios can be used to describe quantum group equivariant noncommutative geometry. We shall in particular define and study a concept of noncommutative gauge groups and their actions on principal connections in this setup by using techniques from topos theory. Examples will be provided via deformation quantisation by Hopf algebra 2-cocycles. On the other hand, we will investigate fuzzy noncommutative geometries and their equivariant generalisations in the spectral triple approach of Alain Connes. We will in particular focus on examples coming from the quantisation of coadjoint orbits, such as complex projective spaces, and construct noncommutative analogs of spin structures for these models, which are certain Hopf-Galois extensions.