The project resulted in 5 research level mathematics publications. By the end of the period, one of these was already published in an algebra journal and another in a mathematical physics journal, with the other three remaining as preprints pending the journal publication process. A key main result was to correct the very definition of the antipode or `inversion’ for a Hopf algebroid, where pre-existing definitions turned out to be too restrictive. With our new weaker notion, we showed that there was a natural Hopf algebroid associated to a quantum principal bundle (previously this was only partially known, at a bialgebroid level), at least for the main classes of interest. These included the case where the fibre of the bundle is a Drinfeld q-deformation quantum group coordinate algebra. The work also showed how such Hopf algebroids can be constructed in the case of a trivial quantum principal bundle by a certain Hopf algebroid twisting procedure. Another main work studied `action Hopf algebroids’, which in nice cases can be seen as the quantum version of an algebra of differential operators on a quantum space. This paper also introduced for the first time a non-Abelian cohomology of Hopf algebroids and, using this, a theory of coquasi-Hopf algebroids. A third paper in this sequence extended the process of putting an `oid’ into key results in Hopf algebra theory, namely to the construction of a certain prebraided category of modules associated to a Hopf algebroid. Such modules for ordinary Hopf algebras are critical for the construction of their noncommutative differential structure, so the same result in the Hopf algebroid case is a step towards their own noncommutative differential geometry. Another paper constructs examples of Hopf algebroids for two classes of quantum principal bundles, namely quantum instanton bundles and a class of quantum homogeneous spaces. A final paper proposes a new formulation of a missing higher category notion of a 2-quantum group, as an application of Hopf algebroids.
All works were disseminated in their preprint form on the arXiv repository. Results were also disseminated in person, with the Researcher giving 14 conference and seminar talks about the results of the project. Also, in July 2023, the Researcher successfully organised a high-level conference `Hopf algebras and noncommutative geometry’ at Queen Mary University of London, with 18 speakers from the UK, USA, Belgium, France, Germany and Italy. This provided a snapshot of the state of the art in the field and of the work of the project in relation to it.
