Matt Daws: Compact quantum metric spaces a la Rieffel
Ulrich Kraehmer: Noncommutative differential geometry and quantum group symmetries
In this survey talk I will attempt to explain where we stand with the study of the noncommutative geometry (in the sense of Alain Connes) of spaces with quantum group symmetry, and why (some) people still care. There will be a brief introduction to the concept of a spectral triple which is the NCG generalisation of a smooth manifold, and then a review of the developments that have taken place over the last two decades. At the end I will describe a few open problems that I personally think might serve as milestones in future grant applications.