Heat trace asymptotics on noncommutative spaces.
We introduce the concept of homotopy equivalence for Hopf Galois extensions and make a systematic study of it. As an application we determine all -Galois extensions up to homotopy equivalence in the case when is a Drinfeld-Jimbo quantum group.
We employ the notion of the universal differential calculus to propose yet another approach to the Hopf-type cohomology of Hopf algebras.