Advances in Quantum and Braided Geometry

We demonstrate our recent general results on the Casimir construction and moduli space of all bicovariant calculi by means of some detailed examples, including finite-difference and 2-jet cacluli on $\R^n$ and full details of the Casimir construction of the 4D calculus of $SU_q(2)$. We likewise demonstrate our previous general constructions with T. Brzezinski of quantum group gauge theory with examples of such nonuniversal differential calculi on spacetime. We outline a notion of quantum homotopy of a quantum space. We indicate a possible application to classical integrable systems.