For the classical forms on differential manifolds, we have near commutativity in the sense that \omega_1 \omega_2=(-1)^{kl}\omega_2\omega_1, if \omega_1 and \omega_2 are forms of dimension k and l, respectively. For the differential calculi considered in Connes' theory of noncommutative geometry this near commutativity no longer holds, but we do have equality under the integral sign: \int \omega_1 \omega_2=(-1)^{kl}\int\omega_2\omega _1, where \int is a "volume integral" on the calculus. Thus, \int is a "trace" or, more precisely, a graded trace. However, the differential calculi introduced by S.L. Woronowicz in the setting of compact quantum groups do not fit into Connes framework, as Woronowicz himself observed. One way in which these calculi differ is that the volume integrals involved are not graded traces. In joint work involving myself and my collaborators Johan Kustermans and Lars Tuset we showed that a volume integral \int in the quantum group setting is typically a twisted graded trace; that is, it satifies the equation \int \omega_1 \omega_2=(-1)^{kl}\int\sigma(\omega_2)\omega_1, for a suitable homomorphism \sigma on the algebra of forms. We invesigated the properties of these KMS-like integrals and obtained existence and essential uniqueness results. We showed that one could relate these twisted graded traces to a theory of twisted cyclic cocycles that we developed that has all the basic properties of Connes' original theory. We have also obtained many other results on differential calculi on quatum groups, including analogues of the Hodge decomposition, de Rham's theorem on the cohomology of a differential calculus and a verson of Poincare duality. The talk outlines some of these and related results.

Our theory indicates that Connes' version of noncommutative geometry is, in some sense, a "semifinite theory". It indicates, as do other developments, the need to consider a "Type III" version of noncommutative geometry to deal with the KMS-type volume integrals, in analogy with the way the modular theory of KMS weights was needed to deal with Type III von Neumann algebras.