Modular spectral triples and KMS states in noncommutative geometry

This thesis investigates the role of dimension in the noncommutative geometry of quantum groups and their homogeneous spaces. We define a generalisation of semifinite spectral triples called modular spectral triples, which replaces the trace with a weight. We prove a resolvent index formula, which computes the index pairing between modular spectral triples and equivariant K-theory. We demonstrate that a modular spectral triple for the Podles sphere has spectral and homological dimension 2. We construct an analogue of a modular spectral triple over quantum SU(2) for which the assumption of bounded commutators fails. We construct a non-trivial twisted Hochschild 3-cocycle for quantum SU(2) using an analytic expression analogous to the Hochschild class of the Chern character for spectral triples. This construction gives the analogous spectral and homological dimension of 3 for quantum SU(2)