Ada Masters

We introduce to spectral noncommutative geometry the notion of tangled spectral triple, which encompasses the anisotropies arising in parabolic geometry as well as the parabolic commutator bounds arising in so-called "bad Kasparov products". Tangled spectral triples incorporate anisotropy by replacing the unbounded operator in a spectral triple that mimics a Dirac operator with several unbounded operators mimicking directional Dirac operators. We allow for varying and dependent orders in different directions, controlled by using the tools of tropical combinatorics. We study the conformal equivariance of tangled spectral triples as well as how they fit into K-homology by means of producing higher order spectral triples. Our main examples are hypoelliptic spectral triples constructed from Rockland complexes on parabolic geometries; we also build spectral triples on nilpotent group C*-algebras from the dual Dirac element and crossed product spectral triples for parabolic dynamical systems.

We extend unbounded Kasparov theory to encompass conformal group and quantum group equivariance. This new framework allows us to treat conformal actions on both manifolds and noncommutative spaces. As examples, we present unbounded representatives of Kasparov's γ-element for the real and complex Lorentz groups and display the conformal SL_q(2)-equivariance of the standard spectral triple of the Podleś sphere. In pursuing descent for conformally equivariant cycles, we are led to a new framework for representing Kasparov classes. Our new representatives are unbounded, possess a dynamical quality, and also include known twisted spectral triples. We define an equivalence relation on these new representatives whose classes form an abelian group surjecting onto KK. The technical innovation which underpins these results is a novel multiplicative perturbation theory. By these means, we obtain Kasparov classes from the bounded transform with minimal side conditions.

This thesis presents a number of new approaches to the treatment of group actions in unbounded Kasparov theory. Its results are motivated by the desire to incorporate into spectral noncommutative geometry several formerly problematic examples. We extend unbounded Kasparov theory to encompass conformal group and quantum group equivariance. We use this, along with tools from geometric group theory, to study the geometry of group C*-algebras and Fell bundles. We prove a nontriviality result for Kasparov modules built from group actions on CAT(0) spaces. We also study the geometry of group extensions using the unbounded Kasparov product.

We introduce a new multiplicative perturbation theory that enables us to treat conformal actions on both manifolds and noncommutative spaces. As examples, we present unbounded representatives of Kasparov's γ-element for the real and complex Lorentz groups and display the conformal SL_q(2)-equivariance of the standard spectral triple of the Podleś sphere. In pursuing descent for conformally equivariant cycles, we are led to a new framework for representing Kasparov classes. Our new representatives, conformally generated cycles, are unbounded, possess a dynamical quality, and also include known twisted spectral triples. We define an equivalence relation on these new representatives whose classes form an abelian group surjecting onto KK-theory.

We also develop a new framework for the treatment of parabolic features in noncommutative geometry, in the form of the notion of tangled cycle. Tangled cycles incorporate anisotropy by replacing the unbounded operator in a higher order cycle that mimics a Dirac operator with several unbounded operators mimicking directional Dirac operators, allowing for varying and dependent orders in different directions, controlled by a weighted graph. Our main examples of tangled cycles fit into two classes: hypoelliptic spectral triples constructed from Rockland complexes on parabolic geometries and Kasparov product spectral triples for nilpotent group C*-algebras and crossed product C*-algebras of parabolic dynamical systems.