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The Baum-Connes conjecture gives a description of the K-theory of the reduced group C*-algebra of a locally compact second countable group. In the case of a connected Lie group G, Connes reformulated the conjecture in terms of a deformation of G provided by a certain continuous field of C*-algebras. The conjecture is known to be true for complex semisimple Lie groups, and in 2008 Higson provided a new proof of this result, using Connes reformulation and an observation due to Mackey about the representation theories of a complex semisimple Lie group and an associated group called the Cartan motion group. In this thesis, we present and prove an analogue of the conjecture for the quantum group quantum SL(2, C) in the spirit of Connes reformulation and Higson's proof. In particular, we define a quantum version of Connes' field, which provides a deformation from quantum SL(2, C) to a quantum analogue of the Cartan motion group. We show that Mackey's observation carries over to the quantum setting, and we then prove an analogue of the conjecture using Higson's method. We also show there is compatibility between the Baum-Connes conjecture for SL(2, C) and our quantum result, in that we can construct a continuous field which encodes Connes' field and our quantum field, as well as a deformation of SL(2, C) to quantum SL(2, C) and a deformation of the Cartan motion group to the quantum Cartan motion group.

The patellofemoral joint (PFJ) is a common source of problems in Orthopaedics and is the source of poorly defined pain and poor function in both normal and replaced knees. Before problems of the PFJ can be fully comprehended a better understanding of the basic form and function of the PFJ is required. The aim of this thesis therefore is to investigate the shape and kinematics of the PFJ and their inter-relations in both normal and replaced knees. The first part of this thesis was concerned with the shape (or form) of the PFJ. Species from the human ancestry over the previous 400 million years were assembled and measurements taken from three dimensional, CT reconstructions, allowing the descriptions of evolutionary changes in the shape, and orientation of the patellofemoral joint in relation to the tibiofemoral compartments. The study chronicled the dramatic changes that occurred in relation to the adoption of the erect bipedal hominin stance which has resulted in varied anatomy at the PFJ, predisposing it to a wide range of pathologies. The articular surface geometry of normal human patellofemoral joints (bone and cartilage) were compared with those of total knee replacements, and patellofemoral joint replacements. Mapping of the trajectory of the apex of the trochlea groove revealed significant differences between native and replaced knees, with the trajectory being orientated laterally in normal knees and either centrally or medially in replaced knees. The second part of this work was concerned with the kinematics (or function) of the PFJ. With current technology it is impossible to measure coronal plane PFJ kinematics with any accuracy in both native and replaced knees. A novel method was developed combining Motion Analysis and UltraSound (MAUS). Validation experiments were undertaken that demonstrated acceptable error (1.8 mm). The MAUS technique was used to show statistically significant differences between the coronal plane kinematics of the patella in normal and replaced knees. In particular in some arthroplasty patients, the patella tracked in the opposite direction to that in normal subjects. The abnormal kinematics were a manifestation of non-anatomical joint replacements. This demonstrates that form and function are closely related. The interaction between form and function in the knee was further investigated using patients with anterior knee pain. Assessment was made of the relationship between patellar subluxation and multiple bony, cartilaginous and soft-tissue factors potentially predisposing to subluxation. The percentage of engagement of the patella in the trochlear groove in knee extension showed the strongest relationship with subluxation, with subjects less than 30% engaged tending to subluxate. This suggests that the most important factor in preventing subluxation is patellar engagement. A clinical study is now required to assess the effect of surgery aimed at improving engagement. The detailed insights into the variability of form and function in the PFJ obtained throughout this thesis will help address pathology in the native knee and guide decisions for new designs of knee replacements. A novel technology has been developed here for measuring patella kinematics which has great potential for future research. The MAUS technique will provide a clinical investigative tool and allow investigation into kinematic abnormalities in other joints.