A variety of techniques used to model metabolic networks are examined, both kinetic (ODE) models and flux balance (FB) models. These models are applied to a case study network describing CO and CO2 metabolism in Clostridium autoethanogenum, bacteria which can produce both ethanol and butanediol from a source of carbon monoxide. ODE and FB methods are also used to model a variety of simpler networks. By comparing the results from these simpler networks, the strengths and weaknesses of each examined method are highlighted, and ultimately, insight is gained into the conclusions that can be drawn from each model. ODE models have commonly been used to model metabolism in both in vivo and in vitro contexts, allowing the dynamic behaviour of wildtype bacteria to be examined, as well as that of mutants. An ODE model is formed for the C. autoethanogenum network. By exploring a range of parameter schemes, the possible long timescale behaviours of the model are fully determined. The model is able to exhibit both steady states, and also states in which metabolite concentrations grow indefinitely in time. By considering the scalings of these concentrations in the long timescale, six different non-steady behaviours are categorised and one steady. For a small range of parameter schemes, the model is able to exhibit both steady and unsteady behaviours in the long timescale, depending on initial conditions. FB methods are also applied to the same network. First flux balance analysis (FBA) is used to model the network in steady state. By imposing a range of constraints on the model, limits on levels of flux in the network that are required for a steady-state are found. In particular, boundaries on the ratio of inputs into the network are calculated, outside of which steady states cannot exist. Comparing the steady state regions predicted by FBA and our ODE model, it is found that the FBA model predicts a wider range of conditions leading to steady state. FBA is only able to observe a network in steady state, so an extension of FBA, known as dynamic flux balance analysis (dFBA), is used to examine non-steady-state behaviours. dFBA predicts similar long term non-steady behaviour to the ODE models, with states in which concentrations of some metabolites are able to grow indefinitely in time. These dFBA states do not precisely match those found by the ODE model, and states that cannot be observed in the ODE model are also found, suggesting that other ODE models for the same network could exhibit different long timescale behaviours. The examples considered clarify the strengths and weaknesses of each approach and the nature of insight into metabolic behaviour each provides.

A class of diffeomorphism invariant gauge theories is studied. The action for this class of theories can be formulated as a generalisation of the well known topological BF-theories with a potential for the B-field or in a pure connection formulation. When the gauge group is chosen to be SU(2) the theory describes gravity. For a larger gauge group G one gets a unified model of gravity and Yang-Mills fields. A background for the theory is chosen which breaks the gauge group G by selecting in it a preferred SU(2) subgroup which describes the gravitational sector. The Yang-Mills sector is described by the part of the gauge group that commutes with this SU(2). Thus, when the action is expanded around this background the spectrum of the linearised theory consists of the usual gravitons plus Yang-Mills fields. In addition, there is a set of massive scalar fields that are charged both under the gravitational and Yang-Mills subgroups. The latter sector is described by the part of the gauge group that does not commute with SU(2). A fermionic Lagrangian is also proposed which can be coupled to the BF plus potential formulation.

The Minkowski question mark function F(x) arises as a real distribution function of rationals in the Farey (alias, Stern-Brocot or Calkin-Wilf) tree. In this thesis we introduce its three natural integral transforms: the dyadic period function G(z), defined in the cut plane; the dyadic zeta function zeta_M(s), which is an entire function; the characteristic function m(t), which is an entire function as well. Each of them is a unique object, and is characterized by regularity properties and a functional equation, which reformulates in its own terms the functional equation for F(x). We study the interrelations among these three objects and F(x). It appears that the theory is completely parallel to the one for Maass wave forms for PSL_2(Z). One of the main purposes of this thesis is to clarify the nature of moments of the Minkowski question mark function.

We propose a new type of state sum model for two-dimensional surfaces that takes into account topology and spin. The definition used - new to the literature - provides a rich class of extended models called spin models. Both examples and general properties are studied. Most prominently, we find this type of model can depend on a surface spin structure through parity alone and we explore explicit cases that feature this behaviour. Further directions for the two dimensional world are analysed: we introduce a source of new information - defects - and show how they can enlarge the class of spin models available.

This thesis considers the registration of shapes, estimation of shape variability and the statistical modelling of human brain magnetic resonance images (MRI). Current shape registration techniques, such as Procrustes analysis, superimpose shapes in order to make inferences regarding the mean shape and shape variability. We apply Procrustes analysis to a subset of the landmarks and give distributional results for the Euclidean distance of a shape from a template. Procrustes analysis is then generalised to minimise a Mahalanobis norm, with respect to a symmetric, positive denite matrix, and the weighted Procrustes estimators for scaling, rotation and translation obtained. This weighted registration criterion is shown, through a simulation study, to reduce the bias and error in maximum likelihood estimates of the mean shape and covariance matrix compared to isotropic Procrustes. A Bayesian Markov chain Monte Carlo algorithm is also presented and shown to be less sensitive to prior information. We consider two MRI data sets in detail. We examine the first data set for large-scale shape dierences between two volunteer groups, healthy controls and schizophrenia patients. The images are registered to a template through modelling the voxel values and we maximise the likelihood over the transformation parameters. Using a suitable labelling and principal components analysis we show schizophrenia patients have less brain asymmetry than healthy controls. The second data set is a sequence of functional MRI scans of an individual's motor cortex taken while they repeatedly press a button. We construct a model with temporal correlations to estimate the trial-to-trial variability in the haemodynamic response using the Expectation-Maximisation algorithm. The response is shown to change with task and through time. For both data sets we compare our techniques with existing software packages and improvements to data pre-processing are suggested. We conclude by discussing potential areas for future research.

We analyse the perturbative expansion of knot invariants related with infinite dimensional representations of sl(2,R) and the Lorentz group taking as a starting point the Kontsevich Integral and the notion of central characters of infinite dimensional unitary representations of Lie Groups. The prime aim is to define C-valued knot invariants. This yields a family of C([h])-valued knot invariants contained in the Melvin-Morton expansion of the Coloured Jones Polynomial. It is verified that for some knots, namely torus knots, the power series obtained have a zero radius of convergence, and therefore we analyse the possibility of obtaining analytic functions of which these power series are asymptotic expansions by means of Borel re-summation. This process is complete for torus knots, and a partial answer is presented in the general case, which gives an upper bound on the growth of the coefficients of the Melvin-Morton expansion of the Coloured Jones Polynomial. In the Lorentz group case, this perturbative approach is proved to coincide with the algebraic and combinatorial approach for knot invariants defined out of the formal R-matrix and formal ribbon elements in the Quantum Lorentz Group, and its infinite dimensional unitary representations.

We investigate generalizations of coherent states as a means of representing the dynamics of excitations of the superconducting ground state. We also analyse the propagation of generalized coherent state wave packets under the Bogoliubov-de Gennes Hamiltonian. The excitations of the superconducting ground state are superpositions of electron and hole quasi-particles described by the Bogoliubov-de Gennes equations, that can only exist at energies outside the band gap. A natural generalization relevant to the excitations of the superconducting ground state is the tensor product of canonical and spin coherent states. This state will quickly become de-localized on phase space under evolution by the Bogoliubov-de Gennes Hamiltonian due to the opposite velocities of the quasi-spin components. We therefore define the electron-hole coherent states which remain localised on phase space over longer times. We show that the electron-hole coherent states though entangled retain many defining features of coherent states. We analyse the propagation of both product and electron hole coherent states in a superconductor with a spatially homogeneous superconducting band gap. The dispersion relation indicates that wavepackets defined on the band gap have a zero group velocity, but we will show that interference effects can create states on the band gap that propagate at the Fermi velocity. We also consider the two semiclassical, short wavelength regimes, hbar→0$ and the large Fermi energy limit mu→infinity. In general these limits produce behaviour analogous to the canonical coherent states except for isolated cases. Finally we analyse the dynamics of the Andreev Reflection of a Gaussian wavepacket incident on a discontinuous normal-superconducting interface. We show that restricting the energy bandwidth of the incident state inside the superconducting band gap precludes the wavepacket from fully entering the superconducting region. We again consider the two semiclassical regimes.

This thesis is broadly split into two parts. In the first part, simple state sum models for minimally coupled fermion and scalar fields are constructed on a 1-manifold. The models are independent of the triangulation and give the same result as the continuum partition functions evaluated using zeta-function regularisation. Some implications for more physical models are discussed. In the second part, the gauge gravity action is written using a particularly simple matrix technique. The coupling to scalar, fermion and Yang-Mills fields is reviewed, with some small additions. A sum over histories quantisation of the gauge gravity theory in 2+1 dimensions is then carried out for a particular class of triangulations of the three-sphere. The preliminary stage of the Hamiltonian analysis for the (3+1)-dimensional gauge gravity theory is undertaken.

Various aspects of three-dimensional spin foam models for quantum gravity are discussed. Spin foam models and graphical calculus are introduced via the Ponzano-Regge model for 3d gravity and some important properties of this model are described. The asymptotic formula for the 6j symbol found by Ponzano and Regge is generalised to include the Ponzano-Regge amplitude for triangulations of handlebodies. Some simple observables are computed in a model for fermions coupled to 3d gravity. The result is a sum over spin foam models with certain vertex amplitudes which are described. An explicit example is given and the vertex amplitudes expressed in terms of 6j symbols. Finally, a group field theory for this spin foam model is described.

This thesis collects my own and collaborative work I have been involved with finding localised systems in quantum field theory that are be useful for quantum information. It draws from many well established physical theories such as quantum field theory in curved spacetimes, quantum optics and Gaussian state quantum information. The results are split between three chapters. For the first results, we set-up the basic framework for working with quantum fields confined to cavities. By considering the real Klein-Gordon field, we describe how to model the non-uniform motion of a rigid cavity through spacetime. We employ the use of Bogoliubov transformations to describe the effects of changing acceleration. We investigate how entanglement can be generated within a single cavity and the protocol of quantum teleportation is affected by non-uniform motion. The second set of results investigate how the Dirac field can be confined to a cavity for quantum information purposes. By again considering Bogoliubov transformations, we thoroughly investigate how the entanglement shared between two cavities is affected by non-uniform motion. In particular, we investigate the role of the Dirac fields charge in entanglement effects. We finally analyse a \one-way-trip" of one of the entangled cavities. It is shown that different types of Dirac field states are more robust against motion than others. The final results looks at using our second notion of localisation, Unruh-DeWitt detectors. We investigate how allowing for a \non-point-like" spatial profile of the Unruh-DeWitt detector affects how it interacts with a quantum field around it. By engineering suitable detector-field interactions, we use techniques from symplectic geometry to compute the dynamics of a quantum state beyond commonly used perturbation theory. Further, the use of Unruh-DeWitt detectors in generating entanglement between two distinct cavities will be investigated.