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This thesis serves as a self-contained review of some recent advances in the study of three-dimensional N=4 quiver gauge theories and their Coulomb branch moduli spaces in particular. Our investigation leverages and develops the Hilbert series and abelianisation approaches and finds them mutually complementary. Their synthe- sis provides an explicit construction of the Coulomb branch with several desirable properties: for example, the global symmetry is made explicit and any complex mass deformation is easily derived. Moreover, it naturally handles two generalisations of quiver gauge theories: non-simply laced quivers and previously unknown wreathed quivers. Many concrete examples are provided to illustrate the concepts.

The currently accepted standard model of cosmology uses general relativity with a ΛCDM matter content to describe the universe on the largest scales. It is an overwhelmingly successful theory, consistent with all observational tests. Despite this, theoretically unsatisfying elements to the theory exist and these have motivated various theories of modified gravity that challenge general relativity. In order to pass the stringent observational tests on a solar system level, the deviation of modified gravities from general relativity must be suppressed. This is known as screening, and different modified gravity theories use different screen- ing mechanisms. We motivate modifying gravity and the need for screening mechanisms. Three explicit models of modified gravity which exhibit screening are presented. These are the Galileon, K-mouflage and Chameleon models. In this thesis we investigate several aspects of these models. We study astrophysical black holes in Galileon and K-mouflage theories. The no-hair theorem of General Relativity states that, under certain specific assumptions, the scalar field is trivial around a black hole. The assumptions going into the no-hair theorem are the absence of external matter and time independence. An astrophysical black hole typically has an accretion disk, so automatically circumvents the no-hair theorem. We display the scalar field profile around such black holes, compute the fifth force and demonstrate that the work done by the fifth force is small compared to the energy lost due to radiation in General Relativity. Further we drop the assumption of a static black hole and investigate the time- dependent solution of the scalar field in both theories. We find exact time-dependent vacuum K-mouflage black hole solutions and further consider time-dependent solutions with an ac- cretion disk. For K-mouflage the solution is similar to the time-independent one whereas the Galileon theories solutions resembles closely the time-dependent vacuum solution. The most general coupling of the scalar field to matter contains both a conformal and disformal term. We investigate the effect of a disformal coupling in K-mouflage theories, calculating the cosmological background evolution of the theory and extending our results on the behaviour of the scalar field around a black hole to include the disformal coupling. We find that large regions of the parameter space provide only percent level deviations from the ΛCDM evolution, despite qualitative differences from the conformal-only case. Often spherical symmetry is assumed to demonstrate the screening of K-mouflage theories. We present preliminary calculations exploring the effect the shape of a source object has on the scalar field it generates. We find that the shape dependence is similar to that of the D-BIon, another theory that screens when the first derivative of the field is large. In particular we find that screening is strongest for planar objects, in contrast to Galileon theories for which screening is entirely absent. We move on from K-mouflage theories to consider Chameleon theories. We propose a logarithmic potential, which differs from the standard power law potential usually assumed, and use observational data to constraint the parameter space of the theory.

In this thesis, we present a systematic method for determining the conditions on the parameters in the action of a parity-preserving gauge theory of gravity about a Minkowski background for it to be free of ghost or tachyon particles. The approach naturally accommodates critical cases in which the parameter values satisfying some critical conditions causing changes of particle contents and may lead to additional gauge invariances. In Chapter 1, we give an overall introduction to the field. We then introduce the systematic method in Chapter 2. The method is implemented as a computer program, and the details of its implementation are presented in Chapter 3. In Chapter 4, we apply the method to investigate the particle content of parity-conserving Poincaré gauge theory (PGT⁺). We find 450 critical cases that are free of ghosts and tachyons and compare the no-ghost-and-tachyon conditions of some critical cases with literature. We also examine the power-counting renormalisability of some of the critical cases of PGT⁺ and clarify the treatment of non-propagating modes in determining whether a theory is power-counting renormalisable (PCR) in Chapter 5. We identify 58 of the ghost and tachyon free PGT⁺ critical cases that are also PCR, of which seven have 2 massless degrees of freedom (d.o.f.) in propagating modes and a massive 0⁻ or 2⁻ mode, 12 have only 2 massless d.o.f., and 39 have only massive mode(s). In chapter 6, we analyse parity-preserving Weyl gauge theory (WGT⁺) in a similar way. Within a subset of WGT⁺, we find 168 critical cases that are free of ghosts and tachyons. We further identify 40 of these cases that are also PCR. Of these theories, 11 have only massless tordion propagating particles, 23 have only a massive tordion propagating mode, and 6 have both. We also repeat our analysis for PGT⁺ and WGT⁺ with vanishing torsion or curvature, respectively. In Chapter 7, we summarise the contents in this thesis and suggest some future work.

From water boiling to the behaviour of magnets, the world around us is often described using the language of Effective Field Theory. This powerful framework is often employed to model a system in terms of its low energy degrees of freedom without much thought being given to the higher energy origins of the model. In this thesis we shall attempt to turn this idea on its head by constructing effective theories by beginning from the high energy theory and then arriving at the low energy description. Our main aim shall be to address this issue in the context of deriving the EFT of perturbations around a background solution of Type IIA Supergravity given by Crampton, Pope and Stelle that involves the non-compact hyperbolic space H 2,2 . Studying the subject of EFTs in the context of Type IIA Supergravity presents it’s own challenges as this theory exists in ten spacetime dimensions, whilst our observed universe is four dimensional. We shall present how to address this issue of dimensional mismatch using Kaluza-Klein methods and discuss aspects of the resulting lower dimensional field with an eye towards achieving lower dimensional diffeomorphism invariance. This treatment shall lead us to the conclusion that a previously undiscovered Stueckelberg field must be accounted for and dealt with if one is to appropriately understand the lower dimensional theory. We also exhibit a generalisation of the solutions of Crampton, Pope and Stelle to include time dependence. Such solutions may have applications to cosmological models.

The work presented in this thesis focuses on the ultra-slow dynamics of localised modes (breathers) for the Discrete Nonlinear Schrödinger Equation (DNLSE), and on the mechanisms which trigger their astronomically slow decays. The thesis starts with a literature review of breathers and the DNLSE, giving information about the theoretical background of the project. It then continues with a description of the numerical methods used throughout the project, and of the general behaviour of the DNLSE under numerical integration. Discrete breathers in Bose-Einstein Condensates in optical lattices or in arrays of optical wave-guides oscillate with frequencies which are much higher than those present in the spectrum of the background. Hence, the interaction between localized breathers and their surroundings is extremely weak, allowing the development of a multiple-time scale perturbation expansion, which is presented in Chapter 4. This analysis will predict a lower bound of the breather drift times and will explain the topological differences between breathers in dimers, trimers and in spatially extended one-dimensional lattices even in the presence of transport from boundary heat baths. These analytical boundaries hold true for lattices of any length, due to the highly localised nature of breathers. We later look closer at the exceedingly slow thermalisation occurring in the DNLSE.We provide evidence that the breather norm is an adiabatic invariant, and this freezes the dynamics of a tall breather. Consequently, relaxation proceeds via rare events,where energy is suddenly released towards the background. A more detailed investigation of these events is provided in Chapter 5. In Chapter 6, a Heterogeneous Multiscale Method is introduced. We take advantage of the fast frequency of the breather oscillation to perform averaging, and to build a numerical integrator which behaves similarly to a predictor-corrector propagator,speeding up the running times by a factor of at least 5.

Understanding the properties of defects, interfaces and other heterogeneities in metal oxides is crucial for enabling new applications. Some properties can be explored in a computer simulation using classical interatomic potentials (IPs) - analytic functions of nuclear positions that provide interatomic forces. In this thesis I have focused on developing, testing and analysing an ionic IP which combines variable-charge (QEq) and polarisable-ion (PI) models, allowing ions' charges and induced dipole moments to vary in response to their changing environments. We aim to improve the limited transferability to heterogeneous environments of the current state-of-the-art, PI-based, IPs. For several bulk and heterogeneous systems, past applications of QEq-based IPs showed no improvement over fixed-charge IPs - at variance with common expectations. Two reasons for this have been suggested in our work - in some systems ionic charges are essentially-constant, in other systems the bonding doesn't adhere well to the limit of the QEq model. Using density functional theory and the QEq-PI IP we calculated and analysed the charge distributions and interatomic forces in BaTiO3 at finite temperature, in a pristine crystal and near neutral O (VO) and Ti (VTi) vacancies. Ionic charges were found to be essentially-constant in the defect-free BaTiO3, but to deviate from their bulk values near the vacancies. The improvement of QEq-PI IP over fixed-charge IP was greater for the system with VO than with VTi, which was explained by the bonding near VO adhering better to the limit of this model. For systems with VO the QEq-PI IP demonstrated a marked improvement in accuracy and transferability over the fixed-charge PI IP; this is the first time, to our knowledge, that a clear improvement over PI IPs has been demonstrated.

The aim of this thesis is the analysis of the spectral stability of plane wave solutions of the 3-wave resonant interaction (3WRI) model, when such solutions undergo localised perturbations. For the first time, we provide a comprehensive topological classification of the spatial stability spectra with respect to the parameters space and the gain functions associated to any stability spectrum. We find that all the stability spectra of the coupled nonlinear Schrödinger (CNLS) system are enclosed in those of the 3WRI system. The topological features of the CNLS stability spectra are gaps on the real axis (solutions not bounded in space), and branches and loops off the real axis (solutions bounded in space which can be linearly unstable in time). New topological components exist in the stability spectra of the 3WRI model: we name such components twisted loops. They are associated with explosive instability (the corresponding solutions blow up in a finite time) and their gain function is non-zero in a whole neighbourhood of the origin. We observe that the gain function associated to the branches is non-zero at low wave numbers, symmetrically located with respect to the zero-value of the wave number, but it is zero at the origin of the plot (linear instability of baseband-type). The gain function associated to the loops is non-zero only away from the origin (linear instability of passband-type). We show that the plane wave solutions of the 3WRI model are linearly unstable in time for any choice of the physical parameters, including those ones associated to the solutions that are explosive. Thus, there is linear instability of the plane wave for any choice of the physical parameters corresponding to a positive gain-function. Finally, we conjecture that the existence of branches in the stability spectra is a necessary condition for the onset of rogue waves ascribable to rational or semi-rational solutions obtained by Darboux Dressing Transformation. Indeed, we observe numerically linear in- stability of plane waves with the subsequent generation of localised structures whose onset, as a result of the perturbation of plane waves, must be investigated further due to the dispersionless nature of the 3WRI system.

Recent experimental progress in the preparation and control of quantum systems has brought to light the importance of Quantum State Tomography (QST) in validating the results. In this thesis we investigate several aspects of QST, whose central problem is to devise estimation schemes for the recovery of an unknown state, given an ensemble of n independent identically prepared systems. The key issues in tackling QST for large dimensional systems is the construction of physically relevant low dimensional state models, and the design of appropriate measurements. Inspired by compressed sensing tomography, in chapters 4, 5 we consider the statistical problem of estimating low rank states (r ≪ d) in the set-up of Multiple Ions Tomography (MIT), where r and d are the rank and the dimension of the state respectively. We investigate how the estimation error behaves with a reduction in the number of measurement settings, compared to ‘full’ QST in two setups - Pauli and random bases measurement designs. We study the estimation errors in this ‘incomplete’ measurement setup in terms of a concentration of the Fisher information matrix. For the random bases design we demonstrate that O(r logd) settings suffice for the mean square error w.r.t the Frobenius norm to achieve the optimal O(1/n) rate of estimation. When the error functions are locally quadratic, like the Frobenius norm, then the expected error (or risk) of standard procedures achieves this optimal rate. However, for fidelity based errors such as the Bures distance we show that no ‘compressive’ recovery exists for states close to the boundary, and it is known that even with conventional ‘full’ tomography schemes the risk scales as O(1/√n) for such states and error functions. For qubit states this boundary is the surface of the Bloch sphere. Several estimators have been proposed to improve this scaling with ‘adaptive’ tomography. In chapter 6 we analyse this problem from the perspective of the maximum Bures risk over all qubit states. We propose two adaptive estimation strategies, one based on local measurements and another based on collective measurements utilising the results of quantum local asymptotic normality. We demonstrate a scaling of O(1/n) for the maximum Bures risk with both estimation strategies, and also discuss the construction of a minimax optimal estimator. In chapter 7 we return to the MIT setup and systematically compare several tomographic estimators in an extensive simulation study. We present and analyse results from this study, investigating the performance of the estimators across various states, measurement designs and error functions. Along with commonly used estimators like maximum likelihood, we propose and evaluate a few new ones. We finally introduce two web-based applications designed as tools for performing QST simulations online.

The rise of quantum technology has put control at the centre of advancements in quantum mechanics. The union of quantum mechanics with mathematical control theory is a meeting that is leading to a much deeper insight into our interaction with the bizarre properties of quantum theory. Often, the study of discrete variable systems is the focus for making this union. Here, we look at how control theory may be applied to the continuous variable theory of Gaussian states. Special emphasis is given to control of the covariance matrix of these states, as it is here that we find the entanglement and entropic properties of the state. We begin by exploring some initial results for the geometry of Gaussian states, revealing different manifold structures dependent on symplectic eigenvalue degeneracy. In this geometrical setting a proposal for an extension of Williamson's theorem is put forward and partially completed. It is often interesting to look at restricted sets of Hamiltonians and ask what transformations can be performed with concatenations of their corresponding unitaries. Controllable systems are those for which the entire group of interest is possible to enact. We explore an uncontrollable system in a single mode and give a physical analysis as to why it behaves this way. This leads to ideas to move forwards for a necessary and sufficient condition for control on the symplectic group that has been conjectured since 1972. Later, we transfer to the question of open dynamics. We focus on a particular and ubiquitous channel known as 'lossy' or 'the attenuation channel'. An equation is derived describing the evolution for the symplectic invariants of a Gaussian state undergoing such dynamics. The equation of the former chapter is used to explore the evolution of entropy and entanglement. Optimal protocols are developed for the manipulation of these properties undergoing lossy dynamics.

This thesis has studied spin chains with Lattice supersymmetry. Connections have been found between Lattice supersymmetry and integrability. In particular, the construction of lattice supercharge has been incorporated into the construction of the quantum inverse scattering method. The open XXZ chain Hamiltonian has been studied, and the construction of the supersymmetric open XXZ chain Hamiltonian by C. Hagendorf and J. Lienardy has been extended. The integrable open XXZ chain Hamiltonian with generic boundary matrices has been studied using TQ equations proposed by W.Yang, R. I. Nepomechie and Y.Zhang, and the integrable open XXZ chain Hamiltonian with diagonal boundary matrices has been studied using the algebraic Bethe ansatz developed by E. K. Sklyanin. For these two types of Hamiltonians, conditions on the boundary parameters have been found where Hamiltonians will have a symmetry in their Bethe equations. This symmetry induces a map SBethe, which shares certain properties with the lattice supercharge SN. An open XXZ chain Hamiltonian has been found which has symmetry with respect to both SN and SBethe. This Hamiltonian is denoted as H(N)XXZ,diagonal. Using the numerical method developed by B.McCoy, the action of the map SBethe of H(N)XXZ,diagonal has been calculated, and the result has confirmed numerically that SBethe is proportional to the lattice supercharge SN. Motivated by this connection, a family of commutation relations between the lattice supercharge of H(N)XXZ,diagonal and sub-matrices of the monodromy matrix corresponding to H(N)XXZ,diagonal have been found and proved using diagrammatic method. Similar commutation relations have also been obtained for the open XXZ chain Hamiltonian with anti-diagonal boundary matrices.

Understanding the magnetic properties of matter is of paramount interest from a principal and technological point of view. For example, transition-metal compounds tuned to a magnetic quantum phase transition often form unconventional superconductivity and exhibit non-Fermi liquid behaviour. The discovery of novel magnetic states such as the skyrmion lattices might lead to new magnetic storage technologies if a full understanding of the magnetic, electronic, and structural material properties is obtained. This thesis presents results on composition and pressure-tuned magnetic compounds: the itinerant ferromagnet Nb(1-x)Fe(2+x) and the insulating helimagnet Cu2OSeO3. The continuous ferromagnetic transition in Fe-rich Nb(1-x)Fe(2+x) can be suppressed and a ferromagnetic quantum phase transition can be reached by replacing Fe with Nb. Nb(1-x)Fe(2+x) follows a trend that nature tends to avoid ferromagnetic quantum critical points. They are typically either replaced by a first order transition or masked by unconventional superconductivity. Nb(1-x)Fe(2+x) offers a predicted third scenario: masking by a spin density wave phase. With neutron diffraction, the extension of the spin density wave phase in the field-temperature phase diagram has been determined. Additionally, cold neutron spectroscopy revealed low-energy magnetic excitations at zero field including soft quasi-elastic scattering in an extended region in reciprocal space, which reflects the proximity of the tuned system to different types of magnetic order. The recently discovered skyrmion lattice in Cu2OSeO3 has created particular interest in this compound as its insulating properties allow the skyrmion lattice to be moved by an electrical field. Moreover, its magnetic ordering temperature increases with pressure, in contrast to the metallic helimagnets. In order to obtain information on the microscopic changes that might be responsible for the strengthening of the magnetic interactions, the structural properties of Cu2OSeO3 have been examined by X-ray diffraction. These studies reveal unusual changes in some Cu-O distances in the low pressure phase that undergoes an irreversible structural transformation to a yet unsolved high-pressure phase at 9.7 GPa. Complementary Raman scattering and infrared absorption measurements confirm subtle changes in the low pressure phase and a marked change in optical properties accompanies the structural phase transition.

In this thesis two questions of equilibrium and non-equilibrium properties in many-body quantum mechanical systems are investigated. The first part is focused on probability distributions of quantum observables in many-body quantum systems. After an introduction to probability distributions, full counting statistics and the transverse field Ising model we give a brief overview over related experiments and then derive an expression for the probability distribution of the transverse field magnetization of a finite subsystem in any Gaussian state. We study the probability distribution in ground and thermal states as well as in a non-equilibrium setting after a quantum quench. We find an analytic expression for the time evolution after the quench and compare to numerics. The second part of the thesis is concerned with the stability of exact quasi-particle excitations of an integrable model after weak integrability breaking perturbations are introduced. For this we first discuss the stability of excitations in integrable systems and then give an introduction to the Heisenberg XXX-model in a magnetic field. After constructing exact excitations we calculate the decay rate in leading order perturbation theory using methods of integrability.

This thesis presents a series of studies into the electronic, thermal and thermoelectric properties of molecular junctions containing single organic molecules. The exploration and understanding the electronic and phononic characteristics of molecules connected to metallic leads is a vital part of nanoscience if molecular electronics is to have a future. This thesis documents a study for various families of organic and organometallic molecules, studied using a combination of density functional theory (DFT), which is implemented in the SIESTA code, and the Green’s function formalism of transport theory. The main results of this thesis are as follows: To elucidate the nature of the high and low conductance groups observed in break-junction measurements of single 4,4-bipyridine molecules, I present a combined experimental and theoretical study of the electrical conductance of a family of 4,4-bipyridine molecules, with a series of sterically-induced twist angles α between the two pyridyl rings. I show that their conductances are proportional to cos2(α), confirming that pi-pi overlap between adjacent rings plays a dominant role. Since both peaks exhibit a cos2 (α) dependence of conductance on torsion angle, this is evidence that the high and low conductances correspond to molecular orientations within the junctions, in which the electrical current passes through the C-C bond linking the pi systems of the two rings. Furthermore, this result demonstrates that the Fermi energy is located within the HOMO-LUMO gap and not close to a transmission resonance. A theoretical investigation into the Seebeck coefficient in pi-stacked molecular junctions is performed using a first principles quantum transport method. Using oligo (phenyleneethynylene) (OPE)-type molecules as a model system, I show that quantum interference produces anti-resonances in the gap between the HOMO and LUMO resonances and the stacking geometry can control the position of this quantum interference feature. The shifting of this resonance enhances the thermopower S is expected when the junction is tuned through a node in the transmission function. We found supramolecular π-π interactions between two molecules changed the sign of thermopower. I have investigated a family molecules with various side branched atoms to study the electron and phonon transport through nanoscale molecular junctions, with a view to understanding the performance thermoelectric materials. My calculations focus on the effect of heteroatoms formed from C, Si, Ge, and Sn on the thermal phonon conductance, electrical conductance, and Seebeck Coefficient. I also examine how the thermoelectric figure of merit is affected by side branched atoms, as the bond length and mass play an important role in determining the thermal phonon conductance of molecular wires. Due to the rigid nature of C-side branching, the thermal phonon conductance decreases monotonically with the bond length and mass, whereas thermal phonon conductance with Si-side branches increases with the length of the bond and mass. The low thermal conductance kel with S-bridging, combined with their higher thermopower and higher electrical conductance leads to a maximum thermoelectric figure of merit of ZT = 1.76, which is several orders of magnitude higher than that of bridges.

In this thesis I explore two pertinent avenues of the $AdS/CFT$ correspondence: the rich, pragmatic context of non-relatavistic holography and the story of holographic multiboundary wormholes and their relation to the profound interplay between bulk geometry and boundary entanglement. In chapter \ref{intro}, I introduce the $AdS/CFT$ correspondence and review the key ideas that motivate and underlie the work in subsequent chapters. In chapter \ref{schleq2} we consider the development of a holographic dictionary for asymptotically locally Schr{\"o}dinger spacetimes for $z < 2$ in a massive vector model in various spatial dimensions. We carry out a linearised analysis of bulk perturbations and identify the boundary data as sources and vevs for the dual stress-energy complex. We verify that a sensible asymptotic expansion of bulk perturbations in sub-leading powers of $r$ exists by expanding them in eigenvalues of the boundary dilatation operator. The third chapter extends the work of the Chapter \ref{schleq2} to the case with $z=2$ in the massive vector model, in various dimensions, where the additional lightlike direction is regarded as internal from the boundary point of view, qualitatively unlike the $z<2$ case. Chapter \ref{hot} considers the entanglement structure of states holographically dual to multiboundary wormholes in the high-temperature limit, in which the thermal scale associated to each boundary is much larger than the $AdS$ scale. We find that the entanglement structure in this limit is almost entirely bipartite in this regime. The fifth chapter investigates the extent to which the results of chapter \ref{hot} generalise to regions of small moduli. We utilise heuristic tensor network methods to construct tensor network models of multiboundary wormhole states built by sewing tensors to Coxeter tilings and their quotients. We find in several cases that we can construct holographic states representing multiboundary wormhole geometries for which the entanglement structure is mostly, or almost entirely bipartite.

In this thesis the finite element method is used to carry out simulations of the impact of elastic-plastic spheres with rigid surfaces and elastic half-spaces. The detailed results that these simulations provide are used to develop new normal and tangential force-displacement models for elastic-plastic materials that are accurate enough to be used in the discrete element method to simulate large numbers of particles. The new force-displacement models are suitable for a wide range of material properties and initial conditions. They perform better than existing force-displacement models from the literature when compared to the results of finite element simulations. The new force-displacement models are used in discrete element simulations of bulk compaction of plastic pellets. These simulations are compared to the results of bulk compaction experiments. Despite some weaknesses in the experimental set up, the simulations match the experimental results until the bed is compressed to 93% of its original height. Beyond this level of compression the results diverge. The simulations are carried out using the new force-displacement models and existing force-displacement models. The different force- displacement models lead to significant differences in the force distributions throughout the pellet bed during compaction, highlighting the importance of the force-displacement model to the results of discrete element simulations.

This thesis takes steps towards the development of a systematic account of the relationships between SUSY quiver gauge theories and the structures of their moduli spaces. Highest Weight Generating functions (“HWGs”), which concisely encode the field content of a moduli space, are introduced and developed to augment the established plethystic techniques for the construction and analysis of Hilbert series (“HS”). HWGs are shown to provide a faithful means of decoding and describing HS in terms of their component fields, which transform in representations of Classical and/or Exceptional symmetry groups. These techniques are illustrated in the context of Higgs branch quiver theories for SQCD and instanton moduli spaces, as a prelude to an account of the quiver theory constructions for the canonical class of moduli spaces represented by the nilpotent orbits of Classical and Exceptional symmetry groups. The known Higgs and/or Coulomb branch quiver theory constructions for nilpotent orbits are systematically extended to give a complete set of Higgs branch quiver theories for Classical group nilpotent orbits and a set of Coulomb branch constructions for near to minimal orbits of Classical and Exceptional groups. A localisation formula (“NOL Formula”) for the normal nilpotent orbits of Classical and Exceptional groups based on their Characteristics is proposed and deployed. Dualities and other relationships between quiver theories, including A series 3d mirror symmetry, are analysed and discussed. The use of nilpotent orbits, for example in the form of T(G) quiver theories, as building blocks for the systematic (de)construction of moduli spaces is illustrated. The roles of orthogonal bases, such as characters and Hall Littlewood polynomials, in providing canonical structures for the the analysis of quiver theories is demonstrated, along with their potential use as building blocks for more general families of quiver theories.

In this thesis we study gauge/gravity duals in the 5d/6d AdS/CFT correspondence. We start with field theories defined on squashed five-spheres with SU(3) × U(1) symmetry. These five-sphere backgrounds are continuously connected to the round sphere. We find a one-parameter family of 3/4 BPS deformations and a two-parameter family of (generically) 1/4 BPS deformations. The gravity duals are constructed in Euclidean Romans F(4) gauged supergravity in six dimensions, and uplift to massive type IIA supergravity. We holographically renormalize the Romans theory, and use our general result to compute the renormalized on-shell actions for the solutions. The results agree perfectly with the large N limit of the dual gauge theory partition function, which we compute using large N matrix model techniques. In addition we compute BPS Wilson loops in these backgrounds, both in supergravity and in the large N matrix model, again finding precise agreement. We conjecture a general formula for the partition function on any five-sphere background, which for fixed gauge theory depends only on a certain supersymmetric Killing vector. We then proceed to study Euclidean Romans supergravity in six dimensions with a non-trivial Abelian R-symmetry gauge field. We show that supersymmetric solutions are in one-to-one correspondence with solutions to a set of differential constraints on an SU(2) structure. As an application of our results we (i) show that this structure reduces at a conformal boundary to the five-dimensional rigid supersymmetric geometry previously studied, (ii) find a general expression for the holographic dual of the VEV of a BPS Wilson loop, matching an exact field theory computation, (iii) construct holographic duals to squashed Sasaki-Einstein backgrounds, again matching to a field theory computation, and (iv) find new analytic solutions to the squashed five-sphere background. We also analyse the classification of gravity duals with zero B-field.

The work presented in this thesis was inspired by precedent results on the Gaudin models (which are integrable) for spins-12 only which, by a change of variables in the algebraic Bethe equations, manage to considerably simplify the numerical treatment of such models. This numerical optimisation is carried out by the construction of determinants, only depending on the previously mentioned variables, for every scalar products appearing in the expression of the mean value of an observable of interest at a given time. By showing it is possible to use the Quantum Inverse ScatteringMethod (QISM), even when the vacuum state is not eigenstate of the transfer matrix, the previous results concerning spins-12 only are generalised tomodels including an additional spin-boson interaction. De facto, this generalisation opened different possible paths of research. First of all, we show that it is possible to further generalise the use of determinants for spin models describing the interaction of one spin of arbitrary normwith many spins-12. We give the method leading to the explicit construction of determinants’ expressions. Moreover, we can extend this work to other Gaudin models where the vacuumstate is not an eigenstate of the transfer matrix. We did this work for spins-12interacting with an arbitrarily oriented magnetic field. Finally, a numerical treatment of systems describing the interaction ofmany spins-12 with a single bosonic mode is presented. We study the time evolution of bosonic occupation and of local magnetisation for two different Hamiltonians, the Tavis-Cummings Hamiltonian and a central spin Hamiltonian. We learn that the dynamics of these systems, relaxing from an initial state to a stationary state, leads to a superradiant-like state for certain initial states.

We studied the impact of graph structure and temperature has on the sizes and lifetimes of crowds (mesoscopic clusters of consensus) that form when due to Ising Spin dynamics. Spin models have proven successful in modelling the formation of global consensus in opinion for large groups of people. We focus on the case where lots of small groups form consensuses but no global consensus forms. Our results can be interpreted in the light of consumer opinion and give an explanation for fads and brand reputation. Data was obtained by performing Monte Carlo simulations of the Ising Model on various graphs, our key control parameter was the inverse temperature β =1/kBT. In the Ising Model each vertex chooses an opinion −1 or +1 each time step based on the opinions of its neighbours. Low β ≪ 1 exhibit as vertices behaving noisily (flipping opinion often), while β → 1 tend to be quieter (flipping rarely). Interactions between neighbours creates feedback loops which determine crowd properties. The simulations we perform start with β_q = 0 for 0 ≤ t < 500, then the temperature is sharply quenched to a value βq ≥ 0 and kept constant for t ∈ (500, 15000]. We analyse the data in the time period 13000 ≤ t ≤ 15000. The simulations were performed on the 2-d rectangular lattice (RL), K-degree 1-d circular lattice (CL), Erdős-Rényi (ER), and 1-d Watts-Strogatz (WS) graphs. We conclude that crowds with multiple unique feedback loops are more robust against noisy vertices and crowd-crowd interactions. On the CL many feedback loops use the same paths which makes it easy for noisy vertices to disrupt the loops simultaneously. However 1-d structure limits crowd-crowd interactions making crowds strong against absorption by their neighbours. On the RL there are many paths between vertices, thus feedback loops can be easily rerouted around noisy vertices and crowds are robust against noise. On weakly connected ER graphs many feedback loops share the same paths and thus crowds are extremely susceptible to noisy vertices. Disorder in the 1-d lattice structure in WS graphs reduces the number of feedback loops in certain regions, this makes some crowds weak against both noisy vertices and neighbouring crowds. On the RL, CL and WS graphs we observe that size of crowds changes like an anomalous diffusion process. The anomalous diffusion exponent α was determined by plotting the root mean square change in crowd size sqrt(< ΔP² >) as a function of the crowd lifetime L. On the RL changes in crowd size were super-diffusive for 0.35 ≲ β ≲ 0.65; crowds grew very large in short periods of time, however such crowds also had short lifetimes as they were constantly replaced by new crowds. This behaviour is similar to the coming and going of fads. For 0.65 ≲ β ≲ 1 crowd sizes changed sub-diffusively, crowds grew in very small amounts to very large sizes over long time periods, this is analogous to the spread of reputation through word of mouth.

In this thesis we explore low energy extensions of Einstein's theory of General Relativity (GR). Initially our motivation will be the cosmological constant problem where large radiative corrections due to quantum field theories minimally coupled to the dynamical metric in GR leads to unacceptably large space-time curvatures. We will discuss the cosmological constant problem in detail, paying special attention to how it affects the global structure of a space-time whose dynamics are dictated by GR. With this in mind we will present and discuss recently proposed global modifications of GR which in the semi-classical limit sequester the radiatively unstable loop corrections to the cosmological constant from the space-time curvature. This is achieved by supplementing the local dynamics of GR with highly non-trivial global constraints, and we demonstrate how this can be achieved in a theory which is manifestly local. In this theory we will also consider the effects of an early universe phase transition on the late time dynamics. Away from global modifications of GR we will also consider local modifications which necessarily involve the propagation of new degrees of freedom. We outline the possible screening mechanisms which, since no new gravitational degrees of freedom have been observed in local environments, are an important feature of any local modification of GR. For one of these mechanisms, namely, the Vainshtein mechanism, we will consider the regime of validity of theories which make use of the Vainshtein mechanism and assess suggestions that one can trust these theories beyond the scale we would naively expect them to become strongly coupled. Following this we will move onto the chameleon mechanism, another example of a screening mechanism, and present a high energy extension motivated by the breakdown in the original chameleon theory in the early universe. The interactions of the resulting DBI chameleon theory will be motivated by our discussion of the Vainshtein mechanism.

Tensoring the field content of two super-Yang-Mills theories results in the field content of a certain supergravity theory, a procedure we call squaring. This thesis investigates how both the local and global internal symmetries enjoyed by the supergravity theory are inherited from the super-Yang-Mills factors. This is part of a much larger framework for studying a supergravity theory through its factorisation into simpler theories. The thesis begins by introducing local and global symmetries in the general context of relativistic field theory. The introduction is a short review on spacetime and internal symmetries in both gravitational and non-gravitational theories with particular focus on the supersymmetric regime. This is followed by the squaring idea and its appearance in various different contexts. After providing a unified description of all super-Yang-Mills theories over the four Normed Division Algebras R,C,H,O, the global internal supergravity symmetries are built with the help of the mathematical construction of the magic square, generalised to what we call the magic pyramid of supergravities. A physical interpretation of the formula reveals the Yang-Mills origin of the symmetries and demonstrates how simultaneous supersymmetry transformations on both factors can contribute to bosonic generators. The analysis is then extended to accommodate more exotic squarings by allowing for the coupling of matter multiplets to the super-Yang-Mills factors. Finally, the focus shifts to local internal symmetries whose linear form is derived directly from the corresponding linear factors. After a general treatment of off-shell squaring in various spacetime dimensions, the possibility of extending the construction to non-linear gravity is discussed.

In this thesis we develop techniques of integrability in the study of dualities between two-dimensional conformal field theories and theories of closed strings on three-dimensional Anti-de Sitter background geometries. For several years after integrability was first applied to the 3d/2d dualities, it was an unanswered question how to incorporate the so-called \massless modes" of these theories into the integrability machinery. Here we tackle this problem in several contexts. We show that in the classical integrable description of closed strings the implementation of the string Virasoro constraints needs to be modified for geometries with multiple factors where massless modes are present. We show further that with the correct implementation of the Virasoro constraints, massless modes can be included in integrability techniques for obtaining quantum corrections to physical quantities such as the energies of string solutions. Lastly, we consider the scattering of fundamental string excitations and derive all-loop expressions for the scattering matrix that includes both massless and massive excitations.

The contribution of this thesis is a novel framework for rewriting in higher categories. Its theoretical foundation is the theory of quasistrict higher categories and the practical realisation is a proof assistant Globular. The framework introduces the notions of diagrams and signatures as new mutually-recursive structures that give the algebraic basis for the approach. These structures are related the notion of an n-polygraph, but allow reasoning about quasistrict higher categorical structures in a way amenable to computer implementation. Building on this, we propose a new definition of a quasistrict 4-category, and prove a result that in a quasistrict 4-category, an adjunction of 1-morphisms gives rise to a coherent adjunction satisfying the butterfly equations.

In this thesis, we examine the construction and characteristics of generalised reflection matrices, within the a_1^(1), a_2^(1) and a_2^(2) integrable affine Toda field theories. In doing so, we generalise the existing finite-dimensional reflection matrices because our construction involves the dressing of an integrable boundary with a defect. Within this framework, an integrable defect's ability to store an unlimited amount of topological charge is exploited, therefore all generalised solutions are intrinsically infinite-dimensional and exhibit interesting features. Overall, further evidence of the rich interplay between integrable defects and boundaries is provided. It is hoped that the generalised solutions presented in this thesis are potential quantum analogues of more general classical integrable boundary conditions.

In the first part of this thesis, we discuss some aspects of the four-dimensional N = 2 superconformal index of theories of class S. We first consider a generalized index on S1 × S3/Zr, and prove S-duality in a particular fugacity slice. We then go on to study the (round) superconformal index in the presence of surface defects. We develop a systematic prescription to compute surface defects labeled by arbitrary irreducible representations of the gauge group and subject those defects to various tests in several different limits. Each of these limits is interesting in its own right, and we go on to explore them in some depth. In the second part of this thesis, we construct the gravity duals of large N supersymmetric gauge theories defined on squashed five-spheres with SU(3) × U(1) symmetry. The gravity duals are constructed in Euclidean Romans F(4) gauged supergravity in six- dimensions, and uplift to massive type IIA supergravity. We compute the partition function and Wilson loop in the large N limit of the gauge theory and compare them to their corresponding supergravity dual quantities. As expected from AdS/CFT, both sides agree perfectly. Based on these results, we conjecture a general formula for the partition function and Wilson loop on any five-sphere background, which for fixed gauge theory depends only on a certain supersymmetric Killing vector. We then go on to construct rigid supersymmetric gauge theories on more general Riemannian five-manifolds. We follow a holographic approach, realizing the manifold as the conformal boundary of the six-dimensional bulk supergravity solution. This leads to a systematic classification of five-dimensional supersymmetric backgrounds with gravity duals.

Our thesis explores aspects of the geometrical work and thought of Isaac Newton in order to better understand and re-evaluate his approach to geometry, and specifically his synthetic methods and the organic description of plane curves. In pursuing this research we study Newton's geometrical work in the context of the changing view of geometry between the late 16th and early 18th centuries, a period defined by the responses of the early modern geometers to a new Latin edition of Pappus' Collectio. By identifying some of the major challenges facing geometers of this period as they attempted to define and practice geometry we are able to contrast Newton's own approach to geometry. The themes emerging from the geometrical thought of early modern geometers provide the mathematical context from which to understand, interpret and re-evaluate the approach taken by Newton. In particular we focus on Newton's profound rejection of the new algebraic Cartesian methods and geometrical philosophies, and the opportunity to focus more clearly on some of his most astonishing geometrical contributions. Our research highlights Newton's geometrical work and examines specific examples of his synthetic methods. In particular we draw attention to the significance of Newton's organic construction and the limitations of Whiteside's observations on this subject. We propose that Newton's organic rulers were genuinely original. We disagree with Whiteside that they were inspired by van Schooten, except in the loosest sense. Further, we argue that Newton's study of singular points by their resolution was new, and that it has been misunderstood by Whiteside in his interpretation of the transformation effected by the rulers. We instead emphasise that it was the standard quadratic transformation. Overall we wish to make better known the importance of geometry in Newton's scientific thought, as well as highlighting the mathematical and historical importance of his organic description of curves as an example of his synthetic approach to geometry. This adds to contemporary discourse surrounding Newton's geometry, and specifically provides a foundation for further research into the implications of Newton's geometrical methods for his successors.

Linking number, writhe and twist are three important measures of a curve's geometry. They have been well studied and their de nitions extended to open curves situated between two horizontal planes [1]. However, many applications of these tools involve geometries that have a curved nature to them [2]. For example, the magnetic coronal-loops in the Sun's atmosphere share a spherical boundary (the photosphere). We reformulate these ideas in a spherical geometry, and then explore the oddities of this curved space to show that our new concept is consistent with its older, at counterpart. The second part of this project concerns a series of datasets from plasma experiments at Basic Plasma Science Facility, UCLA, Los Angeles. These experiments involve the creation of ux ropes inside a large (18m) plasma machine. A strong background magnetic eld is applied which ensures that eld lines travel from one end of the cylindrical device to the other. Due to mutual J B forces, the ux ropes twist and tangle about each other. We study three separate datasets: the rst one involving two ux ropes; the second, three ux ropes; the nal two ux ropes. The last experiment is perhaps the most exciting as the plasma velocity has been recorded. This extra data allows us to employ two di erent non-equivalent concepts of magnetic helicity. First, we use the surface ux formulation that makes various ideal assumptions, discarding several terms in Ohm's law. This is compared to helicity calculated by use of winding numbers { a construction without these ideal assumptions. By examining the di erence of these two results, it is shown that we may arrive at a measure of the resistivity present in the system. The plasma investigations described above rely on being able to seed magnetic eld lines across the length of the machine. This is not a simple process. The dataset itself is spatially non-uniform which makes numerical integration to obtain eld lines di cult. Even before integration is considered, a method to interpolate on our data grid of magnetic ux density is needed. This requires further careful considerations. Any interpolator must ensure that the data remains divergence-free; this requirement imposes conditions on the continuity of the derivatives. We have written a code to perform tricubic spline interpolation, and demonstrate that by using a particular method for xing the coe cients, this level of continuity can be achieved.

In this thesis, we study compactifiations of ten-dimensional heterotic supergravity at O(α'), focusing on the moduli of such compactifications. We begin by studying supersymmetric compactifications to four-dimensional maximally symmetric space, commonly referred to as the Strominger system. The compactifications are of the form M10 = M4 x X, where M4 is four-dimensional Minkowski space, and X is a six-dimensional manifold of what we refer to as heterotic SU(3)-structure. We show that this system can be put in terms of a holomorphic operator D on a bundle Q = T&ast; X ⊕ End(TX) ⊕ End(V ) ⊕ TX, defined by a series of extensions. Here V is the E8 x E8 gauge-bundle, and TX is the tangent bundle of the compact space X. We proceed to compute the infinitesimal deformation space of this structure, given by TM = H(0,1)(Q), which constitutes the infinitesimal spectrum of the lower energy four-dimensional theory. In doing so, we find an over counting of moduli by H(0,1)(End(TX)), which can be reinterpreted as O(α') field redefinitions. In the next part of the thesis, we consider non-maximally symmetric compactifications of the form M10 = M3 x Y , where M3 is three-dimensional Minkowski space, and Y is a seven-dimensional non-compact manifold with a G2-structure. We write X → Y → &reals;, where X is a six dimensional compact space of half- at SU(3)-structure, non-trivially fibered over &reals;. These compactifications are known as domain wall compactifications. By focusing on coset compactifications, we show that the compact space X can be endowed with non-trivial torsion, which can be used in a combination with %α'-effects to stabilise all geometric moduli. The domain wall can further be lifted to a maximally symmetric AdS vacuum by inclusion of non-perturbative effects in a heterotic KKLT scenario. Finally, we consider domain wall compactifications where X is a Calabi-Yau. We show that by considering such compactifications, one can evade the usual no-go theorems for flux in Calabi-Yau compactifications, allowing flux to be used as a tool in such compactifications, even when X is Kähler. The ultimate success of these compactifications depends on the possibility of lifting such vacua to maximally symmetric ones by means of e.g. non-perturbative effects.

In this thesis we develop a full characterization of abelian quantum statistics on graphs. We explain how the number of anyon phases is related to connectivity. For 2-connected graphs the independence of quantum statistics with respect to the number of particles is proven. For non-planar 3-connected graphs we identify bosons and fermions as the only possible statistics, whereas for planar 3-connected graphs we show that one anyon phase exists. Our approach also yields an alternative proof of the structure theorem for the first homology group of n-particle graph configuration spaces. Finally, we determine the topological gauge potentials for 2-connected graphs. Moreover we present an alternative application of discrete Morse theory for two-particle graph configuration spaces. In contrast to previous constructions, which are based on discrete Morse vector fields, our approach is through Morse functions, which have a nice physical interpretation as two-body potentials constructed from one-body potentials. We also give a brief introduction to discrete Morse theory.

We study non-abelian T-duality as a supergravity solution technique and explore its application to holography. We consider well understood geometries which describe the strong coupling regime of minimally supersymmetric gauge theories in 3 and 4 dimensions. We then use non-abelian T-duality to generate new solutions in type-II supergravity and use these to define new gauge theories with interesting dynamics such as confinement. The work contains extensive field theoretic analysis of these new solutions. We explore how the supersymmetry of the "seed" solutions is preserved under T-duality transformations by employing the powerful techniques of G- structures and generalised geometry. As well as giving a geometric description of non-abelian T-duality, this also enables us to extend the duality to cases with calibrated sources. We find that quite generically SU(3)-structures in 6d are mapped to SU(2)-structures. Further we find an intimate relationship between dynamic SU(2)-structures and confinement in these new solutions.

Wave radiation is an important process in many geophysical flows. In particular, it is by wave radiation that flows may adjust to a state for which the dynamics is slow. Such a state is described as “balanced”, meaning there is an approximate balance between the Coriolis force and horizontal pressure gradients, and between buoyancy and vertical pressure gradients. In this thesis, wave radiation processes relevant to these enormously complex flows are studied through the use of some highly simplified models, and a parallel aim is to develop accurate numerical techniques for doing so. This thesis is divided into three main parts. 1. We consider accurate numerical boundary conditions for various equations which support wave radiation to infinity. Particular attention is given to discretely non-reflecting boundary conditions, which are derived directly from a discretised scheme. Such a boundary condition is studied in the case of the 1-d Klein-Gordon equation. The limitations concerning the practical implementation of this scheme are explored and some possible improvements are suggested. A stability analysis is developed which yields a simple stability criterion that is useful when tuning the boundary condition. The practical use of higher-order boundary conditions for the 2-d shallow water equations is also explored; the accuracy of such a method is assessed when combined with a particular interior scheme, and an analysis based on matrix pseudospectra reveals something of the stability of such a method. 2. Large-scale atmospheric and oceanic flows are examples of systems with a wide timescale separation, determined by a small parameter. In addition they both undergo constant random forcing. The five component Lorenz-Krishnamurthy system is a system with a timescale separation controlled by a small parameter, and we employ it as a model of the forced ocean by further adding a random forcing of the slow variables, and introduce wave radiation to infinity by the addition of a dispersive PDE. The dynamics are reduced by deriving balance relations, and numerical experiments are used to assess the effects of energy radiation by fast waves. 3. We study quasimodes, which demonstrate the existence of associated Landau poles of a system. In this thesis, we consider a simple model of wave radiation that exhibits quasimodes, that allows us to derive some explicit analytical results, as opposed to physically realistic geophysical fluid systems for which such results are often unavailable, necessitating recourse to numerical techniques. The growth rates obtained for this system, which is an extension of one considered by Lamb, are confirmed using numerical experiments.

Condensed matter theory is the study of systems at finite density. In this thesis we will attempt to argue that gauge-gravity dualities can give deep and meaningful insights into the behaviour of strongly coupled condensed matter systems. The first three chapters will be a review of material already available in literature. Chapter 1 will introduce holography and the AdS-CFT correspondence. Particularly, in this chapter, the technique for the extraction of diffusion constants for charge and shear stress-energy-momentum fluctuations in a field theory with a holographic dual will be demonstrated. Chapter 2 will summarise relevant literature on the relativistic fluid-gravity correspondence. In the first half of the chapter it will be shown how to calculate the transport coefficients and Navier-Stokes equations for a suitable thermal field theory. The second half of chapter 2 will then be dedicated to extracting the transport coefficients for a strongly coupled field theory dual to a Reissner-Nordstrøm AdS spacetime. In chapter 3 a scaling of the metric and gauge field found in chapter 2 will be taken such that the boundary field theory admits Galilean, as opposed to relativistic, symmetry. Consequently, the governing hydrodynamic equations will be the non-relativistic, incompressible Navier-Stokes. Chapters 4 and 5 represent novel work. In chapter 4 the transport coefficients for a particular strongly coupled thermal field theory with underlying Schrodinger symmetry will be extracted from a charged, asymptotically Schrodinger spacetime. The governing hydrodynamic equations will be compressible with non-relativistic symmetry as opposed to those found via the scaling limit of chapter 3. In chapter 5 we show how knowledge of the transport coefficients of a thermal field theory can be used as a test-bed for numerical methods to explore beyond the hydrodynamic (long wavelength and low frequency) regime. With this in mind we consider Reissner-Nordstrøm AdS4 and determine the two point correlators at arbitrary frequency and momentum. Finally in chapter 6 we summarise the work discussed in this thesis and speculate about further applications of hydrodynamic techniques to strongly coupled condensed matter theories.

The major purpose of this work is to combine the minimal-hidden-photon model with Large Extra Dimensions (LED). This involves confining the Standard- Model photon to a 3-brane, whilst allowing the hidden photon and graviton to occupy the higher-dimensional bulk. After integrating out the extra dimensions both the hidden photon and graviton obtain a tower of massive Kaluza-Klein (KK) modes. The Standard-Model photon obtains no KK modes, in accordance with experiment. The work begins with a discussion of the minimal hidden photon with- out KK modes, including the current constraints. In most cases existing con- straints are simply quoted or rederived, but for some experiments original con- straints are produced. For example new constraints from atomic spectra are produced. Significant modifications are also made to the published constraint from the SN1987a energy-loss experiment. This means properly accounting for the plasma mass of the electron, and also accounting for the modification of the kinetic-mixing parameter in a plasma. Finally constraints are produced for the minimal-hidden-photon model with KK modes.

This thesis presents a mathematical analysis of two classes of behaviours which occur in systems of populations: noise-induced bistability and stochastic patterning. Both behaviours have their origins in the intrinsic stochasticity possessed by a population system due to the discreteness of the individuals: the intrinsic noise. In the study of noise-induced bistability, we analyse a system which exhibits switching between two states. These states do not correspond to fixed points of the corresponding system of deterministic equations, but instead are the states at which the system stochasticity is minimal or vanishing. This feature suggests that the mechanism is intrinsically different to the traditional paradigm of bistability, in which a system with two stable fixed points is subject to noise. Through our mathematical analysis we highlight some characteristic properties of the dynamics, suggesting a way to distinguish, in a real system, the presence of noise-induced bistable states from other types of bistability. Stochastic patterning arises when noise acts on a reaction-diffusion system which exhibits pattern formation via an instability of the homogeneous state. If the system is close to the onset of the instability, whilst still in the stable regime, then patterning occurs due to a combination of stochastic agitation and the exponential decay of the underlying stable homogeneous state. We investigate the case of the stochastic travelling waves on both regular lattices and complex networks. In both cases, a complete analytical treatment is provided via the power spectra of fluctuations. The spirit of the thesis is to propose a simple model which is representative of an observed behaviour, and then solve the model analytically. Numerical simulations are used throughout to verify the accurateness of the analytical approximations. Thus the analytical treatments constitute the core of the work and have two purposes. They are explanatory, in the sense that they help to develop intuition about how the noise leads to a certain behaviour. Moreover, they give quantitative understanding, as we provide the explicit expressions for various quantities (stationary distributions, mean times, etc.). In some cases, the formulas that we have obtained do not rely on the details of the model, so that we would expect them to fit experimental data. In other cases this is not so, yet the analytical treatment may give insight on how to attack more realistic models.

In this thesis we consider a number of different aspects of dynamical locality, an axiom on locally covariant theories proposed by Fewster and Verch that is closely related to the question of whether a theory describes the same physics in all spacetimes. After some introductory material, in Chapters 3 and 4 we examine dynamical locality for the nonminimally coupled scalar field and its enlarged algebra of observables. We show that dynamical locality holds at all masses, including non-zero masses, for the nonminimally coupled scalar field theory. We also demonstrate that dynamical locality holds in the massive minimally coupled and massive conformally coupled cases for the enlarged algebra of observables, and fails to hold in the massless minimally coupled case. In Chapter 5, we discuss a number of categorical structures that can be used in the construction of classical theories that may be quantized using canonical anticommutation relations (CAR), and their subsequent quantization. We prove a number of results pertaining to dynamical locality of classical theories and their CAR-quantized counterparts. In Chapters 6 and 7, we give a simplified version of the locally covariant classical and quantum Dirac theories, using the machinery developed in Chapter 5. We also formulate for the first time versions of these theories that are entirely independent of the choice of a global reference frame for the spacetime, and depend only on an equivalence class of these frames. We demonstrate that both the simplified frame-dependent theories and the frame-independent theories are dynamically local.

In this thesis we explore two different theories for modelling soft matter systems. We start by discussing density functional theory (DFT) and dynamical density functional theory (DDFT) and consider the thermodynamics underpinning these theories as well as showing how the main results may be derived from the microscopic properties of soft matter. We use this theory to set up a model for the evaporation of the solvent from a thin film of a colloidal suspension. The general background for such systems is discussed and we display some of the striking nanostructures which self-assemble during the evaporation process. We show that our theory successfully reproduces some of these patterns and deduce the various mechanisms and transport processes behind the formation of the different structures. In the second part of this thesis we discuss results for a second model; the phase field crystal (PFC) model. The model equations are discussed, showing how they may be derived from DDFT as well as discussing the general background of PFC models. We present some results for the PFC model in its most commonly used form before going on to introduce a modified PFC model. We show how the changes in the model equations are reflected in the thermodynamics of the model. We then proceed by demonstrating how this modified PFC model may be used to qualitatively describe colloidal systems. A two component generalisation of the modified PFC model is introduced and used to investigate the transition between hexagonal and square ordering in crystalline structures. We conclude by discussing the similarities and connections between the two models presented in the thesis.

In this thesis we are going to investigate the behaviour of geodesics in a metric with a singularity known as the ``Mixmaster Universe''. This was motivated from previous work done, where the now well-known AdS/CFT correspondence was used to extract information about an AdS Schwarzschild black hole singularity beyond the horizon by studying correlators on the boundary that correspond to spacelike geodesics which bounce off the singularity. It was then shown that when the singularity was a cosmological one (in this case a Friedmann Robertson Walker cosmology with a Big Crunch), this was no longer possible as it is impossible for spacelike geodesics to bounce off this kind of singularity. This raises the question of whether, when an example of a more general singularity (the ``Mixmaster Universe'') is considered, it is possible for the spacelike geodesics to bounce away from this kind of singularity. This would enable us to potentially extract information about the singularity from the boundary correlators. Unfortunately, it will be shown that bouncing of such geodesics is extremely unlikely (if not impossible) and thus we would be unable to extract information about the singularity in the mixmaster universe using such a technique. We also discuss another aspect of the evolution of the mixmaster universe which shows that it does indeed have a very complicated evolution.

Evolutionary dynamics arise from the interplay of mutation and selection. Fundamentally, these two processes operate at different levels: Mutations modify genetic information (the genotype), which is passed from parent to offspring. Selection is triggered by variation in reproductive success, which depends on the physical properties (the phenotype) of an organism and its environment. Thus the genotype-phenotype map determines if and how mutations can lead to selection. The aim of this dissertation is to incorporate this map explicitly into a theoretical description of evolutionary dynamics. The first part of the analysis presented here is concerned with the static properties of simple models of these maps, which are studied using exhaustive enumeration. The two most important observations are phenotypic bias – some phenotypes are realized by many more genotypes than most other phenotypes – and the existence of neutral spaces – genotypes with the same phenotype can often be reached from each other by single mutational steps. The remainder of the dissertation provides a theoretical description of evolutionary dynamics on and across neutral spaces. Two different mean-field approximations lead to simple analytic results for the first discovery of alternative phenotypes, highlighting the importance of phenotypic bias: Rare phenotypes are hard to find by evolutionary search. These results are used to discuss the relationship of robustness, the ability to withstand mutational change, and evolvability, the ability to create variation through mutation. Several types of fluctuations beyond the mean-field limit are studied, both theoretically and in simulations. The discrete structure of genotype spaces can lead to strong correlations in the spectra of phenotypes produced, increasing the probability that a particular phenotype is fixed in the population quickly after its discovery. Structural correlations between genotypes can increase the effect of phenotypic bias, while the qualitative features of the mean-field description remain valid. All these results highlight that neutral spaces impact evolutionary dynamics in many non-trivial ways, in particular by favouring phenotypes of high accessibly, but comparably low fitness over those phenotypes that are highly fit, but very hard to discover.

This thesis studies quantum integrable structures such as Yangians and quantum affine algebras that arise in and are inspired by the AdS/CFT duality, with a primary emphasis on the exploration of integrable boundaries deeply hidden in the duality. The main goal of this thesis is to find novel algebraic structures and methods that could lead to new horizons in the theory of quantum groups and in the exploration of boundary effects in the gauge/gravity dualities. The main thrust of this work is the exploration of the AdS/CFT worlsheet scattering theory and of integrable boundaries that manifest themselves as Dp-branes (p+1-dimensional Dirichlet submanifolds) which are a necessary part of the superstring theory. The presence of these objects breaks some of the underlying symmetries and leads to boundary scattering theory governed by coideal subalgebras of the bulk symmetry. Here the boundary scattering theory for D3-, D5- and D7-branes is considered in detail, and the underlying boundary Yangian symmetries are revealed. The AdS/CFT worldsheet scattering theory is shown to be closely related to that of the deformed Hubbard chain. This similarity allows us to apply the quantum deformed approach to the boundary scattering theory. Such treatment of the system leads to quantum affine symmetries that manifest themselves in a very elegant and compact form. In such a way the symmetries of distinct boundaries that previously seemed to be unrelated to each other emerge in a uniform and coherent form. The quantum deformed approach also helps us to better understand the phenomena of the so-called secret symmetry. It is called secret due to its peculiar feature of appearing as a level-one generator of the Yangian of the system. However it does not have a Lie algebra (level-zero) analogue. This symmetry is shown to have origins in the quantum deformed model, where it manifest itselfs as two, level-one and level-minus-one, generators of the corresponding quantum affine algebra.