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.

Since the arrival of the personal computer in the early 1980s people have been advocating the use of computer games in aiding teaching and learning. However the increasing prevalence of computers in the early years of the 21st century led to the introduction of the idea of the “digital native” – those who were born since the dawn of the age of the ubiquitous computer. It was then argued widely that this generation would now need to be taught using computer games and that conventional education was not working. This view has been modified since but there still exists arguments for the use of computer games in many areas of education. This thesis looks at the potential benefits of computer games in aiding the teaching of mathematics in primary schools. The particular focus of the research has been whether collaborative computer games if properly designed with the learning outcomes encoded in the game mechanic would be more valuable in promoting engagement with mathematics problem solving than other more conventional methods. A number of hypotheses were developed based on the current theories and designed to be tested for validity. To carry out this investigation a number of studies have been undertaken. A literature review was focused on the methods used to teach mathematics in primary schools, the value of collaboration and the use of computer games in education. This was followed up with a study in a primary school to validate the basic findings from the literature review. A user centred design study began with a trial of a commercial game that was meant to promote collaboration in primary school children’s game play to ascertain what components best promoted collaboration. Interviews with teachers and pupils were also used to develop the ideas behind a game that was suitably themed for the age group. This was then pilot tested for playability and usability along with a dice game that had been adapted from some commercially available games for use in a later controlled experiment in which the effectiveness of the game designed was tested against the control with groups of children from the UK Key Stage 2 (7 to 11 year olds). The hypotheses were evaluated against the results of the controlled experiment. The idea that computer games themselves would always work in promoting learning were disproved but the value of games (both computer and non-computer games) as an adjunct to conventional teaching in collaborative settings was shown to be valuable in promoting engagement with mathematics. It was also clear that these games promoted learning among those who were in the group of low achievers in mathematics.

Geometrically, the main goal of this thesis is to refine the classification of minimal surfaces S with K2/S = 7 and Pg = 4 due to Ingrid Bauer and published in her monograph 'Surfaces with K2/S = 7 and Pg = 4' (cf. [Bauer]). She found that they belong to 10 families according to the behaviour of the canonical map φKs . The 10 families form 3 irreducible components of moduli, but the details of how this happens remained unknown except for a few particular cases. Our treatment consists in studying the abstract canonical model Proj R(S,Ks), where R(S,Ks) := + n ≥ 0 H0(S,0S(nKS)) is the pluricanonical ring. Except when |KS| is base point free, these rings are Gorenstein of codimension ≥ 4. We show that the only previously known deformation family of such rings (constructed by Bauer, Catanese and Pignatelli in [Bauer et al]) relating the 2 families with φKs birational can be recovered using basic arguments about halfcanonical curves. Our techniques also allow us to construct new explicit at families for cases on which φKs is not birational. In particular, we construct a 1-parameter at family of Gorenstein rings with general fibre of codimension 4 and special fibre of codimension 6. At the end we discuss possible applications of our methods to the cases on which |KS| defines a 2-to-1 map to a quadratic surface. We conjecture that the moduli space of surfaces with K2/S = 7 and Pg = 4 is connected.

Twitter has become an important site for national discussions where we can get a new and timely update of the public opinion towards any event. Twitter Sentiment Analysis (TSA) can be an effective method for unpacking the deep insights embodied within the opinions of the public. Recently, various TSA techniques have been developed, but little consideration has gone into emojis, which is a new invention and has been popularly shared by Twitter users from different countries, with various demographic characteristics, and diverse cultural backgrounds. The ubiquitous adoption of emojis on Twitter provides new opportunities to analyse sentiment expressions in a textual context. Emojis should be included when conducting TSA as the meaning of a Twitter post and its sentiment can be identified with greater clarity and accuracy with emojis. This research aims to develop novel approaches that handle emojis properly and tackle current open issues in TSA. Consisting of four phases, this thesis presents a comprehensive and in-depth research work in the field of Emoji Analytics and TSA. Several studies have been conducted to investigate emoji usage on Twitter and evaluate their effects on TSA. The experimental results demonstrate that emojis has become an essential component of Twitter communication and it is an important area of study complementary to TSA, implying promising future research opportunities for TSA. A novel TSA methodological framework that collects, pre-processes, analyses and maps citizen sentiments from Twitter in helping learn citizens’ moods has been implemented and proved to be effective. The novel framework identifies the best setting for TSA when involving emojis, and proposes an effective emoji training heuristic, which is feasible for both ternary and multi-class classification of tweets. Besides, it innovatively includes the visualisation of user-generated contents in a location-based manner on geographical maps, which provides a much easier-to-understand visual representation of the sentiment. The methodological framework has been proved applicable in real-world scenarios and can be used to support research in other fields. Being the first to consider popularity of emojis on Twitter and include them in performing TSA, this research is considered to be a pioneering work in the field, suggesting a new direction for TSA in the era of emojis.

In this thesis I consider the problem of clearing models used for systemic risk assessment in interbank networks. I investigate two extensions of the classical Eisenberg & Noe (2001) model. The first extension permits the analysis of networks with interbank liabilities of several maturities. I describe a clearing mechanism that relies on a fixed-point formulation of the vector of each bank’s liquid assets at each maturity date for a given set of defaulted banks. This formulation is consistent with the main stylised principles of insolvency law, permits the construction of simple dynamic models and furthermore demonstrates that systemic risk can be underestimated by single maturity models. In the context of multiple maturities, specifying a set of defaulted banks is challenging. Two approaches to overcome this challenge are proposed. The algorithmic approach leads to a well-defined liquid asset vector for all financial networks with multiple maturities. The simpler functional approach leads to the definition of the liquid asset vector that need not exist but under a regularity condition does exist and coincides with the algorithmic approach. The second extension concerns the non-uniqueness of clearing solutions. When more than one solution exists, the standard approach is to select the greatest solution. I argue that there are circumstances when finding the least solution is desirable. An algorithm for constructing the least solution is proposed. Moreover, the solution is obtainable under an arbitrary lower bound constraint. In models incorporating default costs, clearing functions can be discontinuous, which renders the problem of constructing the least clearing solution non-trivial. I describe the properties of the construction algorithm by means of transfinite sequences and show that it always terminates. Unlike the construction of the greatest solution, the number of steps taken by the algorithm need not be bounded by the size of the network.

This problem is concerned with the inviscid linear stability of parallel stratified shear layer. Most flows do not form well-defined layers, but have density and/or velocity that varies smoothly and continuously with a spatial coordinate. Even in these cases, dividing the flow into layers may be a useful modelling strategy to simplify the equations of motion. A parallel stratified shear layer is where layers of fluid of different density parallel to one another are moving with different speeds creating shear in between them. If the fluid has constant density then we say it is unstratified or homogeneous. Stratified shear layers can arise in the upper oceans, and this motivates our work. We investigate the temporal, absolute and global stability properties of model stratified flows. In temporal instability the disturbances are assumed to be periodic in the streamwise direction, and propagation properties are not determined, it does determine if a flow is stable or unstable. Absolute instability considers propagation of a spatially localized disturbance and determines whether there is a growth in the rest frame or not. Both temporal and absolute instability assume parallel flow. Global instability takes account of variation of the basic flow in the streamwise direction, and determines if there is a growth in the rest frame when the flow is not parallel. Often shear layers develop slowly in the stream wise direction which justifies a local stability approach, i.e. obtaining dispersion relations based on velocity and density profiles found at particular streamwise positions. Streamwise variation of the basic flow is neglected in the local theory, i.e. the flow is assumed to be parallel. A mixing layer is the region of high shear between two layers of uniform, but different, velocity. The existence of a mixing layer also implies the presence of surrounding uniform (or nearly uniform) flows. Huerre & Monkewitz (1985) showed that mixing layers become locally absolutely unstable if there is a sufficiently strong reverse flow in one of the two streams, and then, disturbances spread and grow both upstream and downstream. Healey (2009) showed that the presence of boundaries parallel to the shear layer can increase the absolute instability so that even mixing layers without reverse flow can become absolutely unstable. We show that for weakly stratified mixing layers typical of the upper ocean, the sea surface and the sea bed can provide the necessary confinement for the creation of local absolute instability. We also showthat absolute instability is sometimes increased by stable stratification. Furthermore, typical bed topographies can create zones of absolute instability parallel to the shoreline that have the potential to act as wave maker regions for global instability. This mechanism could operate in coastal areas with wind blowing offshore. Results are presented for global instabilities of mixing layers where one layer is essentially stationary, a common scenario in geophysical flows. We consider flows where the distance from mixing layer to a boundary varies slowly with the streamwise coordinate, which can create a pocket of absolute instability, and which in turn can produce global instability. Flows of this type can arise, for example, when wind blows over the sea leading to an upper layer moving at nearly uniform velocity lying above an essentially stationary lower layer, with a relatively thin mixing layer between them. We have identified flows that become globally unstable and on the other hand we have found flows that have a region of absolute instability and yet remain globally stable. It is shown here that typical sea bed topographies can generate global instabilities even when stabilizing stratification is included. It is expected that the appearance of global instability would significantly enhance mixing.

Choosing the number of mixture components remains a central but elusive challenge. Traditional model selection criteria can be either overly liberal or conservative when enforcing parsimony. They may also result in poorly separated components of limited practical use. In this thesis, the term parsimony refers to selecting a simpler model by enforcing a separation between the models under consideration, and the term sparsity refers to the ability of penalizing overfitted models leading to well-separated components with non-negligible weight, interpretable as distinct subpopulations. Non-local priors (NLPs) are a family of distributions that encourage parsimony by enforcing a separation between the models under consideration. In this thesis we investigate the use of NLPs to choose the number of components in mixture models. Our main contributions are proposing the use of non-local priors (NLPs) to select the number of components, characterizing the properties of the associated inference (in particular, improved sparsity) and proposing tractable expressions suitable for prior elicitation purposes, simpler and computationally efficient algorithms and practical applications. Chapter 2 develops the theoretical framework. We present NLPs in the context of mixtures and show how they lead to well-separated components that have non-negligible weight, hence interpretable as distinct subpopulations. Moreover we formulate a general NLP class, propose a particular choice leading to tractable expressions and give a theoretical characterization of the sparsity induced by NLPs for choosing the number of mixture components. Although the framework is generic we fully develop multivariate Normal, Binomial and product Binomial mixtures based on a family of exchangeable moment priors. Chapter 3 presents the prior computation and elicitation. We suggest default prior settings based on detecting multi-modal Normal and T mixtures, and minimal informativeness for categorical outcomes where multi-modality is not a natural consideration. The theory and underlying principles in this thesis hold more generally as outlined in Chapter 2, however. Chapter 4 presents the computational framework for model selection and fitting. We propose simple algorithms based on Markov chain Monte Carlo methods and Expectation Maximization algorithms to obtain the integrated likelihood and parameter estimates. Chapters 5-7 contain the simulation studies and applications. In Chapter 5 we compare the performance of our proposal to its local prior counterpart as well as the Bayesian Information Criterion (BIC), the singular Bayesian Information Criterion (sBIC) and the Akaike Information Criterion (AIC). Our results show a serious lack of sensitivity of the Bayesian information criterion (BIC) and insufficient parsimony of the AIC and the local prior counterpart to our formulation. The singular BIC behaved like the BIC in some examples and the AIC in others. In Chapter 6 we explore a computational fast non-local model selection cri- teria and propose a new computational strategy that provides a direct connection between cluster occupancies and Bayes factors with the advantage that Bayes factors allow for more general model comparisons (for instance equal vs unequal covariances in Normal mixtures). This new computational strategy is helpful to discard unoccupied clusters in overfitted mixtures and we remark that the result has interest beyond purely computational purposes, e.g. to set thresholds on empty cluster probabilities in overfitted mixtures. In Chapter 7 we present the applications of this thesis and also offer comparisons to overfitted and repulsive overfitted mixtures. In most examples their performance was competitive but depended on setting the prior parameters adequately to prevent the appearance of spurious components. The number of components inferred under NLPs was closer to the true number (when this was known) and remained robust to prior parameter changes, provided these remain in the range of recommended defaults. In Chapter 8 we have the conclusions and some possible future directions of this work. Finally, in Appendix A we present the proofs of Theorem 1 as well as auxiliary lemmas and corollaries. Appendix B shows the MCMC diagnostics. Appendix C presents the main probability density functions used throughout this thesis.

Since the development of the ensemble Kalman filter, it has seen a wide application to many scientific fields ranging from signal processing to weather forecasting and reservoir simulation. One field which has recently seen a keen interest towards filtering techniques is that of inverse problems. Ensemble-based methods are a popular choice of filtering techniques as they provide a computational advantage over traditional methods whilst retaining a good level of accuracy. This thesis is concerned with developing analysis and numerics of ensemble Kalman inversion (EKI) in the context of Bayesian inverse problems. In particular we are interested in quantifying the uncertainty that can arise for problems where our unknown is defined through geometric features. In the first part of this work we are interested in developing hierarchical approaches for EKI. This motivation is taken from hierarchical computational statistics for Gaussian processes where we are interested in a number of further unknowns such as hyperparameters that define the underlying unknown for the model problem. We present numerics of these hierarchical approaches whilst understanding its long term effect through continuous-time limits. The second part of this work is aimed at improving the computational burden of the forward solver within inverse problems. This improved forward solver is based on the reduced basis method which was designed for parameterized partial differential equations. The final part of the thesis concludes with an application of EKI where we adopt a Bayesian approach of the inverse eikonal equation. Our motivation is to extend the current work to Hamilton-Jacobi equations, where there exists a rich mathematical theory. A key understanding of how to tackle the uncertainty for this equation is addressed.

In the past years, techniques from different areas of mathematics have been successfully applied in extremal combinatorics problems. Examples include applications of number theory, geometry and group theory in Ramsey theory and analytical methods to different problems in extremal combinatorics. By providing an analytic point of view of many discrete problems, the theory of combinatorial limits led to substantial results in many areas of mathematics and computer science, in particular in extremal combinatorics. In this thesis, we explore the connection between combinatorial limits and extremal combinatorics. In particular, we prove that extremal graph theory problemsmay have unique optimal solutions with arbitrarily complex structure, study a property closely related to Sidorenko's conjecture, one of the most important open problems in extremal combinatorics, and prove a 30-year old conjecture of Gyori and Tuza regarding decomposing the edges of a graph into triangles and edges.

This thesis is split into three largely independent chapters. The first concerns the representation theory of Zp[G]-lattices. Specifically, we investigate regulator constants, due to Dokchitser-Dokchitser, which are isomorphism invariants of lattices whose extension of scalars to Qp is self-dual. We first show that when G has cyclic Sylow p-subgroups then regulator constants are strong invariants of permutation modules in a way that can be made precise. Our main result is then that, subject to an additional technical hypothesis on G, this can be combined with existing work of Yakovlev to provide an explicit list of accessible invariants which completely determine, up to isomorphism, any Zp[G]-lattice whose extension to Qp is self-dual. The second chapter is an application of this result in the context of number fields. Given a Galois extension of number fields K=F with Galois group G, the extension of scalars to Zp of the unit group of K modulo its torsion subgroup denes a Zp[G]-lattice. If we assume that G has cyclic Sylow p-subgroups and satisfies the aforementioned hypothesis, then the above result gives a list of invariants which determine the Galois module structure. The main result of this chapter is then that if p divides G at most once, we can explicate these invariants in terms of classical number theoretic objects. For example, in some cases this can be done in terms of capitulation of ideal classes and ramification information. The final (unrelated) topic concerns relative motives over Shimura varieties. Given a Shimura datum (G; Ӿ) and neat open compact subgroup K ≤ G(Af ), denote the corresponding Shimura variety ShK(G;Ӿ) by S. The canonical construction described by Pink shows how to associate variations of Hodge structure on San to representations of G. It is expected that this should be motivic in nature, i.e. that there is a motive over S for every representation of G whose Hodge realisation is the variation of Hodge structure given by the canonical construction. Using mixed Shimura varieties, we show that this can be done functorially for representations with Hodge type {(-1; 0); (0,-1)} and that this is compatible with change of S. When (G;Ӿ) has a chosen PEL-datum, existing work of Ancona allows us to associate a motive over S to any representation of G. We then give results to show that in some cases this compatible with change of S and independent of the choice of PEL-datum.