In this dissertation, the analytic properties and asymptotic behaviour of two and three particle amplitudes have been studied for a large class of potentials using the related mathematical techniques of Fredholm theory and Functional Analysis. The first three sections of this work are devoted to a study of the analytic properties and asymptotic behaviour for large £ of two particle scattering amplitudes using methods based on Predholm theory. In the remaining three sections the analytic properties in the energy and total angular momentum of two and three particle amplitudes are studied using the very elegant methods of Functional Analysis. Especially the partial wave integral equations corresponding to the Fadeev equations are derived and the kernel of these equations is studied to obtain the analyticity properties in the energy and total angular momentum.

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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.

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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.

This thesis concentrates on four and five dimensional black holes and their associated geodesies. Some coordinate charts are presented, which are useful in the analysis of both static and rotating black holes, and their mathematical properties investigated before some methods of solving Einstein's vacuum field equations are examined. The Myers-Perry black hole metric is derived before going on to describe the Inverse Scattering Method of generating new vacuum solutions. The Inverse Scattering Method is used to generate the single and doubly spinning black ring metrics and then the physical properties of these solutions is explored in detail. The latter part of this thesis looks at different ways of visualising geodesies in various spacetimes and examines the pros and cons of each particular method, as well as looking at several examples of geodesies with different parameters. The geodesies of the singly spinning black ring are calculated and it is shown that they cannot in general be analytically integrated. In light of this, some restricted analytic scenarios are investigated with the intention of gaining some insight into how the geodesies behave in the spacetime as a whole. Finally, a method is presented which allows string charges to be added to any vacuum solution to Einstein's equations. The properties of this new charged solution are then compared with the neutral starting solution. The doubly spinning black ring is used as a model to demonstrate how the method can be used to charge up a specific black hole solution and the resulting thermodynamic properties of this charged doubly spinning black ring are then examined.

The local search algorithm WSAT is one of the most successful algorithms for solving the archetypal NP-complete problem of satisfiability (SAT). It is notably effective at solving RANDOM-3-SAT instances near the so-called "satisfiability threshold", which are thought to be universally hard. However, WSAT still shows a peak in search cost near the threshold and large variations in cost over different instances. Why are solutions to the threshold instances so hard to find using WSAT? What features characterise threshold instances which make them difficult for WSAT to solve? We make a number of significant contributions to the analysis of WSAT on these high-cost random instances, using the recently-introduced concept of the <i>backbone </i>of a SAT instance. The backbone is the set of literals which are implicates of and instance. We find that the number of solutions predicts the cost well for small-backbone instances but is much less relevant for the large-backbone instances which appear near the threshold and dominate in the overconstrained region. We undertake a detailed study of the behaviour of the algorithm during search and uncover some interesting patterns. These patterns lead us to introduce a measure of the <i>backbone</i> <i>fragility</i> of an instance, which indicates how persistent the backbone is as clauses are removed. We propose that high-cost random instances for WSAT are those with large backbones which are also backbone-fragile. We suggest that the decay in cost for WSAT beyond the satisfiability threshold, which has perplexed a number of researchers, is due to the decreasing backbone fragility. Our hypothesis makes three correct predictions. First, that a measure of the backbone robustness of an instance (the opposite to backbone fragility) is negatively correlated with the WSAT cost when other factors are controlled for. Second, that backbone-minimal instances (which are 3-SAT instances altered so as to be more backbone-fragile) are unusually hard for WSAT. Third, that the clauses most often unsatisfied during search are those whose deletion has the most effect on the backbone.

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