This thesis lies in the field of constant mean curvature (cmc) hypersurfaces and specifically cmc 1/2 surfaces in the three-manifold H 2 × R. The value 1/2 is the critical mean curvature for H 2 × R, in that there do no exist closed cmc surfaces with mean curvature 1/2 or less. Daniel and Hauswirth have constructed a one-parameter family of complete, cmc 1/2 annuli that are symmetric about a reflection in the horizontal place H 2 × {0}, the horizontal catenoids. In this thesis we prove that these catenoids converge to a singular limit of two tangent horocylinders as the neck size tends to zero. We discuss the analytic gluing construction that this fact suggests, which would create a multitude of cmc 1/2 surfaces with positive genus. The main result of the thesis concerns a key step in such an analytic gluing construction. We construct families of cmc 1/2 annuli with boundary, whose single end is asymptotic to an end of a horizontal catenoid. We produce these families by solving the mean curvature equation for normal graphs off the end of a horizontal catenoid. This is a non-linear boundary value problem, which we solve by perturbative methods. To do so we analyse the linearised mean curvature operator, known as the Jacobi operator. We show that on carefully chosen weighted H¨older spaces the Jacobi operator can be inverted, modulo a finite-dimensional subspace, and provided the neck size of the horizontal catenoid is sufficiently small. Using these linear results we solve the boundary value problem for the mean curvature equation by a contraction mapping argument.

The accelerated expansion of spacetime intuitively points to the existence of new, unknown energy fields pervading the universe, but it is has also spurred the growth of the research field of modified gravity theories. Of these, f(R) theories of gravity is the first and simplest modification to General Relativity, and have been studied extensively for their astrophysical and cosmological predictions. Power law f(R) modifications have been shown to exhibit desirable characteristics, producing the late time accelerated expansion as well as satisfying local tests of gravity. However, there is wide degeneracy among models in this class, and they are known to suffer from cosmological instabilities, which could lead to curvature singularities at finite times. This thesis addresses questions directly relating to model degeneracy and sudden singularities. Cosmologies and cosmological perturbations, resulting from a general broken power law modification to GR are generated, studied and evolved. Simulations are performed using 1+3 space time decomposition of the field equations and a dynamical systems approach to f(R) cosmology. The parameter space of this model, which includes the HuSawicki [6], Starobinsky [96] and Miranda [7] f(R) forms as subclasses, is investigated. It is found that there are regions in the parameter space which are completely singular and bound by continuous curves. We also investigate regions of the parameter space in which the attractive nature of gravity is preserved, and find that these regions intersect. The results of a Markov Chain Monte Carlo analysis significantly narrowed the viable region of the exponent parameter space of the general power law f(R) model. Current cosmological distance data; SNIa (Union 2), BAO (6dFGS, BOSS, SDSS, WiggleZ) as well as the LRG power spectrum (SDSS DR9), were used to obtain these constraints. The best fits are compared with the ΛCDM model, and leads to the conclusion that this class is still a candidate for the gravitational interaction.

Bibliography : p. 161-176. / Propositional systems are deductively closed sets of sentences phrased in the language of some propositional logic. The set of systems of a given logic is turned into an algebra by endowing it with a number of operations, and into a relational structure by endowing it with a number of relations. Certain operations and relations on systems arise from some corresponding base operation or relation, either on sentences in the logic or on propositional valuations. These operations and relations on systems are called power constructs. The aim of this thesis is to investigate the use of power constructs in propositional systems. Some operations and relations on systems that arise as power constructs include the Tarskian addition and product operations, the contraction and revision operations of theory change, certain multiple- conclusion consequence relations, and certain relations of verisimilitude and simulation. The logical framework for this investigation is provided by the definition and comparison of a number of multiple-conclusion logics, including a paraconsistent three-valued logic of partial knowledge.

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Efficient and rigorous source finding techniques are needed for the upcoming large data sets from telescopes like MeerKAT, LSST and the SKA. Most of the current source-finding algorithms lack full statistical rigor. Typically these algorithms use some form of thresholding to find sources, which leads to contamination and missed sources. Ideally we would like to use all the available information when performing source detection, including any prior knowledge we may have. Bayesian statistics is the obvious approach as it allows precise statistical interrogations of the data and the inclusion of all available information. In this thesis, we implement nested sampling and Monte Carlo Markov Chain (MCMC) techniques to develop a new Bayesian source finding technique called FireFly. FireFly employs a technique of switching ‘on’ and ‘off’ sources during sampling to deal with the fact that we don’t know how many true sources are present. It therefore tackles one of the critical questions in source finding, which is estimating the number of real sources in the image. We compare FireFly against a Bayesian evidence-based search method and show on simulated astronomical images that FireFly outperforms the evidence-based approach. We further investigate two implementations of FireFly: the first with nested sampling and the second with MCMC. Our results show that MCMC FireFly has better computational scaling than the nested sampling version FireFly but the nested sampling version of FireFly appears to perform somewhat better than MCMC FireFly. Future work should examine how best to quantify FireFly performance and extend the formalism developed here to deal with multiwavelength data.

Beyond gravitational evidence for dark matter, a set of search techniques are employed in the present thesis within the particle dark matter paradigm. Under the possibility of dark matter annihilating into particles of the Standard Model of Particle Physics, we study the products of annihilation with cosmic-ray detectors, such as AMS, Fermi-LAT and PAMELA, and radio telescopes, such as the SKA. In this work, we focus on the positron fraction measured in the Solar System due to dark matter annihilating in the dark matter galactic halo, but also on radio signals from the Milky Way and dwarf spheroidal galaxies. Our main purpose is to constrain the dark matter parameter space under the light of the latest experimental data for cosmic-rays and the new sensitivities reached in radio astronomy. Furthermore, we discuss some of the most promising locations and synchrotron frequencies to search for dark matter with masses around the TeV scale. The analysis presented in this thesis lies in setting constraints on modelindependent dark matter. However, some specific dark matter candidates in the context of extra-dimensional theories are considered as well. Indeed, brane fluctuations, dubbed branons, are new degrees of freedom appearing in flexible brane-world models. These new fields behave as standard weakly interacting massive particles with a significant associated thermal relic density and would explain dark matter observational features.

Bibliography: leaves 84-88 / We present a collection of results on solitons in low-dimensional classical field theory. We begin by reviewing the geometrical setting of he nonlinear ơ-model and demonstrate the integrability of the theory in two-dimensions on a symmetric target manifold. After reviewing the construction of soliton solutions in the 0(3) ơ-model we consider a class of gauged nonlinear ơ-models on two-dimensional axially-symmetric target spaces. We show that, for a certain choice of self-interaction, these models are all self-dual and analyze the resulting Bogomol'nyi equations in the BPS limit using techniques from dynamical systems theory. Our analysis is then extended to topologically massive gauge fields. We conclude with a deviation into exploring links between four-dimensional self-dual Yang-Mills equations and various lower-dimensional field theories. In particular, we show that at the level of equations of motion, the Euclidean Yang-Mills equations in light-cone coordinates reduce to the two-dimensional nonlinear ơ-model.