Riemann surfaces frequently possess automorphisms which can be exploited to simplify calculations. However, existing computer software (Maple in particular) is designed for maximum generality and has not yet been extended to make use of available symmetries. In many calculations, the symmetries can be most easily used by choosing a specific basis for H₁(Σ,Z) under which the automorphism group acts neatly. This thesis describes a Maple library, designed to be used in conjunction with the existing algcurves, which allows such a choice to be made. In addition we create a visual tool to simplify the presentation of Riemann surfaces as sheeted covers of C and the creation of homology bases suitable for use in the Maple library. Two applications are considered for these techniques, first Klein's curve and then Bring's. Both of these possess maximal symmetry groups for their genus, and this fact is exploited to obtain neat algebraic homology bases. In the Klein case the basis is novel; Bring's is derived from work in the hyperbolic setting by Riera. In both cases previous hyperbolic work and calculations are related to the algebraic results. Vectors of Riemann constants are calculated for both curves, again exploiting the symmetry. Finally this thesis moves back to simpler cases, and presents a general algorithm to convert results from general genus 2 curves into results based on a symmetric basis if one exists. This is applied to algebraic and numeric examples where we discover an elliptic surface covered in each case.

We use the methods of exponential asymptotics to study the solutions of a one dimensional wave equation with a non-constant wave speed c(x,t) modelling, for example, a slowly varying spatio-temporal topography. The equation reads htt(x,t) = (c2(x,t)hx(x,t))x' (1) where the subscripts denote differentiation w.r.t. the parameters x and t respectively. We focus on the exponentially small reflected wave that appears as a result of a Stokes phenomenon associated with the complex singularities of the speed. This part of the solution is not captured by the standard WKBJ (geometric optics) approach. We first revisit the time-independent propagation problem using resurgent analysis. Our results recover those obtained using Meyers integral-equation approach or the Kruskal-Segur (K-S) method. We then consider the time-dependent propagation of a wavepacket, assuming increasingly general models for the wave speed: time independent, c(x), and separable, c1(x)c2(t). We also discuss the situation when the wave speed is an arbitrary function, c(x,t), with the caveat that the analysis of this setup has yet to be completed. We propose several methods for the computation of the reflected wavepacket. An integral transform method, using the Dunford integral, provides the solution in the time independent case. A second method exploits resurgence: we calculate the Stokes multiplier by inspecting the late terms of the dominant asymptotic expansion. In addition, we explore the benefits of an integral transform that relates the coefficients of the dominant solution in the time-dependent problem to the coefficients of the dominant solution in the time-independent problem. A third method is a partial differential equation extension of the K-S complex matching approach, containing details of resurgent analysis. We confirm our asymptotic predictions against results obtained from numerical integration.

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

Some real-world applications involve situations where different physical phenomena acting on very different time scales occur simultaneously. The partial differential equations (PDEs) governing such situations are categorized as "stiff" PDEs. Stiffness is a challenging property of differential equations (DEs) that prevents conventional explicit numerical integrators from handling a problem efficiently. For such cases, stability (rather than accuracy) requirements dictate the choice of time step size to be very small. Considerable effort in coping with stiffness has gone into developing time-discretization methods to overcome many of the constraints of the conventional methods. Recently, there has been a renewed interest in exponential integrators that have emerged as a viable alternative for dealing effectively with stiffness of DEs. Our attention has been focused on the explicit Exponential Time Differencing (ETD) integrators that are designed to solve stiff semi-linear problems. Semi-linear PDEs can be split into a linear part, which contains the stiffest part of the dynamics of the problem, and a nonlinear part, which varies more slowly than the linear part. The ETD methods solve the linear part exactly, and then explicitly approximate the remaining part by polynomial approximations. The first aspect of this project involves an analytical examination of the methods' stability properties in order to present the advantage of these methods in overcoming the stability constraints. Furthermore, we discuss the numerical difficulties in approximating the ETD coefficients, which are functions of the linear term of the PDE. We address ourselves to describing various algorithms for approximating the coefficients, analyze their performance and their computational cost, and weigh their advantages for an efficient implementation of the ETD methods. The second aspect is to perform a variety of numerical experiments to evaluate the usefulness of the ETD methods, compared to other competing stiff integrators, for integrating real application problems. The problems considered include the Kuramoto-Sivashinsky equation, the nonlinear Schrödinger equation and the nonlinear Thin Film equation, all in one space dimension. The main properties tested are accuracy, start-up overhead cost and overall computation cost, since these parameters play key roles in the overall efficiency of the methods.

The content of this thesis can be divided into two broad topics. The first half investigates the deficient values and deficient functions of certain classes of meromorphic functions. Here a value is called deficient if a function takes that value less often than it takes most other values. It is shown that the derivative of a periodic meromorphic function has no finite non-zero deficient values, provided that the function satisfies a necessary growth condition. The classes B and S consist of those meromorphic functions for which the finite critical and asymptotic values form a bounded or finite set. A number of results are obtained about the conditions under which members of the classes B and S and their derivatives may admit rational, or slowly-growing transcendental, deficient functions. The second major topic is a study of real functions -- those functions which are real on the real axis. Some generalisations are given of a theorem due to Hinkkanen and Rossi that characterizes a class of real meromorphic functions having only real zeroes, poles and critical points. In particular, the assumption that the zeroes are real is discarded, although this condition reappears as a conclusion in one result. Real entire functions are the subject of the final chapter, which builds upon the recent resolution of a long-standing conjecture attributed to Wiman. In this direction, several conditions are established under which a real entire function must belong to the classical Laguerre-Polya class LP. These conditions typically involve the non-real zeroes of the function and its derivatives.

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