The Bogomolnyi equation is a PDE for a connection and a Higgs field on a bundle over a 3 dimensional Riemannian manifold. Possible extensions of this PDE to higher dimensions preserving the ellipticity modulo gauge transformations require some extra structure, which is available both in 6 dimensional Calabi-Yau manifolds and 7 dimensional G2 manifolds. These extensions are known as higher dimensional monopole equations and Donaldson and Segal proposed that 'counting' solutions (monopoles) may give invariants of certain noncompact Calabi-Yau or G2 manifolds. In this thesis this possibility is investigated and examples of monopoles are constructed on certain Calabi-Yau and G2 manifolds. Moreover, this thesis also develops a Fredholm setup and a moduli theory for monopoles on asymptotically conical manifolds.

In this thesis a number of related topics in random dynamical systems theory are studied: local attractors and attractor-repeller pairs, the exponential dichotomy spectrum and bifurcation theory. We review two existing theories in the literature on local attractors for random dynamical systems on compact metric spaces and associated attractor-repeller pairs and Morse decompositions, namely, local weak attractors and local pullback attractors. We extend the theory of past and future attractor-repeller pairs for nonautonomous systems to the setting of random dynamical systems, and define local strong attractors, which both pullback and forward attract a random neighbourhood. Some examples are given to illustrate the nature of these different attractor concepts. For linear systems considered on the projective space, it is shown that a local strong attractor that attracts a uniform neighbourhood is an object with sufficient properties to prove an analogue of Selgrade's Theorem on the existence of a unique finest Morse decomposition. We develop the dichotomy spectrum for random dynamical systems and investigate its relationship to the Lyapunov spectrum. We demonstrate the utility of the dichotomy spectrum for random bifurcation theory in the following example. Crauel and Flandoli [Journal of Dynamics and Differential Equations, 10(2):259-274, 1998] studied the stochastic differential equation formed from the deterministic pitchfork normal form with additive noise. It was shown that for all parameter values this system possesses a unique invariant measure given by a globally attracting random fixed point with negative Lyapunov exponent, and hence the deterministic bifurcation scenario is destroyed by additive noise. Here, however, we show that one may still observe qualitative changes in the dynamics at the underlying deterministic bifurcation point, in terms of: a loss of hyperbolicity of the dichotomy spectrum; a loss of uniform attractivity; a qualitative change in the distribution of finite-time Lyapunov exponents; and that whilst for small parameter values the systems are topologically equivalent, there is a loss of uniform topological equivalence.

The basic concepts of Majorana theory were introduced by A. A. Ivanov (2009) as a tool to examine the subalgebras of the Griess algebra V_M from an elementary axiomatic perspective. A Majorana algebra is a commutative non-associative real algebra generated by a finite set of idempotents, called Majorana axes, that satisfy some properties of Conway's 2A-axes of V_M. If G is a finite group generated by a G-stable set of involutions T, a Majorana representation of (G,T) is an algebra representation of G on a Majorana algebra V together with a compatible bijection between T and a set of Majorana axes of V . Ivanov's definitions were inspired by Sakuma's theorem, which establishes that any two-generated Majorana algebra is isomorphic to one of the Norton-Sakuma algebras. Since then, the construction of Majorana representations of various finite groups has given non-trivial information about the structure of V_M. This thesis concerns two main themes within Majorana theory. The first one is related with the study of some low-dimensional Majorana algebras: the Norton-Sakuma algebras and the Majorana representations of the symmetric group of degree 4 of shapes (2A,3C) and (2B,3C). For each one of these algebras, all the idempotents, automorphism groups, and maximal associative subalgebras are described. The second theme is related with a Majorana representation V of the alternating group of degree 12 generated by 11,880 Majorana axes. In particular, the possible linear relations between the 3A-, 4A-, and 5A-axes of V and the Majorana axes of V are explored. Using the known subalgebras and the inner product structure of V , it is proved that neither sets of 3A-axes nor 4A-axes is contained in the linear span of the Majorana axes. When V is a subalgebra of V_M, these results, enhanced with information about the characters of the Monster group, establish that the dimension of V lies between 3,960 and 4,689.

It is the main goal of this text to study certain aspects of time-frequency analysis on the 2n+1-dimensional Heisenberg group. More specifically, we will discuss how the well-studied notions of modulation spaces and Weyl quantization can be extended from the Euclidean space Rn to the Heisenberg group Hn. For quite a long time already this group has served as a good test object to verify which concepts and results from Euclidean (thus Abelian) analysis carry over to simple instances of non-Abelian structures. In the case of the Weyl quantization a reasonable answer for $\H$ was first proposed by A. S. Dynin almost forty years ago, although it was studied in more detail only some twenty years after that by G. B. Folland. We will review the foundations laid by Dynin and Folland and present some new results about left-invariant differential operators and the natural product of symbols, the Moyal product. The special tool for our analysis is a $3$-step nilpotent Lie group to which we will refer as the Dynin-Folland group. As the name suggests it originates in the works of the afore-mentioned authors. The group's unitary irreducible representations are in fact the key to both the Weyl quantization and modulation spaces on Hn. Our results on modulation space on the Heisenberg group are based on H. Feichtinger and K. Grochenig's coorbit theory and a more recent adaption of it by I. and D. Beltictua, which focuses on modulation spaces arising from nilpotent Lie groups. We will use a blend of both approaches and discuss the modulation spaces induced by the Dynin-Folland group, among them a type of modulation spaces on Hn.

Data streams have become ubiquitous over the last two decades; potentially unending streams of continuously-arriving data occur in fields as diverse as medicine, finance, astronomy and computer networks. As the world changes, so the behaviour of these streams is expected to change. This thesis describes sequential methods for the timely detection of changes in data streams based on an adaptive forgetting factor framework. These change detection methods are first formulated in terms of detecting a change in the mean of a univariate stream, but this is later extended to the multivariate setting, and to detecting a change in the variance. The key issues driving the research in this thesis are that streaming data change detectors must operate sequentially, using a fixed amount of memory and, after encountering a change, must continue to monitor for successive changes. We call this challenging scenario "continuous monitoring" to distinguish it from the traditional setting which generally monitors for only a single changepoint. Additionally, continuous monitoring demands that there be limited dependence on the setting of parameters controlling the performance of the algorithms. One of the main contributions of this thesis is the development of an efficient, fully sequential change detector for the mean of a univariate stream in the continuous monitoring context. It is competitive with algorithms that are the benchmark in the single changepoint setting, yet our change detector only requires a single control parameter, which is easy to set. The multivariate extension provides similarly competitive performance results. These methods are applied to monitoring foreign exchange streams and computer network traffic.

High frequency homogenisation is applied to develop asymptotics for waves propagating along interfaces or structured surfaces. The asymptotic method is a two-scale approach fashioned to encapsulate the microstructural information in an effective homogenised macro- scale model. These macroscale continuum representations are constructed to give solutions near standing wave frequencies and are valid even at high frequencies. The asymptotic the- ory is adapted to model dynamic phenomena in functionally graded waveguides and in periodic media, revealing their similarities. Demonstrating the potential of high frequency homogenisation, the theory is extended for treating localisation phenomena in discrete peri- odic media containing localised defects, and for identifying Rayleigh-Bloch waves. In each of the studies presented here the asymptotics are complemented by analytical or numerical solutions, or both.

The aim of this thesis is to investigate two questions which naturally arise in the context of the classification of algebraic varieties. The first project concerns the structure of Mori fibre spaces: these objects naturally appear in the birational classification of higher dimensional varieties and the minimal model program. We ask which Fano varieties can appear as a fibre of a Mori fibre space and introduce the notion of fibre-likeness to study this property. This turns out to be a rather restrictive condition: in order to detect this property, we obtain two criteria (one sufficient and one necessary), which turn into a characterisation in the rigid case. Many applications are discussed and the basis for the classification of fibre-like Fano varieties is presented. In the second part of the thesis, an effective version of Matsusaka's theorem for arbitrary smooth algebraic surfaces in positive characteristic is provided: this gives an effective bound on the multiple which makes an ample line bundle D very ample. A careful study of pathological surfaces is presented here in order to bypass the classical cohomological approach. As a consequence, we obtain a Kawamata-Viehweg-type vanishing theorem for arbitrary smooth algebraic surfaces in positive characteristic.

This thesis is concerned with the effect that boundary-layer instabilities have on laminar- turbulent transition over a commercial aircraft wing. We consider the effect that changing the structure of a wing's surface may have on these instabilities. This thesis is separated into two parts, each concerning a different instability. Firstly our focus is on Tollmien-Schlichting waves; we investigate how abrupt changes may affect boundary-layer transition. The abrupt changes considered are junctions between rigid and porous surfaces. A local scattering problem is formulated; the abrupt changes cause waves to scatter in a subsonic boundary layer. The mechanism is described mathematically by using a triple-deck formalism, while the analysis across the junctions is based in a Wiener- Hopf factorisation. The impact of the wall junctions is characterised by a transmission coefficient, defined as the ratio of the amplitudes of the transmitted and incident waves. From our analysis we determine the effectiveness of porous strips in delaying transition. In the second part of this thesis we concentrate on a curved wing. Over curved sur- faces Görtler vortices may be generated; our focus is on long-wavelength Görtler vortices and the effect of changing curvature. The flow is described using a three-tiered system that balances the displacement and centrifugal forces. Two different problems concerning Görtler vortices are investigated, firstly we consider the effect of slowly varying curvature. Using a WKB approximation we derive multi-scale systems of equations, allowing us to find leading-order analytic solutions. The second problem concerning curved surfaces considers the effect of long-wavelength Görtler vortex-wave interaction. We use vortex-wave interaction theory to describe the evo- lution of this nonlinear interaction over a concave surface, where the curvature is modified in the streamwise direction. Analytical solutions are found for the vortex-induced shear stress and the wave pressure amplitude, using these solutions we solve for the remaining variables numerically.

In this thesis spectral inequalities and trace formulae for discrete and continuous differential operators are discussed. We first investigate spectral inequalities for Jacobi operators with matrix-valued potentials and present a new, direct proof of a sharp inequality corresponding to a Lieb-Thirring inequality for the power 3/2 using the commutation method. For the special case of a discrete Schrödinger operator we also prove new inequalities for higher powers of the eigenvalues and the potential and compare our results to previously established bounds. We then approximate a Schrödinger operator on L²(R) by Jacobi operators on ℓ²(Z) and use the established inequalities to provide new proofs of sharp Lieb-Thirring inequalities for the powers γ = 1/2 and γ = 3/2. By means of interpolation we derive spectral inequalities for Jacobi operators that yield (non-sharp) Lieb-Thirring constants on the real line for powers 1/2 < γ < 3/2. We then consider Schrödinger operators on a finite interval [0,b] with matrix-valued potentials and establish trace formulae of the Buslaev-Faddeev-Zakharov type. The results link sums of powers of the negative eigenvalues to terms dependent on the potential and scattering functions. Finally, we discuss the Berezin inequality, which is well-known on sets of finite measure and find an analogous inequality for the magnetic operator with constant magnetic field on a set whose complement has finite measure. We obtain a similar bound for the Heisenberg sub-Laplacian.

An asymptotic method for predicting stability characteristics, both stationary and travelling crossflow vortices, over a variety of surface variations was created. These include flat, convex and concave curved surfaces. Comparisons were made with two different numerical methods (Parabolised Stability Equations and Velocity-Vorticity) and good agreement, to within 5% of the numerical value of the crossflow mode streamwise growth rate was met for both stationary and travelling modes initially for a flat surface. An additional comparison was made with the streamwise growth rates to observe the impact of including curvature and a small convex curvature surface variation was used. Similar results were achieved for this study also. Likewise results for travelling crossflow modes were with accordance with the numerical values. To understand how effective this disturbance in penetrating the boundary-layer, receptivity analysis was developed to analyse various mechanisms in the production of crossflow vortices. A response function was established from the receptivity analysis to calculate the efficiency of this process. The response function is largest near the leading edge, meaning the disturbance is most effective at propagating into the boundary layer there. This means that the approach qualitatively agrees with other research methods. This is true for all surface curvatures and both crossflow modes. There is an intriguing behaviour the response function exhibits for small concave curvature with travelling modes at a moderate frequency. When we consider moderate spanwise wavenumber, the response function is much larger than other modes or surface variation and this could have repercussions for experiments. Careful consideration is needed for this case and can be avoided with the aid of this research. Finally, an asymptotic theory was created to analyse two-dimensional closed streamlines for secondary instabilities. The first instability analysed was the elliptical instability, due to the links to turbulence and the initial interest in this general problem. The method anticipates the existence of short-wave three-dimensional disturbances on a streamline at a distance away from the centre of the vortex of this secondary instability. There was no limitation in the study for symmetrical known streamlines, the analysis can be extended further to analyse any two-dimensional closed streamline such as separation bubbles. With this in mind an observation was required to test this hypothesis and the approach was tested on the structure inside a cavity, from which the location and behaviour of the disturbance was correctly predicted.