In this thesis we prove attractor existence and its smoothness for several classes of damped wave equations with critical nonlinearity. The term "critical" refers to the fact that behaviour of the solutions is determined not only by the energy but also by some more subtle space-time norms which are known as Strichartz norms. One of the main achievements of the work is the construction of the global attractor to the so called weakly damped wave equation with nonlinearity that admits fifth order polynomial growth. This problem was open from the first part of the 90's and its solution required combination of tools from various branches of mathematics. The ideas that we have developed we apply to several classes of wave equation with non-local damping. In this case the amount of energy dissipation that occurs in a fixed bunch of space depends on the solution in the whole region where the problem is considered. Though this model may seem to be more complicated at first sight, in fact, in this case solutions of the corresponding problem possess better regularizing properties. Finally we would like to remark that the developed ideas have general nature and thus open new opportunities for further investigations. In particular, the newly discovered techniques and ideas have already been successfully implemented for the construction of the global attractor in problems of phase separation.

In this thesis we study partition functions of supersymmetric gauge theories on compact backgrounds in various dimensions, with particular focus on infinite dimensional symmetry algebras encoded in these observables. The compact space partition functions of the considered theories can be decomposed into products of holomorphic blocks which are identified with partition functions on elementary geometries. Partition functions on different compact spaces can be obtained by fusing the holomorphic blocks with pairings reflecting the geometric decomposition of the backgrounds. An example of this phenomenon is given by the S4 partition function of 4d N = 2 theories, which can be written as an integral of two copies of the R4 Nekrasov partition function. Remarkably, the AGT correspondence identifies the S4 partition function of class S theories with Liouville CFT correlators. The perturbative integrand is identified with the product of CFT 3-point functions, while each copy of the non-perturbative instanton partition function is identified with conformal blocks of the Virasoro algebra. In this work we define a class of q-deformed CFT correlators, where chiral blocks are controlled by the q-Virasoro algebra and are identified with R4xS1 instanton partition functions. We derive the 3-point functions for two different q-deformed CFTs, and we show that non-chiral correlators can be identified with S5 and S4xS1 partition functions of certain 5d N = 1 theories. Moreover, particular degenerate correlators are mapped to S3 and S2xS1 partition functions of 3d N = 2 theories. This fits the interpretation of the 3d theories as codimension two defects. We also study 4d N = 1 theories on T2 fibrations over S2. We prove that when anomalies are canceled, the compact space partition functions can be expressed through holomorphic blocks associated to R2xT 2. We argue that for particular theories these objects descend from R4xT 2 partition functions, which we identify with the chiral blocks of an elliptically deformed Virasoro algebra.

We consider second order ordinary differential equations describing periodically forced dynamical systems with one and a half degrees of freedom. First, we study Hill's equation where we investigate boundedness of solutions. We also construct a class of equations, including a class of Hill's equation, for which we can obtain closed form solutions. We continue to analyse the boundedness of solutions to nonlinear systems with sufficiently high and low frequency forcing, utilising averaging techniques and KAM theory. We then focus on the study of dissipative systems with parameters chosen in a region of parameter space for which the equilibrium points of the linearised systems are Lyapunov stable. In dissipative systems, many of the solutions which exist in the absence of dissipation are destroyed, leaving a finite set of attractive solutions. We investigate the basins of attraction of the attractive equilibrium points and periodic orbits. In particular we study how the basins of attraction change when the coefficient of dissipation is allowed to initially vary as a function of time. Although it is the final value of the dissipative coefficient which determines which attractors eventually exist, the sizes of their corresponding basins of attraction are found to depend strongly on the full evolution of the coefficient. We study the dynamics of systems with the dissipative parameter modeled by both linearly increasing and decreasing functions of time with various gradients. In this instance we outline four cases pertaining to the sets of attractors at both the initial and final values of the coefficient of dissipation. For each scenario we state our expectations which are illustrated by means of numerical simulation for the systems of the pendulum with vertically oscillating support and the pendulum with periodically varying length. Further to this, a method which allows the fast numerical computation of basins of attraction for a system with an initially varying coefficient of dissipation is identified. This method is also applied in explaining a phenomenon in which the basins of attraction can drastically change when the coefficient of dissipation is a function of time.

I show that in the generic situations where a biological network, e.g. a protein interaction network, is in fact a subnetwork embedded in a larger bulk network, the presence of the bulk causes not just extrinsic noise but also memory effects. This means that the dynamics of the subnetwork will depend not only on its present state, but also its past. I use projection techniques to get explicit expressions for the memory functions that encode such memory effects, for generic protein interaction networks involving binary and unary reactions such as complex formation and phosphorylation, respectively. Remarkably, in the limit of low intrinsic copy-number noise such expressions can be obtained even for nonlinear dependences on the past. I illustrate the method with examples from a protein interaction network around epidermal growth factor receptor (EGFR), which is relevant to cancer signalling. These examples demonstrate that inclusion of memory terms is not only important conceptually but also leads to substantially higher quantitative accuracy in the predicted subnetwork dynamics. I also study how the presence of Michaelis-Menten reactions affect the behaviour of the subnetwork. While such reactions do not directly t into our framework of unary and binary reactions, I demonstrate that our approach can be generalised to include them. This is done by first introducing enzyme and enzyme complex species and reactions, then constructing the projected equations, and finally taking the limit of fast enzyme reactions that gives back Michaelis-Menten dynamics. I show that this limit can be taken in closed form, leading to a simple procedure that allows the projected equations to be constructed without ever introducing the fast variables explicitly. I then apply projection methods to the analysis of the effects of perturbations in the bulk network, e.g. from gene regulation processes. I show that the resulting behaviour of the linear response can again be decomposed according to a boundary structure, so that the total linear response is split into the eect of the bulk perturbation on the subnetwork boundary and the propagation of the perturbation from there to the rest of the subnetwork. I also use the projection method to find the steady states of the perturbed system in nonlinear response, which makes it possible to analyse biologically relevant scenarios such as knock-down experiments. Finally, I look at the statistics of the random force. I propose a simple approximation of the random force made up of a persistent piece and a random change in the subnetwork initial conditions. I verify that this gives accurate predictions for both the linearised and nonlinear dynamics.

Tailored graph ensembles are a developing bridge between statistical mechanics and biological networks. In this thesis, this concept is used to generate a suite of rigorous mathematical tools to quantify and compare the topology of cellular signalling networks. Earlier published results to quantify the entropy of constrained random graph ensembles are extended by looking at constraints relating to directed graphs, bipartite graphs, neighbourhood compositions and generalised degrees. To incorporate constraints relating to the average number of short loops, a number of innovative techniques are reviewed and extended, moving the analysis beyond the usual tree-like assumption. The generation of unbiased sample networks under some of these new constraints is studied. A series of illustrations of how these concepts may be applied to systems biology are developed. Topological observables are obtained from real biological networks and the entropy of the associated random graph ensemble is calculated. Certain studies on over-represented motifs are replicated and the influence of the choice of null model is considered. The correlation between the topological role of each protein and its lethality is studied in yeast. Throughout, this document aims to promote looking at a network as a realisation satisfying certain constraints rather than just as a list of nodes and edges. This may initially seem to be an abstract approach, but it is in fact a more natural viewpoint within which to consider many fundamental questions regarding the origin, function and design of observed real networks.

String phenomenology aims to explain the physics of the universe in the context of string theory, the leading candidate to unify gravitational and quantum physics. A main ingredient in constructing such models is compactifying the ten-dimensional theory on a six-dimensional manifold, so that one is left with the four non-compact dimensions modelling space-time. Performing this step allows one to reformulate many physical phenomena as properties of the geometry of the compactication manifold, and to nd generic constraints on physical models using methods from algebraic geometry. One such phenomenon in nature are gauge theories, both abelian and non-abelian. In this thesis, we undertake a systematic investigation of the interplay of the two in compactications of F-theory. In F-theory, the relevant compactication spaces are elliptically bered Calabi-Yau manifolds. They are particularly well-suited to the study of gauge symmetries in string phenomenology, since they both allow the existence of exceptional gauge symmetries such as E6 as well as the localization of gauge degrees of freedom on subloci of the compacti-cation manifold. Specically, non-abelian gauge theories are encoded as singularities of the elliptic bration, and the rational sections of the bration specify the abelian part of the gauge group. Using tools from algebraic geometry, we study singularities of elliptic brations with a group of rational sections of rank 1, i.e. with a single abelian gauge factor. In the rst part of the thesis, we rene the spectral cover formalism, which is a way to study local properties of the subloci of the compactication manifold on which gauge degrees of freedom are localized in F-theory. We do so by introducing the spectral divisor. The spectral divisor allows one to construct gauge uxes in F-theory in a purely local description. We exemplify this construction for an elliptic bration with associated gauge group E6. In the second part of the thesis, we use Tate's algorithm to obtain a comprehensive classication of singular bers and an explicit list of possible realizations of F-theory compactications with both an abelian and non-abelian gauge symmetries. This list is complete for low-rank gauge symmetries, which are most relevant for building models of the universe, and thus allows to completely classify all F-theory models with a single abelian gauge factor. In particular, this list includes phenomenologically interesting brations not considered in the literature before.

We discuss the historical evidence for the conjecture that the non-linear realisation of the Kac-Moody algebra E8+++, which will be referred to as E11, describes the extension of eleven-dimensional supergravity known as M-theory. The algebraic background is presented and some of the consequences of the conjecture are explored. In particular, we present the construction of half-BPS branes using the E11 solution generating element with low-level roots before discussing the role of general roots in the solution generating method. The correspondence between roots within E11 and brane solutions is used to reproduce the rules for brane congurations which lead to bound and marginal states. Using these rules, we present the embeddings of simply-laced algebras within E11 with their supergravity solution interpretation. The use of non-linear sigma-models with symmetric spaces to describe the hidden symmetries of gravity, as well as extended gravitational theories, is reviewed. Examples include: the original work of Ehlers, the more general construction of axisymmetric stationary solutions and theories which are consistent truncations of eleven-dimensional supergravity. It is shown that these symmetries generate non-linear transformations of solutions, of which many have well-understood physical interpretations. Applications of the target space symmetries are described and used to generate and transform between solutions. Motivated by the use of null geodesics on symmetric spaces to describe solutions of theories with hidden symmetries, we construct one-dimensional sigma models. These models are built with cosets of normal real forms of the nite, simply-laced algebras and general involution invariant subalgebras. The SL(n;R)=SO(p; q) models, with n 4, are reproduced before we present our work with general An(n), Dn(n) and En(n). Solutions are presented for algebras of low rank and used, iteratively, to construct solutions of arbitrary rank. These models are embedded into A+++ n(n) algebras to generate dual-gravity solutions and E11(11) to generate supergravity solutions with the congurations which are classied in the preliminary material. A special set of maximal co-dimension A2(2) solutions embedded within A+++ n(n) are considered which form a telescopic series of nite algebras within A+ 2(2). These are shown to interpolate between known gravity solutions and exotic objects. We discuss and interpret the gravitational theory extensions which these objects are solutions of. We show that bound states of branes with various algebraic congurations are shown to possess symmetries which are easily described within the sigma model. Several examples are provided and we discuss the role of these symmetries in solution generation.

In this thesis we consider the setting where R is a complete discrete valuation ring of mixed characteristic (0, p) where p > 0 is prime. Let (p, ..., p) denote either the group Z/pZ x ... x Z/pZ or the product of rank p group schemes. Given a degree p Galois cover between R-curves, it is understood when the cover has the structure of a torsor under a finite flat group scheme of rank p. We investigate torsors under group schemes of type (p, ..., p) and establish criteria for their existence. We treat the boundary of the formal fibre and extend our knowledge of the conductor and degree of the different in degree p to the (p, p) setting. We also take the opportunity to explain how this can be naturally extended to the general (p, ..., p) case. Finally, we generalise a local vanishing cycles formula for curves known in the degree p case to Galois groups of type (p, p), relating the genus of two points in terms of just the cover's ramification data and the conductors acting at the boundaries.

It is well known that instabilities in rotational flows, such as those found on planets or in the solar tachocline, lead to the formation of long-lived zonal jets. Pioneered by the work of Rhines, after whom the fundamental length scale of these jets is named, much work has been put into simulating these formations for various situations. These models are often motivated by applications such as the cloud bands of Jupiter, or the geophysical stratospheric polar night jet. The exploration of a driven flow under rotational effects provides a fascinating subject for investigation. Many aspects of fluid behaviour can be observed; from the interaction of mean flows with small-scale turbulence, to the effects of wave-like motion and the transport of potential vorticity (PV). The gradient of PV produces anisotropic behaviour and an inverse energy cascade forming zonal jets with properties governed by the nonlinearity of the system. Starting on the basis of a simple two-dimensional beta-plane system (incompressible Navier-Stokes) under the effects of a body force, we implement a shearing box coordinate system in order to study the competing effects of shear and rotation. We use this in combination with spectral methods to numerically simulate the flow. Following the work of Moffatt, we use the flux of a passive scalar field to calculate and compare the effective diffusivity of the system over a range of the parameter space. In particular, we investigate the effect shear has of disrupting the formation of beta-plane jets, and the resulting modification to transport. We use quasi-linear analysis to further explore these systems. In doing so, we establish important mechanisms bought about by key parameters. We extend the scope of our investigation to include general mean flows. We show relationships between the mean flow and its feedback on the fluid, particularly regarding the perpetuation of zonal jets. We give important modifications to the flow bought about by frictional forces such as viscosity, and show the inherently complicated effect beta has on the mean flow feedback. We make an extension to the above work by looking at the corresponding magnetohydrodynamic system, investigating the effect of adding a magnetic field to a sheared/rotating flow. We find that the magnetic field disrupts beta-plane jets, creating a resonance-like peak in transport, suppressing it when the field strength is increased. We discuss the three predominant quantities governing the feedback for an MHD flow analytically; the Reynolds stress, Lorentz force and magnetic flux. We find that the magnetic flux allows for interactions between the vorticity gradient and magnetic field which potentially allow for zonal features in the mean field; we observe these in our numerical simulations.

Static, spherically symmetric solutions of the Einstein-Maxwell equations in the presence of a cosmological constant are studied, and new classes of solutions are derived. Namely the charged Einstein static universe and the interior and exterior charged Nariai spacetimes, these solutions form a subclass of the RNdS solution with distinct properties. The charged Nariai solutions are then matched at a common boundary. When constructing solutions to gravitational theories it is important that these matter distributions remain in hydrostatic equilibrium. If this equilibrium is lost, with internal gravitational forces dominating internal stresses, the solution will collapse under its gravitational field. An upper bound on the mass-radius ratio Mg/R for charged solutions in de Sitter space is derived, this bound implies hydrostatic equilibrium. The result is achieved by assuming the radial pressure p≥0 and energy density ρ≥0, plus p+2p⊥≤ρ where the tangential pressure p⊥≠ p. The bound provides a generalisation of Buchdahl's inequality, 2M/R ≤8/9, valid for Schwarzschild's solution. In the limit Q→0, Λ→0, the bound reduces to Buchdahl's inequality. Solutions in hydrostatic equilibrium are also considered in modified f(T) gravity. It is shown that the tetrads eⁱμ impact the structure of the field equations, and certain tetrads impose unnecessary constraints. Two particular tetrads are studied in more detail, solutions are then found for both tetrads, and a conservation equation is obtained using an analogous method to obtaining the Tolman-Oppenheimer-Volkoff equation. Although both tetrad fields locally give rise to the spherically symmetric metric, the tetrad fields are not globally well-defined and hence cannot be described as spherically symmetric. We then derive an upper bound on M/R which also implies hydrostatic equilibrium, this yields some constraints on the form of f(T) given a particular tetrad that locally gives rise to the line element ds²=exp(a)dt²-exp(b)dr²-r²dΩ².