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This thesis presents a theoretical analysis of the generation of Tollmien- Schlichting waves in a boundary layer on the wing surface due to external acoustic disturbances and elastic vibrations of the wing itself. An asymptotic approach based on the assumption that the Reynolds number, Re, is large is adopted here. In addition, it is assumed that in the spectrum of these disturbances there are harmonics, which come in resonance with the Tollmien-Schlichting wave on the lower branch of the stability curve. This work is restricted to the cases when the Stokes layer interacts with an isolated roughness, and the flow near the roughness is described by the triple-deck theory. The solution of the triple-deck problem is found in an analytic form and the main concern is with the flow behaviour downstream of the roughness. Further, it is found that there are Tollmien-Schlichting waves forming in the boundary layer behind the roughness, and their amplitudes have been expressed in terms of the receptivity coefficients, which represent the efficiency of the Tollmien-Schlichting wave generation process. The analysis of the boundary layer receptivity to the acoustic disturbances is conducted with an assumption that the flow in the free-stream is in the transonic regime. It is shown that in this situation there are two plane acoustic waves with distinctively different characteristics. One wave always travels downstream in the streamwise direction and has O(1) phase velocity. The second wave phase velocity is an order Re−1/9 quantity and it changes the direction of propagation depending on the Mach number. The analysis has shown that the receptivity coefficient depends on the initial frequency of the perturbations and on the free-stream Mach number. It has been shown that the absolute value of the receptivity coefficient is achieved when the Mach number is one. For the negative values of the parameter, the subsonic behaviour of the receptivity coefficient is recovered. The study of the “slow” moving wave shows that the receptivity coefficient now depends on the shape of the roughness itself as well as acoustic wave parameters and the free-stream Mach number. The third problem considered is a receptivity of the boundary layer on the wing surface due to an elastic vibration of the wing itself. It is found that, when the frequency of the wing surface vibrations is high, the perturbations produced by wing surface vibrations can be described in the framework of “piston” theory. In the flow considered there are two physical mechanisms through which an oscillatory motion of the fluid in the Stokes layer is ex- cited. The first one is the same as for the acoustic problem where the pressure gradient forces the fluid to oscillate in the direction along the wing surface. The second mechanism can only be observed in compressible flows. It was found that there are two Tollmien-Schlichting waves forming in the boundary layer behind the roughness. The first receptivity process is similar to the one studied earlier with the difference that now the flow is considered to be in the subsonic regime. The second receptivity process does not have an analogue in the literature, and it was found that it is large as compared with the first receptivity. This suggests that the receptivity to the wing surface vibrations has to play a major role in the laminar-turbulent transition in the boundary layer.

In this thesis I study various incarnations of rational homotopy theory in the world of arithmetic geometry. In particular, I study unipotent crystalline fundamental groups in the relative setting, proving that for a smooth and proper family of geometrically connected varieties f:X->S in positive characteristic, the rigid fundamental groups of the fibres X_s glue together to give an affine group scheme in the category of overconvergent F-isocrystals on S. I then use this to define a global period map similar to the one used by Minhyong Kim to study rational points on curves over number fields. I also study rigid rational homotopy types, and show how to construct these for arbitrary varieties over a perfect field of positive characteristic. I prove that these agree with previous constructions in the (log-)smooth and proper case, and show that one can recover the usual rigid fundamental groups from these rational homotopy types. When the base field is finite, I show that the natural Frobenius structure on the rigid rational homotopy type is mixed, building on previous results in the log-smooth and proper case using a descent argument. Finally I turn to l-adic étale rational homotopy types, and show how to lift the Galois action on the geometric l-adic rational homotopy type from the homotopy category Ho(Q_l-dga) to get a Galois action on the dga representing the rational homotopy type. Together with a suitable lifted p-adic Hodge theory comparison theorem, this allows me to define a crystalline obstruction for the existence of integral points. I also study the continuity of the Galois action via a suitably constructed category of cosimplicial Q_l-algebras on a scheme.

Due to the availability of the vast amount of graph-structured data generated in various experiment settings (e.g., biological processes, social connections), the need to rapidly identify network structural differences is becoming increasingly prevalent. In many fields, such as bioinformatics, social network analysis and neuroscience, graphs estimated from the same experimental settings are always defined on a fixed set of objects. We formalize such a problem as a labelled graph comparison problem. The main issue in this area, i.e. measuring the distance between graphs, has been extensively studied over the past few decades. Although a large distance value constitutes evidence of difference between graphs, we are more interested in the issue of inferentially justifying whether a distance value as large or larger than the observed distance could have been obtained simply by chance. However, little work has been done to provide the procedures of statistical inference necessary to formally answer this question. Permutation-based inference has been proposed as a theoretically sound approach and a natural way of tackling such a problem. However, the common permutation procedure is computationally expensive, especially for large graphs. This thesis contributes to the labelled graph comparison problem by addressing three different topics. Firstly, we analyse two labelled graphs by inferentially justifying their independence. A permutation-based testing procedure based on Generalized Hamming Distance (GHD) is proposed. We show rigorously that the permutation distribution is approximately normal for a large network, under three graph models with two different types of edge weights. The statistical significance can be evaluated without the need to resort to computationally expensive permutation procedures. Numerical results suggest the validity of this approximation. With the Topological Overlap edge weight, we suggest that the GHD test is a more powerful test to identify network differences. Secondly, we tackle the problem of comparing two large complex networks in which only localized topological differences are assumed. By applying the normal approximation for the GHD test, we propose an algorithm that can effectively detect localised changes in the network structure from two large complex networks. This algorithm is quickly and easily implemented. Simulations and applications suggest that it is a useful tool to detect subtle differences in complex network structures. Finally, we address the problem of comparing multiple graphs. For this topic, we analyse two different problems that can be interpreted as corresponding to two distinct null hypotheses: (i) a set of graphs are mutually independent; (ii) graphs in one set are independent of graphs in another set. Applications for the multiple graphs problem are commonly found in social network analysis (i) or neuroscience (ii). However, little work has been done to inferentially address the problem of comparing multiple networks. We propose two different statistical testing procedures for (i) and (ii), by again using a normality approximation for GHD. We extend the normality of GHD for the two graphs case to multiple cases, for hypotheses (i) and (ii), with two different permutation strategies. We further build a link between the test of group independence to an existing method, namely the Multivariate Exponential Random Graph Permutation model (MERGP). We show that by applying asymptotic normality, the maximum likelihood estimate of MERGP can be analytically derived. Therefore, the original, computationally expensive, inferential procedure of MERGP can be abandoned.

I describe an algebro-geometric theory of skeleta, which provides a unified setting for the study of tropical varieties, skeleta of non-Archimedean analytic spaces, and affine manifolds with singularities. Skeleta are spaces equipped with a structure sheaf of topological semirings, and are locally modelled on the spectra of the same. The primary result of this paper is that the topological space X underlying a non-Archimedean analytic space may locally be recovered from the sheaf |?x| of pointwise valuations of its analytic functions in other words, (X,|?x|) is a skeleton.

In this thesis we address the problem of data-driven coarse-graining, i.e. the process of inferring simplified models, which describe the evolution of the essential characteristics of a complex system, from available data (e.g. experimental observation or simulation data). Specifically, we consider the case where the coarse-grained model can be formulated as a stochastic differential equation. The main part of this work is concerned with data-driven coarse-graining when the underlying complex system is characterised by processes occurring across two widely separated time scales. It is known that in this setting commonly used statistical techniques fail to obtain reasonable estimators for parameters in the coarse-grained model, due to the multiscale structure of the data. To enable reliable data-driven coarse-graining techniques for diffusion processes with multiple time scales, we develop a novel estimation procedure which decisively relies on combining techniques from mathematical statistics and numerical analysis. We demonstrate, both rigorously and by means of extensive simulations, that this methodology yields accurate approximations of coarse-grained SDE models. In the final part of this work, we then discuss a systematic framework to analyse and predict complex systems using observations. Specifically, we use data-driven techniques to identify simple, yet adequate, coarse-grained models, which in turn allow to study statistical properties that cannot be investigated directly from the time series. The value of this generic framework is exemplified through two seemingly unrelated data sets of real world phenomena.

Let $G$ be a semisimple algebraic group over an algebraically closed field $K$, of characteristic $p \geq 0$. A closed subgroup of $G$ is said to be irreducible if it does not lie in any proper parabolic subgroup of $G$. In this thesis we address the following problem: classify the connected irreducible subgroups of $G$, up to conjugacy, where $G$ is of exceptional type. Work of Liebeck and Seitz classifies the conjugacy classes of simple, connected irreducible subgroups of rank at least 2, with a restriction on the characteristic of the underlying field ($p > 7$ is sufficient). When $G$ is of type $F_4$, Stewart has classified the conjugacy classes of simple, connected irreducible subgroups of rank at least 2 in all characteristics. We classify the conjugacy classes of simple, connected irreducible subgroups, of rank at least 2 for $E_6$, $E_7$ and $E_8$. Our approach works in all characteristics, rather than starting from the characteristics excluded in the result of Liebeck and Seitz. We use these classifications to prove corollaries concerning the representation theory of such irreducible subgroups. For example, with one exception, two simple irreducible connected subgroups of rank at least 2 are $G$-conjugate if and only if they have the same composition factors on the adjoint module of $G$. We also consider connected subgroups of rank 1. Work of Lawther and Testerman classifies conjugacy classes of rank 1 connected irreducible subgroups, with a restriction on $p$ ($p>7$ is sufficient). The connected irreducible subgroups of rank 1 were found, in arbitrary characteristic, by Amende for all but $E_8$. We give a new proof of this, finding a set of conjugacy class representatives without repetition. We prove corollaries on the overgroups of irreducible $A_1$ subgroups. For example, we prove that if $p=2$ or $3$ then any irreducible $A_1$ subgroup of $E_7$ is contained in $A_1 D_6$. Finally, consider the semisimple, non-simple connected irreducible subgroups. We classify these, up to conjugacy, for $G_2$, $F_4$ and $E_6$. So in conclusion, we classify conjugacy classes of connected irreducible subgroups of $G_2$, $F_4$ and $E_6$, all simple, connected irreducible subgroups of $E_7$ and all simple, connected irreducible subgroups of $E_8$ of rank at least 2.

This thesis divides very naturally into three chapters, reflecting the three separate areas I have worked in throughout my PhD studies. During my PhD, I published two papers, one relating to the work in the first chapter, and one to that of the second. At the time of submission, the project which the third chapter relates to was still ongoing, with plans for a future publication. The first chapter discusses work done on the real space renormalisation group (RG) for Ising and Potts models in 2 dimensions. Taking inspiration from Hasenbusch's work, a new framework for carrying out the RG is developed, its computational implementation discussed in some detail, as well as the results for a variety of systems and the implications of these results. The numerical scaling of the procedure and the consequences of this for future work are also covered. The results documented in chapter two are purely theoretical and presented in closed form. The research conducted is to do with properties of free interfaces. The second chapter is by and large critical of previous assumptions which have been made about the so-called wave vector dependent, in order to attempt to experimentally measure it and use these measurements to make further predictions. Using some simple toy models, many of these assumptions are shown to be false. In a sense, the goal of the research presented in the chapter two, is not to motivate further research, but to dissuade research in a direction we consider to be misguided, due to the faulty assumptions it is based on. The third chapter covers a small subset of a project concerned with studying correlations within the so-called Abelian Manna Model. The majority of the project involves (computational) Monte Carlo simulations of the dynamics of such systems, but as these results are not ready to present at the time of writing, the chapter is mainly concerned with some analytical results which were derived in order to validate our models for small systems, explain certain quirky phenomena arising from our simulations, and help quantify errors. Finally, there is an appendix which expands upon various topics from the first two chapters.

The subject of this thesis is the application of the Renormalised Perturbation Theory (RPT) to models of magnetic impurities embedded in a non-magnetic host metal. The theoretical description of such models is particularly challenging, for they present strong correlations that render the usual perturbation theory around the non-interacting limit inapplicable. The RPT addresses this di fficulty by incorporating the concept of a quasi-particle into a perturbative framework, and organising the expansion in terms of the quasi-particle parameters of the model rather than the bare parameters; it can thus be carried out regardless of the strength of the interactions. In the present work we present an introduction to the theory and discuss in detail the calculation of the renormalised self-energy expansions for the Anderson impurity model. To cope with the complexity of high-order calculations we develop and implement a computer algorithm to automatically compute the diagrammatic expansion in the renormalised theory to any order. As a demonstration of the usefulness of the theory, we use it to calculate the conductance of a single quantum dot, and of two quantum dots with an inter-dot coupling, to leading order in the quasi-particle interaction. To perform calculations in the renormalised theory it is essential that the values of the renormalised parameters describing the quasi-particles are known. Here we develop a general method for determining them entirely within the RPT framework, which relies on constructing renormalisation flow equations relating the renormalised parameters of two models whose bare parameters differ in infinitesimally. By determining the renormalised parameters for a model with bare parameters that render it amenable to ordinary perturbation theory, and solving the flow equations to relate them to the renormalised parameters of models with progressively stronger correlations, we succeed in deducing the renormalised parameters for models with strong correlations.