We develop a method that is capable of proving lower bounds that are consistent with Manin's conjecture for the number of rational points of bounded height on Fano varieties for which establishing Manin's conjecture is far out of reach with current technology. More specifically we develop the fibration method from the perspective of analytic number theory and as an application we provide correct lower bounds for families of smooth cubic surfaces with a rational line. The key ingredient of our approach is an asymptotic formula with a precise error term for Manin's conjecture for general smooth conics. Using the geometry of conic bundles we transform the average of the Peyre constants of the fibers into divisor sums over values of quintic binary forms. The underlying divisor functions include Dirichlet convolutions of quadratic characters whose modulus is unbounded, an element which is new in the large body of work on divisors sums. In the concluding chapter we combine the fibration method with sieve techniques to attack the saturation number problem. We show that rational points with at most 22 prime factors counted with multiplicity form a Zariski dense subset of each smooth cubic surface with two rational skew lines. This chapter is part of joint work with Yuchao Wang.

Our main focus shall be the use of Fourier analytic methods to count solutions to diophantine equations and inequalities. We begin by using the Hardy-Littlewood circle method to produce mean and variance statistics for the number of solutions to diophantine equations in a thin family. The bulk of this thesis concerns the study of diophantine inequalities, III particular using the Davenport- Heilbronn method. In many cases inequalities may be treated analogously to equations, but sometimes new difficulties arise. Initially, we consider the special case in which a cubic diophantine inequality splits into several parts, providing lower bounds for the number of variables required to ensure that the inequality has a nontrivial solution. Research on diophantine inequalities has previously focussed on homogeneous polynomials with real coefficients. We investigate a new type of inequality problem, involving rational polynomials evaluated at irrationally shifted copies of the integers. The diagonal case gives rise to a new inequalities analogue to Waring's problem, in which sums of shifted powers are considered. Moving onto more general systems of polynomials, we present the first inequalities analogue to Birch's theorem. When it comes to diophantine equations, a popular objective is to demonstrate that an equation has the expected number of solutions. For cubic equations, we supplement the existing literature by showing that these solutions are evenly distributed, in a precise sense. The analytic methods presented here generalise to arbitrary degree. Finally, we consider Waring-Goldbach equations in which the variables are restricted to lie in a prescribed set. Specifically, we show that any subset of the dth powers of primes with positive relative density contains nontrivial solutions to a translation-invariant linear equation in d2 + 1 or more variables, with explicit quantitative bounds.

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In this thesis, we study equations of the form $[x_1,u_1]+[x_2, u_2]+\ldots+[x_k,u_k]=0$ over a free Lie algebra $L$, where $k>1$ and the coefficients $u_1, u_2, \ldots,u_k$ belong to $L$. The starting point of this research is a paper [22], in which the authors embarked on a systematic study of very concrete linear equations over free Lie algebras. They focused on the given equations in the case where $k=2$. We generalise and develop a number of the results on equations with two variables to equations with an arbitrary number of indeterminates. Most of the results refer to the case where the coefficients coincide with the free generators of $L$. Throughout our research, we study some features of the solution space of these equations such as the homogenous structure and the fine homogenous structure. The main achievement in this work is that we give a detailed description of the solution space. Then we obtain explicit bases for some specific fine homogeneous components of the solution space, in particular, we give a basis for the "multilinear'' fine homogenous component. Moreover, we generalise earlier results on commutator calculus using the "language'' of free Lie algebras and apply them to determine the radical and the coordinate algebra of the solution space of the given equations.

Many of the classes of objects studied in geometry are defined by first choosing a class of nice spaces and then allowing oneself to glue these local models together to construct more general spaces. The most well-known examples are manifolds and schemes. The main purpose of this thesis is to give a unified account of this procedure of constructing a category of spaces built from local models and to study the general properties of such categories of spaces. The theory developed here will be illustrated with reference to examples, including the aforementioned manifolds and schemes. For concreteness, consider the passage from commutative rings to schemes. There are three main steps: first, one identifies a distinguished class of ring homomorphisms corresponding to open immersions of schemes; second, one defines the notion of an open covering in terms of these distinguished homomorphisms; and finally, one embeds the opposite of the category of commutative rings in an ambient category in which one can glue (the formal duals of) commutative rings along (the formal duals of) distinguished homomorphisms. Traditionally, the ambient category is taken to be the category of locally ringed spaces, but following Grothendieck, one could equally well work in the category of sheaves for the large Zariski site—this is the so-called ‘functor of points approach’. A third option, related to the exact completion of a category, is described in this thesis. The main result can be summarised thus: categories of spaces built from local models are extensive categories with a class of distinguished morphisms, subject to various stability axioms, such that certain equivalence relations (defined relative to the class of distinguished morphisms) have pullback-stable quotients; moreover, this construction is functorial and has a universal property.

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This work belongs to the theory of fluid flows at high Reynolds number, characterised by low viscosity and, to a lesser extent, high speeds. It is a mathematically intensive field covering most aspects of applied mathematics, from Partial Differential equations to Numerical Analysis and Complex Analysis. Two of the central areas of research within this particular branch of fluid dynamics are the boundary-layer laminar-turbulent transition and the boundary-layer separation. Both of these phenomena have a high impact on the aerodynamic performance of aircraft wings, turbine blades and other fast-moving objects such as rockets. As a consequence, their study is relevant both for industrial applications and to the scientific community as a whole. The mathematical element pertaining to the analysis of these two phenomena is the consistent use of asymptotic methods, such as matched asymptotic expansions and multiple scales analysis. In particular, this work will be based on the viscous-inviscid interaction theory originally discovered by Lin (1955) and further developed by the work of Neiland (1969) and Stewartson & Williams (1969). In the first chapter of this thesis, we will study the initial stages of the laminar-turbulent transition of a compressible boundary layer on a swept wing. The instability of interest will be the stationary crossflow vortex, which is known to be the main instability mode in three-dimensional boundary layers on a swept wing. We will focus on two specific aspects of the transition process, namely the receptivity and linear stability analysis of the flow. The receptivity mechanism introduced in our work is a roughness of size comparable with that of the boundary layer thickness. This justifies our restriction to the study of the inviscid instability of the flow and the use of the impermeability condition on the roughness. The remaining two chapters are concerned with the viscous-inviscid interaction of the boundary layer in the vicinity of a surface discontinuity. We will first study the behaviour of the subsonic flow exposed to the singular pressure gradient dp/dx = κ(x0 - x)-1/3, as x → x0. It forms when the body contour has a point x = x0 near which yw = (x0 - x)5/3. We will show how logarithmic terms need to be included in the solution. We then study a similar problem, this time in the context of an incoming transonic flow near a point of curvature discontinuity. We will show that in both cases the boundary layer experiences an 'extreme acceleration'.