In this thesis we explore extremal, structural, and algorithmic problems involving the partitioning of combinatorial structures. We begin by considering problems from the theory of graph cuts. It is well known that every graph has a cut containing at least half its edges. We conjecture that (except for one example), given any two graphs on the same vertex set, we can partition the vertices so that at least half the edges of each graph go across the partition. We give a simple algorithm that comes close to proving this conjecture. We also prove, using probabilistic methods, that the conjecture holds for certain classes of graphs. We consider an analogue of the graph cut problem for posets and determine which graph cut results carry over to posets. We consider both extremal and algorithmic questions, and in particular, we show that the analogous maxcut problem for posets is polynomial-time solvable in contrast to the maxcut problem for graphs, which is NP-complete. Another partitioning problem we consider is that of obtaining a regular partition (in the sense of the Szemeredi Regularity Lemma) for posets, where the partition respects the order of the poset. We prove the existence of such order-preserving, regular partitions for both the comparability graph and the covering graph of a poset, and go on to derive further properties of such partitions. We give a new proof of an old result of Frankl and Furedi, which characterises all 3-uniform hypergraphs for which every set of 4 vertices spans exactly 0 or 2 edges. We use our new proof to derive a corresponding stability result. We also look at questions concerning an analogue of the graph linear extension problem for posets.

This thesis establishes a generalised setting with which to unify the study of finite local complexity (FLC) patterns. The abstract notion of a pattern is introduced, which may be seen as an analogue of the space group of isometries preserving a tiling but where, instead, one considers partial isometries preserving portions of it. These inverse semigroups of partial transformations are the suitable analogue of the space group for patterns with FLC but few global symmetries. In a similar vein we introduce the notion of a collage, a system of equivalence relations on the ambient space of a pattern, which we show is capable of generalising many constructions applicable to the study of FLC tilings and Delone sets, such as the expression of the tiling space as an inverse limit of approximants. An invariant is constructed for our abstract patterns, the so called patternequivariant (PE) homology. These homology groups are defined using infinite singular chains on the ambient space of the pattern, although we show that one may define cellular versions which are isomorphic under suitable conditions. For FLC tilings these cellular PE chains are analogous to the PE cellular cochains [47]. The PE homology and cohomology groups are shown to be related through Poincare duality. An efficient and highly geometric method for the computation of the PE homology groups for hierarchical tilings is presented. The rotationally invariant PE homology groups are shown not to be a topological invariant for the associated tiling space and seem to retain extra information about global symmetries of tilings in the tiling space. We show how the PE homology groups may be incorporated into a spectral sequence converging to the Cech cohomology of the rigid hull of a tiling. These methods allow for a simple computation of the Cech cohomology of the rigid hull of the Penrose tilings.

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This thesis uses primary and secondary sources to study advances in complex number theory during the 17th and 18th Centuries. Some space is also given to the early 19th Century. Six questions concerning their rules of operation, usage, symbolism, nature, representation and attitudes to them are posed in the Introduction. The main part of the thesis quotes from the works of Descartes, Newton, Wallis, Saunderson, Maclaurin, d'Alembert, Euler, Waring, Frend, Hutton, Arbogast, de Missery, Argand, Cauchy, Hamilton, de Morgan, $ylvester and o~hers, mainly in chronological order, with comment and discussion. More attention has been given tp algebraists, the originators of most advances in complex numbers, than to writers in trigonometry, calculus and analysis, who tended to be users of them. The last chapter summarises the most imoortant points and considers the extent to which the six questions have been resolved. The most important developments during the period are identified as follows the advance in status of complex numbers from 'useless' to 'useful' their interpretation by Wallis, Argand and Gauss in arithmetic, geometric and algebraic ways the discovery that they are essential for understanding polynomials and logarithmic, exponential and trigonometric functions the extension of trigonometry, calculus and analysis into the complex number field the discovery that complex numbers are closed under exponentiation, and so under all algebraic operations partial reform of nomenclature and symbolism the eventual extension of complex number theory to n dimensions In spite of the advances listed above, it is noted that there was a continued lack of confidence in complex numbers and avoidance of them by some mathematicians, particularly in England.

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The main goal of this thesis is to demonstrate the adaptation of the Hardy-Littlewood circle method to the function field setting. Particular emphasis is placed on counting points on rather general varieties defined over function fields. With suitable notions for integral points and heights in this setting, asymptotic formulae are obtained for the number of integral points on these varieties with bounded heights, subject to certain conditions on the varieties and function fields at hand. Under the same conditions, weak approximation is also established for those varieties that are smooth. This counting problem is then specialised to the case of a cubic hypersurface, in which similar results are obtained, with further refinements available in terms of the invariant of the cubic involved. The thesis also addresses the particular case of Waring's problem for cubes over the integers. The representability of positive integers as the sum of four cubes, two of which are small, is investigated. A lower bound is obtained for how small these two cubes call be without impeding the representation of almost all natural numbers this way. An asymptotic formula is finally established for the number of such representations for almost all positive integers.