The aim of the thesis is to work towards a many-valued logic over a commutative unital quantale and, at the same time, towards a generalisation of coalgebraic logic enriched over a commutative unital quantale Ω. This is done by noticing that the contravariant powerset adjunction can be generalised to categories enriched over a commutative unital quantale. From here we define categorical algebras for the monad generated by this adjunction. We finish by showing that these categorical algebras are algebras over Set with operations and equations, and show that in some cases we can restrict the arity of those operations to be finite.

In topology, homotopy theory can be put into an algebraic framework. The most complete such framework is that of a Quillen model Category [[15], [5]]. The usual class of coarse spaces appears to be too small to be a Quillen model category. For example, it lacks a good notion of products. However, there is a weaker notion of a cofibration category due to Baues [[1], [2]]. The aim in this thesis is to look at notions of cofibration category within the world of coarse geometry. In particular, there are several sensible notions of the structure of a coarse version of a cofibration category that we define here. Later we compare these notions and apply them to computations. To be precise, there are notions of homotopy groups in a Baues cofibration category. So we compare these groups as well for the different structures we have defined, and to the more concrete notion of coarse homotopy groups defined also in [10]. Going further, there is an abstract notion of a cell complex defined in the context of a cofibration category. In the coarse setting, we prove such cell complexes have a more geometric definition, and precisely we prove that a coarse CW-complex is a cell complex. The ultimate goal of such computations is a version of the Whitehead theorem relating coarse homotopy groups and coarse homotopy equivalences for cell complexes. Abstract versions of the Whitehead theorem are known for cofibration categories [1], so we relate these abstract results to something more geometric. Another direction of the thesis involves Quillen model categories. As already mentioned, there are obstructions to the class of coarse spaces being a Quillen model category; there is no apparent way to define category-theoretic products of coarse spaces. However, such obvious objections vanish if we add extra spaces to the coarse category. These extra spaces are termed non-unital coarse spaces in [9]. We have proved most of Quillen axioms but the existence of limits in one of our categories.

This thesis presents CiGRAM, a model of complex networks ith known modular structure that is capable of generating realistic graph topology. Much of the recent focus on module detection has been geared towards developing new algorithms capable of detecting biologically significant clusters. However, evaluating clusterings detected by different methods shows that there is little topological agreement or consensus in terms of meta-data despite most methods discovering modules with significant ontology. In this thesis an approach to modelling complex networks with ground-truth modular structure is presented. This approach is capable of generating graphs with heterogeneous degree distributions, high clustering coefficients and assortative degree correlations observed in real data but often ignored in existing benchmarks. Moreover, the model for modular structure concludes that non-modular random graphs are indistinguishable from modules. This model can be tuned to fit many empirical biological and non-biological datasets through fitting target graph summary statistics. The ground-truth structure allows the evaluation of module extraction algorithms in a domain specific context. Furthermore, it was found that degree assortativity appears to negatively impact several module extraction methods such as the popular infomap and modularity maximisation methods. Results presented disagree with other benchmark models highlighting the potential for future research into improving existing methods in ways that challenge assumptions about the detectability of modules.

This thesis explores the combinatorial and topological properties of tiling spaces associated to 1-dimensional symbolic systems of aperiodic type and their associated algebraic invariants. We develop a framework for studying systems which are more general than primitive substitutions, naturally partitioned into two instances: in the first instance we allow systems associated to sequences of substitutions of primitive type from a finite family of substitutions (called mixed substitutions); in the second instance we concentrate on systems associated to a single substitution, but where we entirely remove the condition of primitivity. We generalise the notion of a Barge-Diamond complex, in the one-dimensional case, to any mixed system of symbolic substitutions. This gives a way of describing the associated tiling space as an inverse limit of complexes. We give an effective method for calculating the Cech cohomology of the tiling space via an exact sequence relating the associated sequence of substitution matrices and certain subcomplexes appearing in the approximants. As an application, we show that there exists a system of substitutions on two letters which exhibit an uncountable collection of minimal tiling spaces with distinct isomorphism classes of Cech cohomology. In considering non-primitive substitutions, we naturally divide this set of substitutions into two cases: the minimal substitutions and the non-minimal substitutions. We provide a detailed method for replacing any non-primitive but minimal substitution with a topologically conjugate primitive substitution, and a more simple method for replacing the substitution with a primitive substitution whose tiling space is orbit equivalent. We show that an Anderson-Putnam complex with a collaring of some appropriately large radius suffices to provide a model of the tiling space as an inverse limit with a single map. We apply these methods to effectively calculate the Cech cohomology of any substitution which does not admit a periodic point in its subshift. Using its set of closed invariant subspaces, we provide a pair of invariants which are each strictly finer than the usual Cech cohomology for a substitution tiling space.

In the post-war years, the theory of 3-dimensional manifolds has developed tremendously. On the one hand, Bing and Moise have proved that 3-manifolds can be triangulated, and that the Hauptvermutung (that any two triangulations of the same space are combinatorially equivalent) is true for 3-manifolds. On the other hand, Papakyriakopoulos has proved Dehn's Lemma, and, using ideas of Papakyriakopoulos, Whitehead has proved the Sphere Theorem. As a result of this concerted attack from two different directions, the theory of 3-manifolds has become an extremely interesting and fruitful field of study. It seems as though we are well on the way to solving the two main problems in the field:- the Poincaré Conjecture, and the classification of closed 3-manifolds. In this thesis, some theorems connected with 3-manifolds are proved. The most important theorem is the Projective Plane Theorem (6.1), in which it is proved that elements of the second homotopy group of a 3-manifold can be represented, in a certain sense, by 2-spheres or projective planes in the manifold. The Projective Plane Theorem is, perhaps, an important tool in the problem of classifying non-orientable 3-manifolds. The entire thesis depends on the Projective Plane Theorem, except for Chapters I and III. In Chapter I, the linking of n-spheres in (n+2)-space is dealt with. In Chapter Ill, non-orientable compact 3-manifolds, with finite fundamental groups are considered, with the aim of proving that there is essentially only one such 3-manifold. The reader is warned that a different definition of a 3-manifold is adopted in each chapter. This is in the interest of brevity and clarity. The author hopes that no confusion will arise. The definition appropriate in each chapter is given in the introduction to that chapter. The exact hypotheses about the 3-manifold, required for each theorem, are given just before the statement of the theorem. The follov/ing conventions are used throughout the thesis:- i) "M" denotes a 3-manifold; ii ) "X ̃" denotes some covering space of the topological space X; iii ) "G" denotes a group; iV) "O" denotes the group with only one element, or the unit element of a group which is definitely abelian, or the integer zero; v) "l" denotes the unit element of a (possibly) non-abelian group, or the integer one; vi ) "Homotopic to zero" means "homotopic to the constant map "; vii) '"The zero map" of one group into another group denotes the map which sends all elements into the trivial element. The author would like to thank Dr. E. C. Zeeman most warmly for his constant help and encouragement during the writing of this work. He could not have hoped for a better teacher. The author has found Dr. Zeeman's comments, suggestions and keen interest in his work invaluable and inspiring. The author has also had many very interesting and useful conversations with Dr. J. F. Adams, to whom he is most grateful.

A non-trivial 'Euler characteristic' is exhibited on a class of closed three dimensional manifolds. This is done by taking a class of Seifert fibre spaces, these being in 1- 1 correspondence with their fundamental groups - in-this case infinite cyclic central extensions of Fuchsian groups. By observation of-their finite index subgroups, or initially their presentations courtesy of the Reidemeister - Schreier approach, one can obtain a group -theoretic Euler characteristic which is in general non-trivial.

We study combinatorial properties of ordinals under the order topology, focusing on the subspaces, partition properties and autohomeomorphism groups of countable ordinals. Our main results concern topological partition relations. Let n be a positive integer, let κ be a cardinal, and write [X] n for the set of subsets of X of size n. Given an ordinal β and ordinals αi for all i ∈ κ, write β →top (αi) n i∈κ to mean that for every function c : [β] n → κ (a colouring) there is some subspace X ⊆ β and some i ∈ κ such that X is homeomorphic to αi and [X] n ⊆ c −1 ({i}). We examine the cases n = 1 and n = 2, defining the topological pigeonhole number P top (αi) i∈κ to be the least ordinal β (when one exists) such that β →top (αi) 1 i∈κ , and the topological Ramsey number Rtop (αi) i∈κ to be the least ordinal β (when one exists) such that β →top (αi) 2 i∈κ . We resolve the case n = 1 by determining the topological pigeonhole number of an arbitrary sequence of ordinals, including an independence result for one class of cases. In the case n = 2, we prove a topological version of the Erd˝os–Milner theorem, namely that Rtop (α, k) is countable whenever α is countable and k is finite. More precisely, we prove that Rtop(ω ω β , k + 1) ≤ ω ω β·k for all countable ordinals β and all positive integers k. We also provide more careful upper bounds for certain small ordinals, including Rtop(ω + 1, k + 1) = ω k + 1, Rtop(α, k) < ωω whenever α < ω2 , Rtop(ω 2 , k) ≤ ω ω and Rtop(ω 2 + 1, k + 2) ≤ ω ω·k + 1 for all positive integers k. Outside the partition calculus, we prove a topological analogue of Hausdorff’s theorem on scattered total orderings. This allows us to characterise countable subspaces of ordinals as the order topologies of countable scattered total orderings. As an application, we compute the number of subspaces of an ordinal up to homeomorphism. Finally, we study the group of autohomeomorphisms of ω n ·m+1 for finite n and m. We classify the normal subgroups contained in the pointwise stabiliser of the limit points. These subgroups fall naturally into D (n) disjoint sets, each either countable or of size 22 ℵ0 , where D (n) is the number of ⊆-antichains of P ({1, 2, . . . , n}). Our techniques span a variety of disciplines, including set theory, general topology and permutation group theory.

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In toric topology it is important to have a way of constructing Delzant polytopes, which have canonical combinatorial data. Furthermore we have examples of quasitoric manifolds with representatives in all even dimensions. These examples give rise to sequences of polytopes with related combinatorial invariants. In this thesis we intend to formalise the concept of a family of polytopes, which will behave in a similar way to the quotient spaces of quasi-toric manifolds. We will then compute certain combinatorial invariants in this context. Recently, polytope theory has developed to include the ring P with homogeneous polynomial invariants and an important operator d, which takes a polytope to the disjoint union of its facets. We will examine our families against this background and extend the polynomial invariants and d to entire families. In particular we will introduce a method to calculate polynomial invariants of families by the use of partial differential equations. We will also look at some polytopes called nestohedra, which arise from building sets. These nestohedra give us a construction of Delzant polytopes. We will show that it is possible to calculate d for any given nestohedra directly from its building set. We will also show that the canonical characteristic function of a nestohedron, F, which is a facet of a nestohedron, P, agrees with the characteristic function of F as a facet of P. We will see that nestohedra naturally form families. We will end this work by combining the work on nestohedra with the work on families and calculating the combinatorial invariants of some families of nestohedra.

I begin by developing a procedure for the construction of a Seifert surface, using Seifert's algorithm, and the calculation of a Seifert matrix for a knot from a suitable encoding of a knot diagram. This procedure deals with the inherent indeterminacy of the diagram encoding and is fully implementable. From a Seifert matrix one can form a presentation matrix for the Alexander module of a knot and calculate generators for the Alexander ideals. But to use the Alexander ideals to their full potential to distinguish pairs of knots one needs a Gröbner basis type theory for A = Z[t,t(-1)], the ring of Laurent polynomials with integer coefficients. I prove the existence of what I call Châtelet bases for ideals in A. These are types of Gröbner bases. I then develop an algorithm for the calculation of a Châtelet basis of an ideal from any set of generators for that ideal. This is closely related to Buchberger's algorithm for Gröbner bases in other polynomial rings. Using these algorithms and the knot diagram tables in the program Knotscape I calculate Châtelet bases for the Alexander ideals of all prime knots of up to 14 crossings. We determine the number of distinct ideals that occur and find examples of pairs of mutant knots distinguished by the higher Alexander ideals but not by any of the polynomials of Alexander, Jones, Kauffman or HOMFLY.