A homotopy n-type is a topological space which has trivial homotopy groups above degree n. Every space can be constructed from a sequence of such homotopy types, in a sense made precise by the theory of Postnikov towers, yielding improving `approximations' to the space by encoding information about the first n homotopy groups for increasing n. Thus the study of homotopy types, and the search for models of such spaces that can be fruitfully investigated, has been a central problem in homotopy theory. Of course, a homotopy 0-type is, up to weak homotopy equivalence (isomorphism of homotopy groups), a discrete set. It is well-known that a connected 1-type can be represented, again up to weak homotopy equivalence, as the classifying space of its fundamental group: this is the geometric realization of the simplicial set that is the nerve of the group regarded as a category with one object. Another way to phrase this is that the homotopy category of 1-types obtained by localizing at maps which are weak homotopy equivalences | formally adding inverses for these | is equivalent to the skeleton of the category of groups. In [Mac Lane and Whitehead] it was proved that connected homotopy 2-types can be modeled, in the sense described above, by crossed modules of groups. A crossed module is equivalently what in [Loday] is called a 1-cat-group, but now often referred to as a cat1.

In this thesis we focus on two topics. For the first we introduce a row version of Kadison and Kastler's metric on the set of C*-subalgebras of B(H). By showing C*-algebras have row length (in the sense of Pisier) of at most two we show that the row metric is equivalent to the original Kadison- Kastler metric. We then use this result to obtain universal constants for a recent perturbation result of Ino and Watatani, which states that succiently close intermediate subalgebras must occur as small unitary perturbations, by removing the dependence on the structure of inclusion. Roydor has recently proved that injective von Neumann algebras are Kadison-Kastler stable in a non-self adjoint sense, extending seminal results of Christensen. We prove a one-sided version, showing that an injective von Neumann algebra which is nearly contained in a weak*-closed non-self adjoint algebra can be embedded by a similarity close to the natural inclusion map. This theorem can then be used to extend results of Cameron et al. by demonstrating Kadison-Kastler stability of certain crossed products in the non self-adjoint setting. These crossed products can be chosen to be non-amenable.

Combinatorial group theory abounds with geometrical techniques. In this thesis we apply some of them to three distinct areas. In Chapter 1 we present all of the techniques and background material neccessary to read chapters 2,3,4. We begin by defining complexes with involutary edges and define coverings of these. We then discuss equivalences between complexes and use these in §§1.3 and 1.4 to give a way (the level method) of simplifying complexes and an application of this method (Theorem 1.3). We then discuss star-complexes of complexes. Next we present background material on diagrams and pictures. The final section in the chapter deals with SQ-universality. The.basic discussion of complexes is taken from notes, by Pride, on complexes without involutary edges, and modified by myself to cover complexes with involution. Chapters 2,3, and 4 are presented in the order that the work for them was done. Chapters 2,3, alld 4 are intended (given the material in chapter 1) to be self contained, and (iv) each has a full introduction. In Chapter 2 we use diagrams and pictures to study groups with the following structure. (a) Let r be a graph with vertex set V and edge set E. We assume that no vertex of r is isolated. (b) For each vertex VEV there is a non-trivial group Gv ' (c) For each edge e-{u,v}EE there is a set Se of cyclically reduced elements of Gu*Gv , each of length at least two. We define Ge to be the quotient of Gu*Gv by the normal closure of Se. We let G be the quotient of *Gv by the normal closure of VEV S- USe. For convenience, we write eEE The above is a generalization ofa situation studied by Pride [35], where each Gv was infinite cyclic.' Let e-{u,v} be an edge of r. We will say that Ge has property-Wk if no non-trivial element of Gu*Gv of free product length less than or equal to 2k is in the kernel of the natural epimorphism (v) We will work with one of the following: (I) Each Ge has property-W2 (II) r is triangle-free and each Ge has property-WI' Assuming that (I) or (II) holds we: (i) prove a Freihietssatz for these groups; (ii) give sufficient conditions for the groups to be SQ-universal; (iii) prove a result which allows us to give long exact sequences relating the (co)-homology G to the (co)-homology of the groups The work in Chapter 2 is in some senses the least original. The proofs are extensions of proofs given in [35] and [39] for the case when each Gv is infinite cyclic. However. there are some technical difficulties which we had to overcome. In chapter 3 we use the two ideas of star-complexes and coverings to look at NEC-groups. An NEC (Non-Euclidean Crystallographic) group is a discontinuous group of isometries (some of which may be (vi) orientation reversing) of the Non-Euclidean plane. According to Yilkie [46], a finitely generated NEC-group with compact orbit space has a presentation as follows: Involutary generators: Yij (i,j)EZo Non-involutary generators: 6i (iElf), tk (l~~r) (*) Defining paths: (YijYij+,)mij (iElf, l~j~n(i)-l) where In Hoare, Karrass and Solitar [22] it is shown that a subgroup of finite index in a group with a presentation of the form (*), has itself a presentation of the form (*). In [22] the same authors show that a subgroup of infinite ingex in a group with a presentation of the form (*) is a free product of groups of the following types: (A) Cyclic groups. (vii) (B) Groups with presentations of the form Xl' ... 'Xn involutary. (e) Groups with presentations of the form Xi (iEZ) involutary. We define what we mean by an NEe-complex. (This involves a structural re$triction on the form of the star-complex of the complex.) It is obvious from the definition that this class of complexes is clo$ed under coverings, so that the class of fundamental groups of NEe-complexes is trivially closed under taking subgroups. We then obtain structure theorems for both finite and infinite NEe-complexes. We show that the fundamental group of a finite NEe-complex has a presentation of the form (*) and that the fundamental group of an infinite NEe-complex is a free product of groups of the forms (A). (B) and (e) above. We then use coverings to derive some of the results on normal subgroups of NEe-groups given in [5] and [6]. , (viii) In chapter 4 we use the techniques of coverings and diagrams. to stue,iy the SQ-universau'ty of Coxeter groups. This is a problem due to B.H. Neumann (unpublished). see [40]. A Coxeter pair is a 2-tup1e (r.~) where r is a graph (with vertex set V(r) and edge set E(r» and ~ is a map from E(r) to {2.3.4 •.•• }. We associate with (r.~) the Coxeter group c(r,~) defined by the presentation tr(r,~)-<v(r);(xy)~({X'Y}) ((x,y}eE(r»>, where each generator is involutary. Following Appel and Schupp [1] we say that a Coxeter pair is of large type if 2/Im~. I conjecture that if (r,~) is of large type with IV(r)I~3 and r not a triangle with all edges mapped to 3 by ~. then C(r,~) is SQ-universa1. In connection with this conjecture we firstly prove (Theorem 4.1), Let (r,~) be a Coxeter pair of large type. Suppose (A) r is incomplete on at least three vertices, or (B) r is complete on at least five vertices and for 1 < - 2 (ix) Then C(r,~) is SQ-universal. Secondly we prove a result (Theorem 4.2) which shows: If (r,~) is a Coxeter pair with IV(r)I~4 and hcf[~(E(r»] > 1, then C(r,~) is either SQ-universal or is soluble of length at most three. Moreover our Theorem allows us to tell the two possibilities apart. The proof of this result leads to consideration of the following question: If a direct sum of groups is SQ-universal, does this imply that one of the summands is itself SQ-universal? We show (in appendix B) that the answer is "yes" for countable direct sums. We consider the results in chapter 4 and its appendix to be the most significant part of this thesis

Chapter 1 covers some basic notions and results from Algebraic Topology such as CW-complexes, homotopy and homology groups of a space in general and cellular homology for CW-complexes in particular. Also we give some basic ideas from abstract reduction systems and some supporting material such as several order relations on a set and the Knuth-Bendix completion procedure. There are only two original results of the author in this chapter, Theorem 1.4.5 and Theorem 1.7.3. The material related to Topology and Homological Algebra can be found in [12], [21], [40], [62], [82], [91] and [92]. The material related to reduction systems can be found in [5] and [11]. The original work of the author is included in Chapters 2, 3 and 4 apart from Section 3.2 which contains general notions from Category Theory, Section 3.5.2 which contains an account of the work in [67] and Section 4.1 which contains some basics from Combinatorial Semigroup Theory. The results of Section 4.2 are part of [83] which is accepted for publication in the International Journal of Algebra and Computation. The material related to Category Theory can be found in [59], [64], [66], [67], [74], [75], [76], [82] and [93]. The material related to Semigroup Theory is in [24] and [34].In Chapter 2 we show that for every monoid S which is given by a finite and complete presentation P = P[x, r], and for every n ~ 2, there is a chain of CW-complexes such that ~n has dimension n, for every 2 ~ s ~ n the s-skeleton of ~n is ~s and F acts on ~n. This action is called translation. Also we show that, for 2 ~ s ~ n, the open s-cells of ~n are in a 1-1 correspondence with the s-tuples of positive edges of V with the same initial. For the critical s-tuples, the corresponding open s-cells are denoted by Ps-I and the set of their open translates by F.Ps-I.F. The following holds true. if s ~ 3 if s = 2, where U stands for the disjoint union. Also, for every 2 ~ s ~ n - 1, there exists a cellular equivalence "'s on Ks = (~s X ~8)(s+1) such that Ks/ "'s= (V, PI, ... ,Ps-I) and the following is an exact sequence of (ZS, ZS)-bimodules where (D, Pl, ... , Ps-2) = V if s = 2. Using the above short exact sequences, we deduce that S is of type bi-FPn and that the free fi~ite resolution of'lS is S-graded. In Chapter 3 we generalize the notions left-(respectively right)-FPn and bi-FPn for small categories and show that bi-FPn implies left-(respectively right)-FPn . Also we show that another condition, which was introduced by Malbos and called FPn , implies bi-FPn . Since the name FPn is confusing, we call it here f-FPn for a reason which will be made clear in Section 3.1. Restricting to monoids, we show that, if a monoid is given by a finite and complete presentation, then it is of type f-FPn . Lastly, for every small category C, we show how to construct free resolutions of ZC, at lea..'lt up to dimension 3, using some geometrical ideas which can be generalized to construct free resolutions of ZC of any length. vi In Chapter 4 we study finiteness conditions of ~onoids of a combinatorial nature. We show that there are semigroups S in which min'R., is independent of other conditions which S may satisfy such as being finitely generated, periodic, inverse, E-unitary and even from the finiteness of the maximal subgroups of S. We also relate the congruences of a monoid with the finiteness condition minQ, and show that, if S is a monoid which satisfies minQ, then every congruence JC on S which contains Q is of finite index in S. If a semigroup satisfies minQ and has all its maximal subgroups locally finite, then we show that it is finite. Lastly, we show that, for trees of completely O-simple semigroups, the local finiteness of its maximal subgroups implies the local finiteness of the semigroups.

The theory of outer automorphism groups allows us to better understand groups through their symmetries, and in this thesis we approach outer automorphism groups from two directions. In the first direction we start with a class of groups and then classify their outer automorphism groups. In the other direction we start with a broad class of groups, for example finitely generated groups, and for each group Q in this class we construct a group G_Q such that Q is related, in a suitable sense, to the outer automorphism group of G_Q. We give a list of 14 groups which precisely classifies the outer automorphism groups of one-ended two-generator, one-relator groups with torsion. We also describe the outer automorphism groups of such groups which have more than one end. Combined with recent algorithmic results of Dahmani–Guirardel, this work yields an algorithm to compute the outer automorphism group of a two-generator, one-relator group with torsion. We prove a technical theorem which, in a certain sense, writes down a specific subgroup of the outer automorphism group of a particular kind of HNN-extension. We apply this to prove two main results. These results demonstrate a universal property of triangle groups and are as follows. Fix an arbitrary hyperbolic triangle group H. If Q is a finitely generated group then there exists an HNN-extension G_Q of H such that Q embeds with finite index into the outer automorphism group of G_Q. Moreover, if Q is residually finite then G_Q can be taken to be residually finite. Secondly, fix an equilateral triangle group H = ⟨a, b; a^i, bi, (ab)^i⟩ with i > 9 arbitrary. If Q is a countable group then there exists an HNN-extension G_Q of H such that Q is isomorphic to the outer automorphism group of G_Q. The proof of this second main result applies a theory of Wise underlying his recent work leading to the resolution of the virtually fibering and virtually Haken conjectures. We prove a technical theorem which, in a certain sense, writes down a specific subgroup of the outer automorphism group of a semi-direct product H.

Some adverse drug reactions (ADRs) are not detected before marketing approval is given because clinical trials are not suited for their detection, for various reasons [5, 23]. Drug regulatory bodies therefore weigh the potential benefits of a drug against the harms and allow drugs to be marketed if felt that the potential benefits far outweigh the harms [26,48]. Associated adverse events are subsequently monitored through various means including reports submitted by health professionals and the general public in what is commonly referred to as spontaneous reporting system (SRS) [19, 23, 69]. The resulting database contains thousands of adverse event reports which must be assessed by expert panels to see if they are bona fide adverse drug reactions, but which are not easy to manage by virtue of the volume [6]. This thesis documents work aimed at developing a statistical model for assisting in the identification of bona fide drug side-effects using data from the United States of America’s Food and Drugs Administration’s (FDA) Spontaneous Reporting System (otherwise known as the Adverse Event Reporting System (AERS)) [28]. Four hierarchical models based on the Conway-Maxwell-Poisson (CMP) distribution [43,78] were explored and one of them was identified as the most suitable for modeling the data. It compares favourably with the Gamma Poisson Shrinker (GPS) of DuMouchel [19] but takes a dimmer view of drug and adverse event pairs with very small observed and expected count than the GPS. Two results are presented in this thesis; the first one, from a preliminary analysis, presented in Chapter 2, shows that problems such as missing values for age and sex that militate against the optimal use of SRS data, enumerated in the literature, remain. The second results, presented in Chapter 5, concern the main focus of the research mentioned in the previous paragraph.

The theory of cometary panspermia is tested in the wake of two reported falls in Tissint, Morocco on July 18, 2011 and in the central province of Polonnaruwa, Sri Lanka on December 29, 2012. Samples of the Tissint and Polonnaruwa stones were studied using a variety of laboratory procedures and equipment including ICP-OES, GC-MS, SEM, EDAX, CHN, FTIR, Raman Spectroscopy, XRD and Optical Spectroscopy. Results of Tissint show the presence of several 5-50μm pyrite grains rimmed by a layer of reduced organic carbon with graphitisation levels consistent with other Martian meteorites. A complex precursor carbon inventory is demonstrated with peak temperatures ~ 250 OC and elemental ratios typical of high volatility bituminous coals. A theoretical model of the ecology of arsenic on early Mars is then developed and discussed involving microbial reduction of Fe-oxides. This hypothesis is shown to be supported by SEM observations of spherical chains of pits, with morphologies distinct from abiotic alteration features but closely comparable to biologically mediated microstructures created by Fe- and S-oxidising microbes. The contribution of core-mantle grains to mid-IR emission features is then modelled using extinction and scattering efficiencies for composite spheres based on the Guttler extension of the Mie formulae. Results show that kerogen-pyrite grains closely adhere to observed 9-13μm emission characteristics observed in the Trapezium nebula. Results of studies on Polonnaruwa show a highly porous Si-K-rich, Al-depleted, amorphous melt enclosing trace (commonly <1μm) anorthoclase, albite, anorthite and quartz. Bound H2O < 0.03wt% indicates origin from hypervelocity impact. SEM analysis revealed several fossil microorganisms similar to acritarchs, hystrichospheres and diatoms. Geologic age of the stones is determined by N/C atomic ratio depletion that indicate the presence of embedded fossil remains that date back to at least ~300 Ma. Triple oxygen isotope analysis provide values of Δ17O = - 0.335 with δ17O = 8.978 ± 0.050 and δ18O = 17.816 ± 0.100 that is shown to be consistent with non-terrestrial sources. Results are seen to substantially support the theory of cometary panspermia.

Social networks are increasingly being investigated in the context of individual behaviours. Research suggests that friendship connections have the ability to influence individual actions, change personal opinions and subsequently impact upon personal wellbeing. This thesis aims to investigate the effects of social networks, through the use of Agent Based Simulation (ABS) and Link Prediction (LP) methods. Three main investigations form this thesis, culminating in the development of a new simulation-based approach to Link Prediction (PageRank-Max) and a model of behavioural spread through a connected population (Behavioural PageRank-Max). The first project investigates the suitability of ABS to explore a connected social system. The Peter Principle is a theory of managerial incompetence, having the potential to cause detrimental effects to system efficiency. Through the investigation of a theoretical hierarchy of workplace social contacts, it is observed that the structure of a social network has the ability to impact system efficiency, demonstrating the importance of social network structure in conjunction with individual behaviours. The second project aims to further understand the structure of social networks, through the exploration of adolescent offline friendship data, taken from 'A Stop Smoking in Schools Trial' (ASSIST). An initial analysis of the data suggests certain factors may be pertinent in the formation of school social networks, identifying the importance of centrality measures. An ABS aiming to predict the evolution of the ASSIST social networks is created, developing an algorithm based upon the optimisation of an individual's eigen-centrality - termed PageRank-Max. This new approach to Link Prediction is found to predict ASSIST social network evolution more accurately than four existing prominent LP algorithms. The final part of this thesis attempts to improve the PageRank-Max method, by placing particular emphasis upon specific individual attributes. Two new methods are developed, the first restricting the search space of the algorithm (Behavioural Search), while the second alters its calculation process by applying specific attribute weights (Behavioural PageRank-Max). The results demonstrate the importance of individual attributes in adolescent friendship selection. Furthermore, the Behavioural PageRank-Max offers an approach to model the spread of behaviours in conjunction with social network structure, with the value of this being evaluated against alternative models.

This thesis is concerned with fluid-structure interaction problems using the immersed boundary method (IBM). Fluid-structure interaction problems can be classified into two categories: a remeshing approach and a fixed-grid approach. Both approaches consider the fluid and structure separately and then couple them together via suitable interface conditions. A common choice of remeshing approach is the Arbitrary-Eulerian-Lagrangian (ALE) technique. Whilst the ALE method is a good choice if deformations are small, it becomes computationally very expensive if deformations are large. In such a scenario, one turns to a fixed-grid approach. However, the issue with a fixed-grid approach is the enforcement of the interface conditions. An alternative to the remeshing and fixed-grid approach is the IBM. The IBM considers the immersed elastic structure to be part of the surrounding fluid by replacing the immersed structure with an Eulerian force density. Therefore, the interface conditions are enforced implicitly. This thesis applies the finite element immersed boundary method (IBM) to both Newtonian and Oldroyd-B viscoelastic fluids, where the fluid variables are approximated using the spectral element method (hence we name the method the spectral element immersed boundary method (SE-IBM)) and the immersed boundary variables are approximated using either the finite element method or the spectral element method. The IBM is known to suffer from area loss problems, e.g. when a static closed boundary is immersed in a fluid, the area contained inside the closed boundary decreases as the simulation progresses. The main source of error in such a scenario can be found in the spreading and interpolation phases. The aim of using a spectral element method is to improve the accuracy of the spreading and interpolation phases of the IBM. We illustrate that the SE-IBM can obtain better area conservation than the FE-IBM when a static closed boundary is considered. Also, the SE-IBM obtains higher order convergence of the velocity in the L2 and H1 norms, respectively. When the SE-IBM is applied to a viscoelastic fluid, any discontinuities which occur in either the velocity gradients or the pressure, introduce oscillations in the polymeric stress components. These oscillations are undesirable as they could potentially cause the numerics to break down. Finally, we consider a higher-order enriched method based on the extended finite element method (XFEM), which we call the eXtended Spectral Element Method (XSEM). When XSEM is applied to the SE-IBM with a viscoelastic fluid, the oscillations present in the polymeric stress components are greatly reduced.

The role of the financial mathematician is to find solutions to problems in finance through the application of mathematical theory. The motivation for this work is specification of models to accurately describe the price evolution of a risky asset, a risky asset is for example a security traded on a financial market such as a stock, currency or benchmark index. This thesis makes contributions in two classes of models, namely activity time models and integer valued models, by the discovery of new real valued and integer valued stochastic processes. In both model frameworks applications to option pricing are considered.