We study the fields of fractions and the Poisson spectra of polynomial Poisson algebras. First we investigate a Poisson birational equivalence problem for polynomial Poisson algebras over a field of arbitrary characteristic. Namely, the quadratic Poisson Gel'fand-Kirillov problem asks whether the field of fractions of a Poisson algebra is isomorphic to the field of fractions of a Poisson affine space, i.e. a polynomial algebra such that the Poisson bracket of two generators is equal to their product (up to a scalar). We answer positively the quadratic Poisson Gel'fand-Kirillov problem for a large class of Poisson algebras arising as semiclassical limits of quantised coordinate rings, as well as for their quotients by Poisson prime ideals that are invariant under the action of a torus. In particular, we show that coordinate rings of determinantal Poisson varieties satisfy the quadratic Poisson Gel'fand-Kirillov problem. Our proof relies on the so-called characteristic-free Poisson deleting derivation homomorphism. Essentially this homomorphism allows us to simplify Poisson brackets of a given polynomial Poisson algebra by localising at a generator. Next we develop a method, the characteristic-free Poisson deleting derivations algorithm, to study the Poisson spectrum of a polynomial Poisson algebra. It is a Poisson version of the deleting derivations algorithm introduced by Cauchon [8] in order to study spectra of some noncommutative noetherian algebras. This algorithm allows us to define a partition of the Poisson spectrum of certain polynomial Poisson algebras, and to prove the Poisson Dixmier-Moeglin equivalence for those Poisson algebras when the base field is of characteristic zero. Finally, using both Cauchon's and our algorithm, we compare combinatorially spectra and Poisson spectra in the framework of (algebraic) deformation theory. In particular we compare spectra of quantum matrices with Poisson spectra of matrix Poisson varieties.

In this publication, we investigate the groups H(m, n, p, q |k) defined by the presentation < x, y|x^m, y^n, (xy)^p, (xy^k)^q > and determine finiteness and infiniteness for these parameters as far as possible using geometric arguments, principally through pictures and curvature. We state and subsequently prove theorems relating to infiniteness, and also discuss troublesome cases where spherical pictures may arise. We also provide a list of known finite groups and unresolved cases in Appendix A.

This thesis is devoted to the study of various properties of Banach function algebras. We are particularly interested in the study of antisymmetric decompositions for uniform algebras and regularity of Banach function algebras. We are also interested in the study of Swiss cheese sets, essential uniform algebras and characterisations of C(X) among its subalgebras. The maximal antisymmetric decomposition for uniform algebras is a generalisation of the celebrated Stone-Weierstrass theorem and it is a powerful tool in the study of uniform algebras. However, in the literature, not much attention has been paid to the study of closed antisymmetric subsets. In Section 1.7 we give a characterisation of all the closed antisymmetric subsets for the disc algebra on the unit circle, and we use this characterisation to give a new proof of Wermer’s maximality theorem. Then in Section 4.1 we give characterisations of all the closed antisymmetric subsets for normal uniform algebras on the unit interval or the unit circle. The two types of regularity points, the R-point and the point of regularity, are important concepts in the study of regularity of Banach function algebras. In Section 3.2 we construct two examples of compact plane sets X, such that R(X) has either one R-point while having no points of regularity, or R(X) has one point of continuity while having no R-points. There are the first known examples of natural uniform algebras in the literature which show that R-points and points of continuity can be different. We then use properties of regularity points to study R(X) which is not regular while having no non-trivial Jensen measures. We also use properties of regularity points in Section 4.2 to study small exceptional sets for uniform algebras. In Chapter 2 we study Swiss cheese sets. Our approach is to regard Swiss cheese sets “abstractly”: we study the family of sequences of pairs of numbers, where the numbers represent the centre and radius of discs in the complex plane. We then give a natural topology on the space of abstract Swiss cheeses and give topological proofs of various classicalisation theorems. It is standard that the study of general uniform algebras can be reduced to the study of essential uniform algebras. In Chapter 5 we study methods to construct essential uniform algebras. In particular, we continue to study the method introduced in [26] to show that some more properties are inherited by the constructed essential uniform algebra from the original one. We note that the material in Chapter 2 is joint work with J. Feinstein and S. Morley and is published in [28, 27]. The material in Chapter 3 is joint work with J. Feinstein and is published in [32]. Section 4.2 contains joint work with J. Feinstein.

Using the higher tame symbol and Kawada and Satake’s Witt vector method, A.N. Parshin developed class field theory for positive characteristic higher local fields, defining reciprocity maps separately for the tamely ramified and wildly ramified cases. We prove reciprocity laws for these symbols using techniques of Morrow for the Witt symbol and Romo for the higher tame symbol. We then extend this method of defining a reciprocity map to the case of positive characteristic local- global fields associated to points and curves on an algebraic surface over a finite field.

Combinatorial group theory is a part of group theory that deals with groups given by presentations in terms of generators and defining relations. Many techniques both algebraic and geometric are used in dealing with problems in this area. In this thesis, we adopt the geometric approach. More specifically, we use so-called pictures over relative presentations to determine the asphericity of such presentations. We remark that if a relative presentation is aspherical then group theoretic information can be deduced. In Chapter 1, the concept of relative presentations is introduced and we state the main theorems and some known results. In Chapter 2, the concept of pictures is introduced and methods used for checking asphericity are explained. Excluding four unresolved cases, the asphericity of the relative presentation $\mathcal{P}$= $\langle G, x|x^{m}gxh\rangle$ for $m\geq2$ is determined in Chapter 3. If $H=\langle g, h\rangle$ $\leq G$, then the unresolved cases occur when $H$ is isomorphic to $C_{5}$ or $C_{6}$. The main work is done in Chapter 4, in which we investigate the asphericity of the relative presentation $\mathcal{P}$= $\langle G, x|xaxbxcxdxexf\rangle$, where the coefficients $a, b, c, d, e, f\in G$ and $x \notin G$ and prove the theorems stated in Chapter 1.

This thesis details research I have carried out in the field of quantum walks, which are the quantum analogue of classical random walks. Quantum walks have been shown to offer a significant speed-up compared to classical random walks for certain tasks and for this reason there has been considerable interest in their use in algorithmic settings, as well as in experimental demonstrations of such phenomena. One of the most interesting developments in quantum walk research is their application to spatial searches, where one searches for a particular site of some network or lattice structure. There has been much work done on the creation of discrete- and continuous-time quantum walk search algorithms on various lattice types. However, it has remained an issue that continuous-time searches on two-dimensional lattices have required the inclusion of additional memory in order to be effective, memory which takes the form of extra internal degrees of freedom for the walker. In this work, we describe how the need for extra degrees of freedom can be negated by utilising a graphene lattice, demonstrating that a continuous-time quantum search in the experimentally relevant regime of two-dimensions is possible. This is achieved through alternative methods of marking a particular site to previous searches, creating a quantum search protocol at the Dirac point in graphene. We demonstrate that this search mechanism can also be adapted to allow state transfer across the lattice. These two processes offer new methods for channelling information across lattices between specific sites and supports the possibility of graphene devices which operate at a single-atom level. Recent experiments on microwave analogues of graphene that adapt these ideas, which we will detail, demonstrate the feasibility of realising the quantum search and transfer mechanisms on graphene.

This thesis considers two modelling frameworks for interaction networks in biology. The first models the interacting species qualitatively as discrete variables, with the regulatory graphs expressing their mutual influence. Circuits in the regulatory structure are known to be indicative of some asymptotic behaviours. We investigate the relationship between local negative circuits and sustained oscillations, presenting new examples of Boolean networks without local negative circuits and admitting a cyclic attractor. We then show how regulatory properties of Boolean networks can be investigated via satisfiability problems, and use the technique to examine the role of local negative circuits in networks of small dimension. To enable the application of Boolean techniques to the study of multivalued networks, a mapping of discrete networks to Boolean can be considered. The Boolean version, however, is defined only on a subset of the Boolean states. We propose a method for extending the Boolean version that preserves both the attractors and the regulatory structure of the network. Chemical reaction network theory models the dynamics of species concentrations via systems of ordinary differential equations, establishing connections between the network structure and the dynamics. Some results assume mass action kinetics, whereas biochemical models often adopt other rate forms. We propose algorithms for elimination of intermediate species, that can be used to find whether a mass action network simplifies to a given chemical system. We then consider the problem of identification of generalised mass action networks that give rise to a given mass action dynamics, while displaying useful structural properties, such as weak reversibility. In particular, we investigate systems obtained by preserving the reaction vectors of the mass action network, and outline a new algorithmic approach.

The Jiang-Su algebra Z and the notion of Z-stability (i.e. tensorial absorption of the Jiang-Su algebra) are now widely acknowledged to be of particular importance in the classification and structure theory of separable nuclear C*-algebras. The key results in this thesis are early attempts to explore Z-stability outside the constraints of unital and of nuclear C*-algebras. Standard unitisations of a separable Z-stable C*-algebra are not Z-stable and we therefore explore possible unitisations that preserve Z-stability. We construct the minimal Z-stable unitisation of a separable Z-stable C*-algebra and show that it satisfies an appropriate universal property. An interesting area in which to exploit Z-stability outside of the context of nuclear C*-algebras is the so-called Kadison’s similarity problem. We show that the tensor product of two separable unital C*-algebras has Kadison’s similarity property if one of them is nuclear and admits a unital *-homomorphism from (the building blocks of) the Jiang-Su algebra. An immediate consequence of this is that any separable unital Z-stable C*-algebra also has this property.

Dendrites are the most visually striking parts of neurons. Even so many neuron models are of point type and have no representation of space. In this thesis we will look at a range of neuronal models with the common property that we always include spatially extended dendrites. First we generalise Abbott’s “sum-over-trips” framework to include resonant currents. We also look at piece-wise linear (PWL) models and extend them to incorporate spatial structure in the form of dendrites. We look at the analytical construction of orbits for PWL models. By using both analytical and numerical Lyapunov exponent methods we explore phase space and in particular we look at mode-locked solutions. We will then construct the phase response curve (PRC) for a PWL system with compartmentally modelled dendrites. This sets us up so we can look at the effect of multiple PWL systems that are weakly coupled through gap junctions. We also attach a continuous dendrite to a PWL soma and investigate how the position of the gap junction influences network properties. After this we will present a short overview of neuronal plasticity with a special focus on the spatial effects. We also discuss attenuation of distal synaptic input and how this can be countered by dendritic democracy as this will become an integral part of our learning mechanisms. We will examine a number of different learning approaches including the tempotron and spike-time dependent plasticity. Here we will consider Poisson’s equation around a neural membrane. The membrane we focus on has Hodgkin-Huxley dynamics so we can study action potential propagation on the membrane. We present the Green’s function for the case of a one-dimensional membrane in a two-dimensional space. This will allow us to examine the action potential initiation and propagation in a multi-dimensional axon.

In this thesis we investigate and develop analytic models for polymer nucleation and other barrier crossing problems. Our most broadly appealing method for certain multi dimensional barrier crossing problems is a one-dimensional projection which includes a novel technique to extract rate kinetics from simulations [M J Hamer et al., Soft Matter, 2012, 8, 11396-11408]. The scenarios we expect our method to be potentially useful are situations where barrier crossings are rare, and the dominant mechanism is through a series of unlikely incremental steps. The rate kinetics extraction technique is also reliant on the equilibrium energy barrier being relevant to non-equilibrium system, but is not appropriate when strong kinetic contributions dominate the process, and enable crossings over highly unfavourable energetic pathways. We explore and significantly enhance the Graham-Olmsted (GO) polymer nucleation simulation [R S Graham and P D Olmsted, Phys. Rev. Lett., 2009, 103, 115702], producing a combinatorial calculation to obtain exact energy landscapes from it’s basic stochastic rules of monomer attachment [M J Hamer et al., J. Non-Newton. Fluid., 2010, 165, 1294-1301]. We apply our rate kinetics extraction technique to the GO model and find that for most flow rates in purely long chain melts, nuclei tend to grow along similar paths over energy landscapes. The technique reveals a clear signature when this pattern is disobeyed, as in the case of blends of long and short chain polymer melts, some of which display highly anisotropic growth. In addition, we design several one-dimensional barrier crossing models with distinct characteristics, predicting the average and the distribution of crossing times with great accuracy. That finally enables us to completely describe the GO simulation’s nucleation rates with analytic theory, by presenting a model of polymer nucleation featuring crystal rotation, which vastly impacts nucleation rates when polymer melts are subject to flow.