<p> This thesis gives characterizations for all the injective and projective objects in the categories of compact, totally disconnected, distributive topological lattices, and compact, distributive topological lattices. It also contains known results concerning distributive lattices and Hausdorff topological spaces.</p> / Thesis / Master of Science (MSc)

<p> This thesis gives basic properties of the newly defined topological properties local paracompactness and strong local paracompactness. An example is given to show that they do not coincide in T2 spaces; another example is given of a strongly locally paracompact T2 space which is neither locally compact nor paracompact. The existence of a one point paracompactification analagous to the Aleksandrov one point compactification is constructed and proved for strongly locally paracompact T2 spaces. Also considered are conditions under which these two properties are preserved under closed maps and heredity.</p> / Thesis / Master of Science (MSc)

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The study of topological superconductivity has been at the forefront of condensed matter physics for the past few years. Topological superconductors are predicted to have odd parity pairing and host so called Majorana fermions, which are not only of fundamental importance, but also proposed to be building blocks for fault-tolerant quantum computing. In this dissertation, we use point contact spectroscopy to study the pairing symmetry of candidate topological superconducting materials. We study proximity induced superconductivity in the topological insulator Bi2Se3 by a superconducting niobium tip, and propose a model to explain its features in point contact spectra. We further study the nature of the superconductivity in highly doped superconducting topological insulators, including CuxBi2Se3 and Sn1-xInxTe, using both a normal metal gold tip and a superconducting niobium tip. For CuxBi2Se3, we observe a robust zero-bias conductance peak (ZBCP) in the differential conductance spectra with the gold point contact, while with the niobium point contact we find the height of the peak exhibiting unusual non-monotonic temperature dependence. We argue that both observations cannot be explained by Andreev reflection within the standard Blonder-Tinkham-Klapwijk (BTK) model, but signify unconventional superconductivity in the material. For Sn1-xInxTe samples, we observe ZBCP in the differential conductance spectra with the gold point contact, while with the niobium point contact, the temperature dependence of ZBCP is monotonic as expected from conventional theory, leaving the nature of the superconductivity of Sn1-xInxTe still an open question.

Among a number of fundamental issues, the origin of inertial mass remains one of the major open problems in particle physics. Furthermore, topological effects related to non perturbative field configurations are poorly understood in those gauge theories of direct relevance to our physical universe. Motivated by such issues, this Thesis provides a deeper understanding for the appearance of topological effects in abelian gauge field theories, also in relation to the existence of a mass gap for the gauge interactions. These effects are not accounted for when proceeding through gauge fixings as is customary in the literature. The original Topological-Physical factorisation put forth in this work enables to properly identify in topologically massive gauge theories (TMGT) a topological sector which appears under formal limits within the Lagrangian formulation. Our factorisation then allows for a straightforward quantisation of TMGT, accounting for all the topological features inherent to such dynamics. Moreover dual actions are constructed while preserving the gauge symmetry also in the presence of dielectric couplings. All the celebrated mass generation mechanisms preserving the gauge symmetry are then recovered but now find their rightful place through a network of dualities, modulo the presence of topological terms generating topological effects. In particular a dual formulation of the famous Nielsen-Olesen vortices is constructed from TMGT. Within a novel physically equivalent picture, these topological defects are interpreted as dielectric monopoles.

We are living in an era of digital electronics. The number of robots have already exceeded the human population of the entire earth. An article in the Guardian newspaper dated 30th March 2018 suggests that 10 million UK workers will be jobless within 15 years as they will be replaced by robots. These astonishing facts shed light on the importance of knowledge and how important it is to use it wisely for our benefit without ultimately destroying us. Knowledge in all forms is accessible without going to a library or buying a newspaper. Furthermore to access information, we often use sleek devices such as smart phones, using highly developed multimedia platforms which consume large amounts of power. In 2016, IBM found that humans create 2.5 quintillion bytes of data daily. Since high computing usage is related to large power consumption, the basic building block of electronics i.e. the transistor is required to be more power efficient. This is now possible through spintronics, where the spin of an electron is exploited instead of the charge. A new class of exotic materials called topological insulators are predicted to exhibit efficient spintronic applications. These materials can conduct spin polarised current on their surface while remaining completely insulting from the inside. Moreover, doping topological insulators with magnetic impurities unlocks new avenues for spin memory devices in the form of a single spin polarized dissipationless conduction channel. In topological insulators, there is always a contribution from the inside (bulk) in addition to surface conduction, thereby yielding charge transport rather than spin transport. On this basis, the aim of my PhD was to explore techniques to grow, characterize, fabricate and measure devices on topological Dirac materials, with the hope to experimentally distinguish the bulk from the surface states and also exploit their exotic properties arising from opening of the bulk band gap by intentional magnetic doping. Samples consisted of thin films of Bi2Se3, Sb2Te3, Cr doped Sb2Te3, bilayers of Dy doped Bi2Te3/Cr doped Sb2Te3 and Cd3As2 nanowires. It was found that a seed layer of an undoped topological insulator was a crucial first step to ensure high quality growth by molecular beam epitaxy, followed by the desired stoichiometry. By physically doping Sb2Te3 with Cr, a successful control of the magnetic and electrical properties such as coercivity, anomalous Hall resistance RA xy, Curie temperature Tc, carrier density and mobility were achieved. A substitutional Cr doping ranging from 7.5% to 38% was attained revealing a Tc reaching up to 186 K. Gated electrical measurements displayed a change in RA xy and carrier density by ~ 50% on applicating of just -3 V gate bias in a sample with 29% doping. A comparison between electrical transport, Magneto-optical Kerr effect and terahertz time domain spectroscopy measurements revealed that the mechanism of magnetization was RKKY mediated. Furthermore, the bilayer structure displays a clear exchange bias coupling arising from the proximity of the antiferromagnetic Dy doped Bi2Te3 layer with the ferromagnetic Cr doped Sb2Te3 layer. Electrical transport measurements on Bi2Se3 Hall bars fabricated using Ar+ milling and wet chemical etching were compared. The results showed a more bulk type response in the chemical etched sample even though Ar+ milling was responsible for creating more disorder in the system leading to a higher carrier density and lower mobility. A thickness dependent study on Sb2Te3 thin films revealed a single conducting channel associated with a coupled surface and bulk state for a 12 nm sample, compared to, two conducting channels associated with the top and bottom surfaces for the 25 nm sample. Electrical transport on Dirac semimetal Cd3As2 nanowires reveal an ultra-high mobility of 56884 cm2V-1s-1 at 1.8 K from analysis of Shubnikov-de Haas oscillations. By studying various Dirac materials, new avenues for practical device applications can be explored.

M.Sc. (Physics) / The way we live has been fundamentally changed by technological innovations based on optical, electronic and magnetic materials. Without the continuous increase of scienti c understanding on phenomena that occur in materials, together with the processing and synthesis of materials, these technological revolutions would be impossible. Thus, the search of new materials is still the key driving force for the continuous blooming of modern technology...

In this thesis we study descriptive-set-theoretic and measure-theoretic properties of Polish groups, with a thematic emphasis on the contrast between groups which are locally compact and those which are not. The work is divided into three major sections. In the first, working jointly with Robert Kallman, we resolve a conjecture of Gleason regarding the Polish topologization of abstract groups of homeomorphisms. We show that Gleason's conjecture is false, and its conclusion is only true when the hypotheses are considerably strengthened. Along the way we discover a new automatic continuity result for a class of functions which behave like but are distinct from functions of Baire class 1. In the second section we consider the descriptive complexity of those subsets of the permutation group S? which arise naturally from the classical Levy-Steinitz series rearrangement theorem. We show that for any conditionally convergent series of vectors in Euclidean space, the sets of permutations which make the series diverge, and diverge properly, are ?03-complete. In the last section we study the phenomenon of Haar null sets a la Christensen, and the closely related notion of openly Haar null sets. We identify and correct a minor error in the proof of Mycielski that a countable union of Haar null sets in a Polish group is Haar null. We show the openly Haar null ideal may be distinct from the Haar null ideal, which resolves an uncertainty of Solecki. We show that compact sets are always Haar null in S? and in any countable product of locally compact non-compact groups, which extends the domain of a result of Dougherty. We show that any countable product of locally compact non-compact groups decomposes into the disjoint union of a meager set and a Haar null set, which gives a partial positive answer to a question of Darji. We display a translation property in the homeomorphism group Homeo+[0,1] which is impossible in any non-trivial locally compact group. Other related results are peppered throughout.