The main objective of this thesis is to introduce the concept of 'singular chains on topological stacks'. The idea is to functorially associate to a topological stack, a simplicial set which captures its homotopy type. This will allow us to compute the singular homology and cohomology of topological stacks. Noohi and Behrend have given several approaches to this problem, however all of these approaches rely on the choice of an atlas for a topological stack. We shall show that our new approach agrees with the existing approaches but has the advantage of being functorial. Noohi has introduced weak equivalences and brations of topological stacks. In analogy to the singular chains functor for topological spaces, we shall show that the functor Sing preserves the weak equivalences and brations de ned by Noohi under certain ` brancy conditions'. In the second part, we shall push the analogy with the topological singular chains further by considering the adjunction with the geometric realization and the associated counit. We develop a corresponding (but weaker) notion for topological stacks. We shall give a method for computing the homotopy type of a stack which has a groupoid presentation. Finally, we shall compute the homotopy type of certain mapping stacks and develop the totalization of a cosimplicial topological stack. We shall indicate how this (using the approach of Cohen and Jones) gives a method for computing the string topology of a topological stack.

Chan, Wai-Kit. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaves 52-53). / Abstracts in English and Chinese. / Abstract --- p.ii / Acknowledgement --- p.iii / Introduction --- p.iv / Chapter 1 --- Preliminaries --- p.1 / Chapter 1.1 --- C*-algebras --- p.1 / Chapter 1.2 --- Topological groups --- p.8 / Chapter 2 --- Property T for topological groups --- p.18 / Chapter 2.1 --- Definitions and some basic properties --- p.18 / Chapter 2.2 --- Hereditary properties --- p.23 / Chapter 2.3 --- A characterization for property T --- p.26 / Chapter 2.4 --- Examples --- p.32 / Chapter 3 --- Property T for C*-algebras --- p.34 / Chapter 3.1 --- Countable discrete groups and their group C*- algebras --- p.34 / Chapter 3.2 --- Property T and nuclearity --- p.46 / Bibliography --- p.52

archives@tulane.edu / The discovery of topological semimetals has attracted enormous interest since they not only possess many unusual exotic properties, but also offer a fertile ground for searching for new fermions in the low energy spectrum. The first established example of a topological state of matter is the quantum Hall effect, which supports a gapless edge state protected by topological invariance. Later the concept of topology has been extended to describe electronic band structure of solid state materials and this effort leads to discoveries of many new topological quantum states, such as Dirac cone state in graphene, quantum spin Hall insulator states in semiconductor quantum wells, 3D topological insulators, etc. The recently discovered Dirac/Weyl semimetals can be viewed as a 3D analog of graphene. This thesis work aims to discover new Dirac/Weyl semimetals through single crystal synthesis and characterization. This thesis is organized as follows: In chapter 1, I will first briefly review several basic concepts of topological properties and introduce a few prototype topological semimetals related to my thesis work. Since one important part of my thesis work involves single crystal growth of topological semimetals, I will introduce the crystal growth methods used in my research in chapter 2. In chapters 3, 4 and 5, I will present my experimental discoveries of new topological semimetals, including YSn2, CaSn3 and TbPtBi. I will not only show property characterization of these material, but also discuss their underlying physics. For YSn2, my work reveals that its slightly distorted square lattice of Sn generates multiple topologically non-trivial bands, one of which likely hosts nodal line and tunable Weyl semimetal state induced by the Rashba spin-orbit coupling (SOC) and proper external magnetic field. The quasiparticles described as relativistic fermions from these bands are manifested by nearly zero mass, and non-trivial Berry phases probed in de Haas–van Alphen (dHvA) oscillations. The dHvA study also reveals YSn2 has a complex Fermi surface (FS), consisting of several 3D and one 2D pocket. Our first principle calculations show the point-like 3D pocket at Y point on the Brillouin zone boundary hosts the possible Weyl state. Our findings establish YSn2 as a new interesting platform for observing novel topological phases and studying their underlying physics. In the study of CaSn3, we not only found it possesses non-trivial band topology, but also discovered its intrinsic superconductivity at 1.178 K. Its topological fermion properties, including the nearly zero quasi-particle mass and the non-trivial Berry phase accumulated in cyclotron motions, were revealed from the dHvA quantum oscillation studies of this material. Our findings make CaSn3 a promising candidate for exploring new exotic states arising from the interplay between non-trivial band topology and superconductivity, e.g., topological superconductivity. For the Half-Heusler compound TbPtBi, we have studied its field-induced Weyl semimetal state. We have observed remarkable transport signatures of its Weyl state, including the chiral anomaly, intrinsic anomalous Hall effect (AHE), and in-plane Hall effect. Moreover, we found TbPtBi exhibits a much larger AHE than the previously reported field-induced Weyl semimetal state in GdPtBi. The distinct aspect of TbPtBi is that Tb ions carry greater magnetic moments than Gd ions in GdPtBi (9.0B/Tb vs.7.0B/Gd). We find that such a moment increase in TbPtBi drastically enhances its AHE, with its anomalous Hall angle reaching as large as 0.50-0.76 in its antiferromagnetic (AFM) state. This finding not only strongly supports that the Zeeman effect due to the large exchange field from 4f electrons plays a critical role in creating the field-included Weyl state, but also provides clear evidence for the theoretical prediction that the intrinsic anomalous Hall conductivity is proportional to the separation of the Weyl points with opposite chirality. / 1 / Yanglin Zhu

Thesis (Ph. D.)--Oregon State University, 1969. / Typescript. Includes bibliographical references (leaf 69). Also available on the World Wide Web.

Thesis (Ph.D.)--York University, 2004. Graduate Programme in Mathematics. / Typescript. Includes bibliographical references (leaves 77-83) and index. Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://wwwlib.umi.com/cr/yorku/fullcit?pNQ99185

<p>The main purpose of this thesis is to study the interplay between relational structures and topology , and to portray pairings in terms of some finite poset models and order preserving maps. We show the interrelations between the categories of topological spaces, closure spaces and relational structures. We study the 4-point non-Hausdorff model S4 weakly homotopy equivalent to the circle S1. We study pairings of some objects in the category of relational structures similar to the multiplication S4 x S4- S4 S4 fails to be order preserving for posets. Nevertheless, applying a single barycentric subdivision on S4 to get S8, an 8-point model of the circle enables us to define an order preserving poset map S8 x S8- S4. Restricted to the axes, this map yields weak homotopy equivalences S8 x S8, we obtain a version of the Hopf map S8 x S8s - SS4. This model of the Hopf map is in fact a map of non-Hausdorff double map cylinders.</p>

This thesis is devoted to application of topological ideas to Galois theory. In the fi rst part we obtain a characterization of branching data that guarantee that a regular mapping from a Riemann surface to the Riemann sphere having this branching data is invertible in radicals. The mappings having such branching data are then studied with emphasis on those exceptional properties of these mappings that single them out among all mappings from a Riemann surface to the Riemann sphere. These results provide a framework for understanding an earlier work of Ritt on rational functions invertible in radicals. In the second part of the thesis we apply topological methods to prove lower bounds in Klein's resolvent problem, i.e. the problem of determining whether a given algebraic function of n variables is a branch of a composition of rational functions and an algebraic function of k variables. The main topological result here is that the smallest dimension of the base-space of a covering from which a given covering over a torus can be induced is equal to the minimal number of generators of the monodromy group of the covering over the torus. This result is then applied for instance to prove the bounds k is at least n/2 in Klein's resolvent problem for the universal algebraic function of degree n and the answer k = n for generic algebraic function of n variables of degree at least 2n.

Thesis (M.A.)--University of North Texas, August, 2009. / Title from title page display. Includes bibliographical references.

Thesis--Université de Paris XI. / Includes bibliographical references.

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