Thesis (Ph. D.)--University of Wisconsin--Madison, 1968. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 61-64). Available also in a digital version from Dissertation Abstracts.

Thesis (Ph. D.)--Florida State University, 2004. / Advisor: Dr. Wolfgang Heil, Florida State University, College of Arts and Sciences, Dept. of Mathematics. Title and description from dissertation home page (viewed Sept. 28, 2004). Includes bibliographical references.

A Polish space is a separable topological space that can be metrized by means of a complete metric. A subset A of a Polish space X is analytic if there is a Polish space Z and a continuous function f : Z —> X such that f(Z)= A. After proving that each uncountable Polish space contains a non-Borel analytic subset we conclude that there exists a universally measurable non-Borel set.

Thesis (Ph. D.)--University of North Texas, May, 2008. / Title from title page display. Includes bibliographical references.

Thesis (M.Sc.) -- Memorial University of Newfoundland. 1973. / Typescript. Bibliography : leaves 45. Also available online.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: Within lattice theory, an interesting question asked is whether a given abstract lattice may be represented concretely as subsets of a closure system on a topological space. This is true for boolean algebras, bounded distributive lattices and arbitrary bounded lattices. In particular, there are a multitude of ways to represent bounded lattices. We present some of these ideas, as well as an analysis of the differences between them. We further investigate the attempts that were made to extend the above representations to lattices endowed with operators, in particular the work done on bounded distributive lattices with operators. We then make a new contribution by extending this work to arbitrary bounded lattices with operators. We also show that the so-called sufficiency operator has a relational representation in the bounded lattice case. / AFRIKAANSE OPSOMMING: Binne die raamwerk van tralie teorie word die vraag soms gevra of ’n gegewe tralie konkreet veteenwoordig kan word as subversamelings van ’n afsluitingssisteem op ’n topologiese ruimte. Die voorgenoemde is waar vir, onder andere, boolse algebras, begrensde distributiewe tralies en algemene begrensde tralies. Daar is veral vir begrensde tralies menigte maniere om hul te verteenwoordig. Ons bied sommige van hierdie idees voor, asook ’n analiese van die verskille daarin teenwoordig. Verder ondersoek ons ook sommige van die maniere waarop tralies tesame met operatore verteenwoordig kan word. Ons sal spesiale aandag gee aan distributiewe tralies met operatore, soos gedoen in, met die idee om die voorgenoemde uit te brei na algemene begrensde tralies met operatore. Ons toon dan verder aan dat die sogenoemde voldoende operator ook ’n relasionele verteenwoordiging het in die begrensde tralie geval.

Orientador: Marcos Benevuto Jardim / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-08T04:40:41Z (GMT). No. of bitstreams: 1 Frejlich_Pedro_M.pdf: 852390 bytes, checksum: bcde0cd89c0bcfe3b2515b5cfe48e528 (MD5) Previous issue date: 2007 / Resumo: Construímos a transformada de Nahm de um fibrado de Higgs estável de grau nulo sobre uma superfície de Riemann de gênero pelo menos 2. Para tanto, empregamos a Teoria do Índice de Atiyah-Singer e um vanishing theorem que segue da hipótese de estabilidade do fibrado. O principal resultado é que o fibrado transformado é hiperholomorfo e sem fatores planos. Desse modo não só recuperamos os resultados algébricos de [7] e os de [12] para o cos q = 0 como também provamos uma descrição mais detalhada da estrutura geométrica da transformada ¿ o que, aliada às técnicas de [10] sugere que ela possa ser invertida. Palavras-chave: Superfícies de Riemann, Fibrados Estáveis, Teoria do Índice, Transformada de Nahm, Transformada de Fourier-Mukai, Variedades Hiper-Kähler, Variedades Abelianas, Conexões Hiperholomorfas / Abstract: We construct the Nahm transform of a stable, degree-zero Higgs bundle on a Riemann surface of genus at least 2. Atiyah-Singer¿s index theorem is the basic tool employed, along with a vanishing theorem which is due to the stability hypothesis. Our main result is that the transformed bundle is hyperholomorphic and without flat factors. This not only recovers the algebraic results of [7] and that of [12] for the cos q = 0, but also provides a more detailed description of the geometric structure of the transformed bundle. Such results suggest that this Nahm transform can be inverted, cf. [10]. Key-words:Riemann surfaces, Stable bundles, Index Theory, Nahm Transform, Fourier-Mukai Transform, Hyperk¨ahler manifolds, Abelian varieties, Hyperholomorphic connections / Mestrado / Geometria Diferencial/Geometria algebrica / Mestre em Matemática

In this thesis, we discuss three main projects which are related to Polish groups and their actions on standard Borel spaces. In the first part, we show that the complexity of the classification problem of continua is Borel bireducible to a universal orbit equivalence relation induce by a Polish group on a standard Borel space. In the second part, we compare the relative complexity of various types of classification problems concerning subspaces of [0,1]^n for all natural number n. In the last chapter, we give a topological characterization theorem for the class of locally compact two-sided invariant non-Archimedean Polish groups. Using this theorem, we show the non-existence of a universal group and the existence of a surjectively universal group in the class.

This dissertation deals with three topics in descriptive set theory. First, the order topology is a natural topology on ordinals. In Chapter 2, a complete classification of order topologies on ordinals up to Borel isomorphism is given, answering a question of Benedikt Löwe. Second, a map between separable metrizable spaces X and Y preserves complete metrizability if Y is completely metrizable whenever X is; the map is resolvable if the image of every open (closed) set in X is resolvable in Y. In Chapter 3, it is proven that resolvable maps preserve complete metrizability, generalizing results of Sierpiński, Vainštein, and Ostrovsky. Third, an equivalence relation on a Polish space has the Laczkovich-Komjáth property if the following holds: for every sequence of analytic sets such that the limit superior along any infinite set of indices meets uncountably many equivalence classes, there is an infinite subsequence such that the intersection of these sets contains a perfect set of pairwise inequivalent elements. In Chapter 4, it is shown that every coanalytic equivalence relation has the Laczkovich-Komjáth property, extending a theorem of Balcerzak and Głąb.