Thesis (Ph. D.)--University of Hawaii at Manoa, 2003. / Includes bibliographical references (leaves 51-52).

<p>Gersten-Witt cohomologies of Hirzebruch surfaces are computed.</p> / Dissertation

Not available / text

The thesis Witt Groups of Complex Varieties studies and compares two related cohomology theories that arise in the areas of algebraic geometry and topology: the algebraic theory of Witt groups, and real topological K-theory. Specifically, we introduce comparison maps from the Grothendieck-Witt and Witt groups of a smooth complex variety to the KO-groups of the underlying topological space and analyse their behaviour. We focus on two particularly favourable situations. Firstly, we explicitly compute the Witt groups of smooth complex curves and surfaces. Using the theory of Stiefel-Whitney classes, we obtain a satisfactory description of the comparison maps in these low-dimensional cases. Secondly, we show that the comparison maps are isomorphisms for smooth cellular varieties. This resultapplies in particular to projective homogeneous spaces. By extending knowncomputations in topology, we obtain an additive description of the Witt groups of all projective homogeneous varieties that fall within the class of hermitian symmetric spaces.

[pt] Estudamos a variedade M3 de curvas de gênero três em característica dois. Para cada uma destas curvas calculamos seus possíveis números de pontos de Weierstrass, seus pesos, normalizações de muitos loci no espaço de moduli, entre outras coisas. Tratamos ainda do conceito de ponto de Galois. / [en] We study the variety M3 of curves of genus three in characteristic two. For each of the curves we compute the possible number of Weierstrass points, their weights, normalizations of many loci in the moduli space, and so on. We also deal with the concept of a Galois point.

This paper examines the probability that a random polynomial of specific degree over a field has a specific number of distinct roots in that field. Probabilities are found for random quadratic polynomials with respect to various probability measures on the real numbers and p-adic numbers. In the process, some properties of the p-adic integer uniform random variable are explored. The measure Witt ring, a generalization of the canonical Witt ring, is introduced as a way to link quadratic forms and measures, and examples are found for various fields and measures. Special properties of the Haar measure in connection with the measure Witt ring are explored. Higher-degree polynomials are explored with the aid of numerical methods, and some conjectures are made regarding higher-degree p-adic polynomials. Other open questions about measure Witt rings are stated. / Graduation date: 1999

Let R be a connective ring spectrum and let M be an R-bimodule. In this paper we prove several results that relate the K-theory of R⋉M and T[superscript M, subscript R] to a “topological Witt vectors” construction W(R; M), where R ⋉ M is the square-zero extension of R by M and T [superscript M, subscript R] is the tensor algebra on M. Our main results include a desciption of the Taylor tower of K(R ⋉ (−)) and the derived functor of K̃(TR(−)) on the category of R-bimodules in terms of the Taylor tower of W(R;−). W(R;−) has an easily described Taylor tower, given explicitly by Lindenstrauss and McCarthy in [17]. Our main results serve as generalizations of the results for discrete rings in [17, 18] and also extend the computations by Hesselholt and Madsen [15] showing that π₀(TR(R; p)) is isomorphic to the p-typical Witt vectors over R when R a commutative ring. / text

Thesis (Ph. D.)--Oregon State University, 1999. / Typescript (photocopy). Includes bibliographical references (leaf 66). Also available on the World Wide Web.

Thesis (doctoral)--Universität, Hannover, 2004.

Orientador: Antonio Jose Engler / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-14T13:32:53Z (GMT). No. of bitstreams: 1 PapaNeto_Angelo_D.pdf: 939822 bytes, checksum: 40aedb2e25f1f22956a7ff182038cae3 (MD5) Previous issue date: 2007 / Resumo: Um corpo ordenado é uma estrutura algébrica similar à do corpo dos números reais. No entanto, ao contrário dos reais, um corpo arbitrário F pode admitir mais de uma ordem, inclusive um número infinito e não enumerável de ordens. A cada elemento x do corpo F podemos associar uma forma quadrática binária [1, x], chamada 1-forma de Pfister. Os elementos de F = F 0} representados por [1, x], constituem um grupo que chamamos grupo de valores da forma e denotamos por D[1,x]. Um elemento d S F é chamado rígido se D[1, d] = F2 U dF2 , onde F2 é o subgrupo de F formado pelos quadrados. Um elemento d é dito birígido se d e -d são rígidos. O presente trabalho tem como objetivo principal obter um teorema de estrutura para o anel de Witt (das classes de equivalência de formas quadráticas) de um corpo ordenado F admitindo um elemento rígido que não é birígido e que é negativo em relação à pelo menos uma das ordens do corpo. Mais precisamente, obtemos uma decomposição do anel de Witt de F como produto de anéis de Witt de duas extensões H ¿ F e K ¿ F, ambas contidas no fecho quadrático de F. Os anéis de Witt de H e K têm estrutura mais simples que a do anel de Witt de F. Obtemos os corpos H e K construindo subgrupos Rd e Sd associados ao elemento rígido d e exigindo que valha uma propriedade de decomposição: F = Rd· Sd. O corpo H é uma henselização de F relativa a um anel de valorização (A;mA) de F tal que Rd = (1 + mA) F2 . O corpo K é pitagórico e tem espaço de ordens XK homeomorfo ao espaço X/Sd das ordens de F que contém Sd. Obtemos ainda uma condição necessária e suficiente para que ocorra a decomposição F = Rd · Sd, que depende do grupo de valores e do corpo de resíduos do anel de valorização A. / Abstract: An ordered field is an algebraic structure like the field of real numbers. However, while the field of real numbers have only one ordering, an arbitrary ordered field F may have more than one ordering, and also a infinite and uncountble number of orderings is allowed. To each element x Î F one can associate an binary quadratic form [1, x], called Pfister 1-fold form. The set of elements in F = F 0} which are represented by [1, x] is a group D[1,x], called value group of [1,x]. An element d S F is called rigid if D[1, d] = F2 U dF2, where F 2 denotes the subgroup of squares in F . An element d is called birigid if d and -d are both rigid. The main purpose of this thesis is to prove an structure theorem for Witt ring (of equivalence classes of quadratic forms) of an ordered field F with a rigid element which is not birigid and is negative in at least one ordering of F, that is, we get a decomposition of the Witt ring of F as a product of Witt rings of extensions H ¿ F and K ¿ F, both inside the quadratic closure of F. The Witt rings of H and K have a simpler structure than Witt ring of F. We get fields H and K by builting subgroups Rd and Sd associated to the rigid element d and making the addicional assumption that F = Rd·Sd holds. The field H is a henselization of F relative to a valuation ring (A;mA) of F such that Rd = (1 + mA) F2. The pythagorean field K has space of orderings XK homeomorphic to X/Sd, the space of orderings of F which contain Sd. Moreover, we settle an necessary and suficient condiction to decomposition F = Rd·Sd holds, relative to value group and residue field of valuation ring A. / Doutorado / Algebra / Doutor em Matemática