published_or_final_version / Mathematics / Master / Master of Arts

Thesis (M.A.)--University of Hong Kong, 1966. / Also available in print.

This thesis is concerned with the calculus of alternating differential forms on a manifold. It establishes that this calculus is of a purely algebraic nature by developing its precise analogue for an arbitrary commutative algebra with unit over a field. / Thesis / Master of Science (MSc)

The aim of this work is to introduce differential forms on Euclidean space. The theory of differential forms provides a way of abstracting integration by formalising differentials over which an integral can be taken. The work builds towards Stokes’ Theorem for which a proof is given. Finally, using Stokes’ Theorem, three famous integral theorems from vector analysis are derived.

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No abstract provided. / Thesis / Doctor of Philosophy (PhD) / Scope and contents: Two topics are studied in this thesis. The first topic is concerned with the relation between the categories of complexes over two algebras when there is a unitary algebra homomorphism from one to the other. The second topic deals with differential forms. A certain finiteness theorem for the module of integral differential forms is studied.

Made available in DSpace on 2015-04-22T22:16:11Z (GMT). No. of bitstreams: 1 Dissertacao Walter Lucas Pinto.pdf: 3826919 bytes, checksum: 2d46d1c5f92ab885832ed054b123f3f5 (MD5) Previous issue date: 2009-02-27 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this paper we will study about Ribaucour Transformations and we will use the definition proposed by K.Tenenblat in [1]. We will study concerning the geometric aspects of such transformations and we will obtain as applications some Dupin surfaces in the R3 that are associated by a such Transformation. We will also present the visualization of those surfaces using resources of graphic computation. / Neste trabalho faremos um estudo acerca de Transformações de Ribaucour e usaremos a definição proposta por K.Tenenblat em [1]. Estudaremos acerca dos aspectos geométricos de tais transformações e obteremos como aplicações algumas superfícies de Dupin no R3 que estão associadas por uma tal Transformação. Apresentaremos também a visualização dessas superfícies usando recursos de computação gráfica.

This thesis consists of six chapters and deals with four topics related to De Rham Theory on semialgebraic sets. The first topic deals with L-infinity cohomology on semialgebraic sets. We introduce smooth L-infinity differential forms on a singular (semialgebraic) space X in Rn. Roughly speaking, a smooth L-infinity differential form is a collection of smooth forms on disjoint smooth subsets (stratification) of X with matching tangential components on the adjacent strata and of bounded size (in the metric induced from Rn). We identify the singular homology of X as the homology of the chain complex generated by semialgebraic singular simplices, i.e. continuous semialgebraic maps from the standard simplex into X. Singular cohomology of X is defined as the homology of the Hom dual to the chain complex of the singular chains. Finally, we prove a De Rham type theorem establishing a natural isomorphism between the singular cohomology and the cohomology of smooth L-infinity forms. The second topic is a construction of a Lipschitz deformation retraction on a neighborhood of a point in a semialgebraic set with estimates on its derivatives. Such a deformation retraction is the key to the results of the first and the third topics. The third topic is related to Poincare inequality on a semialgebraic set. We study Poincare type Lp inequality for differential forms on a compact semialgebraic subset of Rn for p >> 1. First we derive a local inequality by using a Lipschitz deformation retraction with estimates on its derivatives from the second topic and then we extend it to a global inequality by employing a technique developed in the appendix. As a consequence we obtain an isomorphism between Lp cohomology and singular cohomology of a normal compact semialgebraic set. The final topic is in the appendix. It deals with an explicit proof of Poincare type inequality for differential forms on compact manifolds. We prove the latter inequality by means of a constructive 'globalization' method of a local Poincare inequality on convex sets. The appendix serves as a model case for the results of the third topic in Chapter 5.

This thesis consists of six chapters and deals with four topics related to De Rham Theory on semialgebraic sets. The first topic deals with L-infinity cohomology on semialgebraic sets. We introduce smooth L-infinity differential forms on a singular (semialgebraic) space X in Rn. Roughly speaking, a smooth L-infinity differential form is a collection of smooth forms on disjoint smooth subsets (stratification) of X with matching tangential components on the adjacent strata and of bounded size (in the metric induced from Rn). We identify the singular homology of X as the homology of the chain complex generated by semialgebraic singular simplices, i.e. continuous semialgebraic maps from the standard simplex into X. Singular cohomology of X is defined as the homology of the Hom dual to the chain complex of the singular chains. Finally, we prove a De Rham type theorem establishing a natural isomorphism between the singular cohomology and the cohomology of smooth L-infinity forms. The second topic is a construction of a Lipschitz deformation retraction on a neighborhood of a point in a semialgebraic set with estimates on its derivatives. Such a deformation retraction is the key to the results of the first and the third topics. The third topic is related to Poincare inequality on a semialgebraic set. We study Poincare type Lp inequality for differential forms on a compact semialgebraic subset of Rn for p >> 1. First we derive a local inequality by using a Lipschitz deformation retraction with estimates on its derivatives from the second topic and then we extend it to a global inequality by employing a technique developed in the appendix. As a consequence we obtain an isomorphism between Lp cohomology and singular cohomology of a normal compact semialgebraic set. The final topic is in the appendix. It deals with an explicit proof of Poincare type inequality for differential forms on compact manifolds. We prove the latter inequality by means of a constructive 'globalization' method of a local Poincare inequality on convex sets. The appendix serves as a model case for the results of the third topic in Chapter 5.

The theory of Riemann surfaces is quite old, consequently it is well developed. Riemann surfaces originated in complex analysis as a means of dealing with the problem of multi-valued functions. Such multi-valued functions occur because the analytic continuation of a given holomorphic function element along different paths leads in general to different branches of that function. The theory splits in two parts; the compact and the non-compact case. The function theory developed on these cases are quite dissimilar. The main difficulty one encounters in the compact case is the scarcity of global holomorphic functions, which limits one’s study to meromorphic functions. This however is not an issue in non-compact Riemann surfaces, where one enjoys a vast variety of global holomorphic functions. While the function theory of compact Riemann surfaces is centered around the Riemann-Roch theorem, which essentially tells us how many linearly independent meromorphic functions there are having certain restrictions on their poles, the function theory developed on non-compact Riemann surface engages tools for approximation of functions on certain subsets by holomorphic maps on larger domains. The most powerful tool in this regard is the Runge’s approximation theorem. An intriguing application of this is the Gunning-Narasimhan theorem, which says that every connected open Riemann surface has an immersion into the complex plane. The main goal of this project is to prove Runge’s approximation theorem and illustrate its effectiveness in proving the Gunning-Narasimhan theorem. Finally we look at an analogue of Gunning-Narasimhan theorem in the case of a compact Riemann surface.