Clifford algebras have been studied for many years and their algebraic properties are well known. In particular, all Clifford algebras have been classified as matrix algebras over one of the three division algebras. But Clifford Algebras are far more interesting than this classification suggests; they provide the algebraic basis for a unified language for physics and mathematics which offers many advantages over current techniques. This language is called geometric algebra - the name originally chosen by Clifford for his algebra - and this thesis is an investigation into the properties and applications of Clifford's geometric algebra. The work falls into three broad categories: - The formal development of geometric algebra has been patchy and a number of important subjects have not yet been treated within its framework. A principle feature of this thesis is the development of a number of new algebraic techniques which serve to broaden the field of applicability of geometric algebra. Of particular interest are an extension of the geometric algebra of spacetime (the spacetime algebra) to incorporate multiparticle quantum states, and the development of a multivector calculus for handling differentiation with respect to a linear function. - A central contention of this thesis is that geometric algebra provides the natural language in which to formulate a wide range of subjects from modern mathematical physics. To support this contention, reformulations of Grassmann calculus, Lie algebra theory, spinor algebra and Lagrangian field theory are developed. In each case it is argued that the geometric algebra formulation is computationally more efficient than standard approaches, and that it provides many novel insights. - The ultimate goal of a reformulation is to point the way to new mathematics and physics, and three promising directions are developed. The first is a new approach to relativistic multiparticle quantum mechanics. The second deals with classical models for quantum spin-I/2. The third details an approach to gravity based on gauge fields acting in a fiat spacetime. The Dirac equation forms the basis of this gauge theory, and the resultant theory is shown to differ from general relativity in a number of its features and predictions.

Singular Riemannian foliations in spheres provide local models for an extensive kind of singular Riemannian foliations, whose theory contributes in the understanding of Riemannian manifolds. Hence the importance of studying and classifying them, a research subject that still remains open. In 2014, Marco Radeschi constructed indecomposable singular Riemannian foliations of arbitrary codimension, most of them inhomogeneous, which generalized all known examples of that type so far. The present dissertation is a detailed study of his work, along with observations about the progress made on this dynamic field since that paper was published. Besides introducing preliminary notions and examples on singular Riemannian foliations, isometric actions and Clifford theory, it is explained a construction of inhomogeneous isoparametric hypersurfaces, due to Ferus, Karcher and Münzner, that was a fundamental framework for the results of Radeschi. After that, it is described exhaustively the construction of Clifford and composed foliations in spheres, which are the examples that Radeschi created using Clifford systems. In the sequel it is established an extraordinary bijective correspondence between Clifford foliations (merely geometric objects) and Clifford systems (purely algebraic objects). This text finishes examining the relations of homogeneity properties among FKM, Clifford and composed foliations. / Folheações Riemannianas singulares em esferas fornecem modelos locais para folheações Riemannianas singulares mais gerais, cuja teoria contribui na compreensão de variedades Riemannianas. Daí a sua importança de estudá-los e classificá-los, uma área de pesquisa que se mantém aberta. Em 2014, Marco Radeschi construiu folheações Riemannianas singulares indecomponíveis de codimensão arbitrária, a maioria delas não homogêneas, que generalizaram todos os exemplos conhecidos desse tipo até então. A presente dissertação é um estudo detalhado desse trabalho, junto com observações sobre avanços que se têm feito neste dinâmico campo desde a publicação do artigo. Após introduzir as noções e exemplos preliminares de folheações Riemannianas singulares, ações isométricas e teoria de Clifford, é explorada uma construção de hipersuperfícies isoparamétricas não homogêneas, devida a Ferus, Karcher e Münzner (FKM), que foi peça fundamental para os resultados de Radeschi. Em seguida, descreve-se minuciosamente a construção de folheações composta e de Clifford em esferas, que são os exemplos que o autor mencionado anteriormente gerou usando sistemas de Clifford. Continuando com a análise dessas novas folheações Riemannianas singulares, estabelece-se uma extraordinária correspondência biunívoca entre folheações de Clifford (objetos meramente geométricos) e sistemas de Clifford (objetos puramente algébricos). Este texto termina examinando as relações das propriedades de homogeneidade entre folheações FKM, compostas e de Clifford.

Singular Riemannian foliations in spheres provide local models for an extensive kind of singular Riemannian foliations, whose theory contributes in the understanding of Riemannian manifolds. Hence the importance of studying and classifying them, a research subject that still remains open. In 2014, Marco Radeschi constructed indecomposable singular Riemannian foliations of arbitrary codimension, most of them inhomogeneous, which generalized all known examples of that type so far. The present dissertation is a detailed study of his work, along with observations about the progress made on this dynamic field since that paper was published. Besides introducing preliminary notions and examples on singular Riemannian foliations, isometric actions and Clifford theory, it is explained a construction of inhomogeneous isoparametric hypersurfaces, due to Ferus, Karcher and Münzner, that was a fundamental framework for the results of Radeschi. After that, it is described exhaustively the construction of Clifford and composed foliations in spheres, which are the examples that Radeschi created using Clifford systems. In the sequel it is established an extraordinary bijective correspondence between Clifford foliations (merely geometric objects) and Clifford systems (purely algebraic objects). This text finishes examining the relations of homogeneity properties among FKM, Clifford and composed foliations. / Folheações Riemannianas singulares em esferas fornecem modelos locais para folheações Riemannianas singulares mais gerais, cuja teoria contribui na compreensão de variedades Riemannianas. Daí a sua importança de estudá-los e classificá-los, uma área de pesquisa que se mantém aberta. Em 2014, Marco Radeschi construiu folheações Riemannianas singulares indecomponíveis de codimensão arbitrária, a maioria delas não homogêneas, que generalizaram todos os exemplos conhecidos desse tipo até então. A presente dissertação é um estudo detalhado desse trabalho, junto com observações sobre avanços que se têm feito neste dinâmico campo desde a publicação do artigo. Após introduzir as noções e exemplos preliminares de folheações Riemannianas singulares, ações isométricas e teoria de Clifford, é explorada uma construção de hipersuperfícies isoparamétricas não homogêneas, devida a Ferus, Karcher e Münzner (FKM), que foi peça fundamental para os resultados de Radeschi. Em seguida, descreve-se minuciosamente a construção de folheações composta e de Clifford em esferas, que são os exemplos que o autor mencionado anteriormente gerou usando sistemas de Clifford. Continuando com a análise dessas novas folheações Riemannianas singulares, estabelece-se uma extraordinária correspondência biunívoca entre folheações de Clifford (objetos meramente geométricos) e sistemas de Clifford (objetos puramente algébricos). Este texto termina examinando as relações das propriedades de homogeneidade entre folheações FKM, compostas e de Clifford.

La L-théorie classique d'un anneau commutatif est construite à partir des formes quadratiques sur cet anneau modulo une relation d'équivalence lagrangienne. Nous construisons la L-théorie dérivée, à partir des formes quadratiques $n$-décalées sur un anneau commutatif dérivé. Nous montrons que les formes $n$-décalées qui admettent un lagrangien possèdent une forme standard. Nous montrons des résultats de chirurgie pour la L-théorie dérivée, qui permettent de réduire une forme quadratique décalée en une forme plus simple équivalente. On compare la L-théorie dérivée avec la L-théorie classique.On définit un champ dérivé des formes quadratiques dérivées, et un champ dérivé des lagrangiens dans une forme, qui sont localement algébriques de présentation finie. On calcule les complexes tangents, et on trouve des points lisses. On montre un résultat de rigidité pour la L-théorie : la L-théorie d'un anneau commutatif est isomorphe à celle d'un voisinage hensélien de cet anneau. Enfin, on définit l'algèbre de Clifford d'une forme quadratique n-décalée, qui est une déformation d'une algèbre symétrique en tant qu'E_k-algèbre. On montre un affaiblissement de la propriété d'Azumaya pour ces algèbres, dans le cas d'un décalage nul n=0, qu'on appelle semi-Azumaya. Cette propriété exprime la trivialité de l'homologie de Hochschild du bimodule de Serre. / The classical L-theory of a commutative ring is built from the quadratic forms over this ring modulo a lagrangian equivalence relation.We build the derived L-theory from the n-shifted quadratic forms on a derived commutative ring. We show that forms which admit a lagrangian have a standard form. We prove surgery results for this derived L-theory, which allows to reduce shifted quadratic forms to equivalent simpler forms. We compare classical and derived L-theory.We define a derived stack of shifted quadratic forms and a derived stack of lagrangians in a form, which are locally algebraic of finite presentation. We compute tangent complexes and find smooth points. We prove a rigidity result for L-theory : the L-theory of a commutative ring is isomorphic to that of any henselian neighbourhood of this ring.Finally, we define the Clifford algebra of a n-shifted quadratic form, which is a deformation as E_k-algebra of a symmetric algebra. We prove a weakening of the Azumaya property for these algebras, in the case n=0, which we call semi-Azumaya. This property expresses the triviality of the Hochschild homology of the Serre bimodule.

Philosophiae Doctor - PhD / The aim of this dissertation is to calculate character tables of group extensions. There are several well–developed methods for calculating the character tables of group extensions. In this dissertation we study the method developed by Bernd Fischer, the so–called Fischer–Clifford matrices method, which derives its fundamentals from the Clifford theory. We consider only extensions G of the normal subgroup K by the subgroup Q with the property that every irreducible character of K can be extended to an irreducible character of its inertia group in G, if K is abelian. This is indeed the case if G is a split extension, by a well-known theorem of Mackey. A brief outline of the classical theory of characters pertinent to this study, is followed by a discussion on the calculation of the conjugacy classes of extension groups by the method of coset analysis. The Clifford theory which provide the basis for the theory of Fischer-Clifford matrices is discussed in detail. Some of the properties of these Fischer-Clifford matrices which make their calculation much easier are also given. As mentioned earlier we restrict ourselves to split extension groups G in which K is always elementary abelian. In this thesis we are concerned with the construction of the character tables of certain groups which are associated with Fi₂₂ and Sp₈ (2). Both of these groups have a maximal subgroup of the form 2⁷: Sp₆ (2) but they are not isomorphic to each other. In particular we are interested in the inertia groups of these maximal subgroups, which are split extensions. We use the technique of the Fischer-Clifford matrices to construct the character tables of these inertia groups. These inertia groups of 2⁷ : Sp₆(2), the maximal subgroup of Fi₂₂, are 2⁷ : S₈, 2⁷ : Ο⁻₆(2) and 2⁷ : (2⁵ : S₆). The inertia group of 2⁷ : Sp₆(2), the affine subgroup of Sp₈(2), is 2⁷ : (2⁵ : S₆) which is not isomorphic to the group with the same form which was mentioned earlier.

Thesis (Ph. D.)--University of Oregon, 2007. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 130-131). Also available for download via the World Wide Web; free to University of Oregon users.

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas, Programa de Pós-Graduação em Física, Florianópolis, 2014. / Made available in DSpace on 2015-02-05T21:09:01Z (GMT). No. of bitstreams: 1 328126.pdf: 592017 bytes, checksum: 7fc49c2c540d01b9d8ba814d8319d3d1 (MD5) Previous issue date: 2014 / Este trabalho consiste em um estudo sistemático dos usos do formalismo de Vlasov e sua aplicação à matéria nuclear a partir da Função de Wigner, que será estudada extensivamente no desenvolvimento deste trabalho. Entre as propriedades que podem ser calculadas, estamos especialmente interessados nas relações de dispersão a que a matéria nuclear descrita pelo modelo o - w - p obedece. Resultados para as relações de dispersão já foram obtidos emum artigo usando a função de Wigner, embora com um formalismo diferente,conhecido como função geratriz. Como será mostrado, os resultados para ambos os casos são idênticos, fato que comprova a viabilidade do formalismo de Vlasov. Este trabalho é um estudo introdutório e aborda a base necessária para a aplicação do formalismo de Vlasov nas mais diversas situações.<br> / Abstract : This work consists in a systematic study on the Vlasov formalism uses and its application to nuclear matter using the Wigner function, which will be studied extensively in this work. Among the properties that can be calculated, weare particularly interested in the dispersion relations that the nuclear matter described by the model o - w - p obeys. Results for the dispersion relations have been obtained in one paper using the Wigner function, although with a different formalism, known as generating function. As will be shown, the results for both cases are identical, fact that proves the viability of the Vlasov formalism. This work is an introductory study and approach the necessary basis for the application of the Vlasov formalism in different situations.

No description available.

Orientador: Nathan Jacob Berkovits / Banca: Victor de Oliveira Rivelles / Banca: Andrei Yuryevich Mikhaylov / Resumo: Em supersimetria padrão, construimos um superespaço com parâmetros 'x POT. 'mu'', 'teta', 'teta BARRA' (anti)comutantes e supercampos que dependem destes, então impomos vínculos e após definirmos supercampos quirais e vetorias, que são exemplos interessantes, e enfim construímos uma lagrangeana supersimétrica. Vamos aqui colocar condições mais fracas sobre os parâmetros do superespaço: agora tais parâmetros que antes eram anticomutantes vão formar uma álgebra de Clifford {'teta' POT. 'alfa'', 'teta' POT. 'beta'} = 'C POT. 'alfa' 'beta'' (1) e seguindo um procedimento análogo, com algumas definições adicionais, extenderemos nossa lagrangeana para o caso de um superespaço não-anticomutativo / Abstract: In standard supersymmetry, we build a superspace with parameters 'x POT. 'mu'', 'teta', 'teta BARRA' that (anti)commute and superfields that depend on them, then we impose constraints and define chiral and vector superfields, which are interesting examples, from which we build a supersymmetric Lagrangian. We will now impose a weaker condition on the superspace of the parameters: the ones that were anticommuting will now form a Clifford algebra {'teta' POT. 'alfa'', 'teta' POT. 'beta'}= 'C POT. 'alfa' 'beta'' (2) and, following an analogous procedure, with additional definitions, we will generalize our Lagrangian for the case of the non-anticommutative superspace / Mestre

Graduation date: 1994