Thesis (doctoral)--Université de Paris-Sud, Orsay, 1982. / Bibliography: p. 93-96.

The construction of characteristic classes via the curvature form of a connection is one motivation for the refinement of integral cohomology by de Rham cocycles -- known as differential cohomology. We will discuss the analog in the case of a group action on the manifold: We will show the compatibility of the equivariant characteristic class in integral Borel cohomology with the equivariant characteristic form in the Cartan model. Motivated by this understanding of characteristic forms, we define equivariant differential cohomology as a refinement of equivariant integral cohomology by Cartan cocycles.

We consider the regularization of some equivariant Euler classes of certain infinite-dimensional vector bundles over a finite-dimensional manifold M using the framework of zeta-regularized products [35, 53, 59]. An example of such a regularization is the Atiyah–Witten regularization of the T-equivariant Euler class of the normal bundle v(TM) of M in the free loop space LM [2]. In this thesis, we propose a new regularization procedure — W-regularization — which can be shown to reduce to the Atiyah–Witten regularization when applied to the case of v(TM). This new regularization yields a new multiplicative genus (in the sense of Hirzebruch [26]) — the ^Γ-genus — when applied to the more general case of a complex spin vector bundle of complex rank ≥ 2 over M, as opposed to the case of the complexification of TM for the Atiyah–Witten regularization. Some of its properties are investigated and some tantalizing connections to other areas of mathematics are also discussed. We also consider the application of W-regularization to the regularization of T²- equivariant Euler classes associated to the case of the double free loop space LLM. We find that the theory of zeta-regularized products, as set out by Jorgenson–Lang [35], Quine et al [53] and Voros [59], amongst others, provides a good framework for comparing the regularizations that have been considered so far. In particular, it reveals relations between some of the genera that appeared in elliptic cohomology, allowing us to clarify and prove an assertion of Liu [44] on the ˆΘ-genus, as well as to recover the Witten genus. The ^Γ₂-genus, a new genus generated by a function based on Barnes’ double gamma function [5, 6], is also derived in a similar way to the ^Γ-genus. / Thesis (Ph.D.) - University of Adelaide, School of Mathematical Sciences, 2008

We consider the regularization of some equivariant Euler classes of certain infinite-dimensional vector bundles over a finite-dimensional manifold M using the framework of zeta-regularized products [35, 53, 59]. An example of such a regularization is the Atiyah–Witten regularization of the T-equivariant Euler class of the normal bundle v(TM) of M in the free loop space LM [2]. In this thesis, we propose a new regularization procedure — W-regularization — which can be shown to reduce to the Atiyah–Witten regularization when applied to the case of v(TM). This new regularization yields a new multiplicative genus (in the sense of Hirzebruch [26]) — the ^Γ-genus — when applied to the more general case of a complex spin vector bundle of complex rank ≥ 2 over M, as opposed to the case of the complexification of TM for the Atiyah–Witten regularization. Some of its properties are investigated and some tantalizing connections to other areas of mathematics are also discussed. We also consider the application of W-regularization to the regularization of T²- equivariant Euler classes associated to the case of the double free loop space LLM. We find that the theory of zeta-regularized products, as set out by Jorgenson–Lang [35], Quine et al [53] and Voros [59], amongst others, provides a good framework for comparing the regularizations that have been considered so far. In particular, it reveals relations between some of the genera that appeared in elliptic cohomology, allowing us to clarify and prove an assertion of Liu [44] on the ˆΘ-genus, as well as to recover the Witten genus. The ^Γ₂-genus, a new genus generated by a function based on Barnes’ double gamma function [5, 6], is also derived in a similar way to the ^Γ-genus. / Thesis (Ph.D.) - University of Adelaide, School of Mathematical Sciences, 2008

Orientador: Caio Jose Colletti Negreiros / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-10T10:04:33Z (GMT). No. of bitstreams: 1 Grama_LinoAndersondaSilva_M.pdf: 862309 bytes, checksum: ac6a88c1ff96ef74d7a840ce591336f5 (MD5) Previous issue date: 2008 / Resumo: O principal objetivo desta dissertação 'e apresentar a construção e a classificação das aplicações harmônicas de S2 em U(n), baseado nas idéias de K.Uhlenbeck. Apresentamos um exemplo de aplicação harmônica em U(4) e provamos que tal exemplo 'e, de fato, uma aplicação harmônica não-holomorfa na variedade de Grassman G2(C4), de 2-planos em C4.Demonstramos o teorema de Valli sobre o espectro da energia e, por fim, parametrizamos o conjunto Harm(S2, U(n)), de todas aplicações harmônicas de S2 em U(n), fornecendo uma classifica¸c¿ao para tais aplicações, seguindo o trabalho de J.C.Wood / Abstract: This dissertation is concerned with the construction and classification of harmonic maps from S2 on U(n), according to K. Uhlenbeck. We construct an example of harmonic map on U(4) and prove that this example is, in fact, a non-holomorphic harmonic map in the Grassmann manifold G2(C4) of 2-plans on C4. We also prove the theorem of Valli on the spectrum of energy and, finally, describe the arametrization of the space Harm(S2, U(n)), of all harmonics maps from S2 in U(n), provide the classification for such maps, following the work of J.C.Wood / Mestrado / Mestre em Matemática

This dissertation consists of two distinct parts, the first concerning S^1-equivariant cohomology of loop spaces and the second concerning stability in partial differential equations. In the first part of this dissertation, we study the existence of S^1-equivariant characteristic classes on certain natural infinite rank bundles over the loop space LM of a manifold M. We discuss the different S^1-equivariant cohomology theories in the literature and clarify their relationships. We attempt to use S^1-equivariant Chern-Weil techniques to construct S^1-equivariant characteristic classes. The main result is the construction of a sequence of S^1-equivariant characteristic classes on the total space of the bundles, but these classes do not descend to the base LM. In addition, we identify a class of bundles for which a single S^1-equivariant characteristic class does admit an S^1-equivariant Chern-Weil construction. In the second part of this dissertation, we study the Maslov index as a tool to analyze stability of steady state solutions to a reaction-diffusion equation in one spatial dimension. We show that the path of unstable subspaces associated to this equation is governed by a matrix Riccati equation whose solution S develops singularities when changes in the Maslov index occur. Our main result proves that at these singularities the change in Maslov index equals the number of eigenvalues of S that increase to +∞ minus the number of eigenvalues that decrease to -∞.

The construction of characteristic classes via the curvature form of a connection is one motivation for the refinement of integral cohomology by de Rham cocycles -- known as differential cohomology. We will discuss the analog in the case of a group action on the manifold: We will show the compatibility of the equivariant characteristic class in integral Borel cohomology with the equivariant characteristic form in the Cartan model. Motivated by this understanding of characteristic forms, we define equivariant differential cohomology as a refinement of equivariant integral cohomology by Cartan cocycles.

O teorema clássico de Borsuk-Ulam nos dá informações à respeito de aplicações \'S POT. n\' \'SETA\' \'R POT. n\', no qual \'S POT. n\' é um \'Z IND. 2\' -espaço livre. O teorema afirma que existe pelo menos uma órbita que é enviada em um único ponto em \'R POT. n\'. Dold [9] estendeu este problema para o contexto de fibrados, considerando aplicações f : S (E) \'SETA\' \'E POT. \'prime\'\' nos quais preservam fibras; aqui, S (E) denota o espaço total do fibrado em esfera sobre B associado ao fibrado vetorial E \'SETA\' B e \'E POT. \'prime\'\' \'SETA\' B é o outro fibrado vetorial. O objetivo desse trabalho é provar esta versão do teorema de Borsuk-Ulam obtida por Dold, chamada versão parametrizada do teorema de Borsuk-Ulam. Nós também provamos uma versão cohomológica deste problema / The classical Borsuk-Ulam Theorem gives information about maps \'S POT. n\' \'ARROW\' \'R POT. n\' where \'S POT. n\' has a free action of the cyclic group \'Z IND. 2\'. The theorem states that there is at least one orbit which is sent to a single point in \'R POT. n\'. Dold [9] extended this problem to a fibre-wise setting, by considering maps f : S (E) \'ARROW\' \' E POT. prime\' which preserve fibres; here, S (E) denotes the total space of the sphere bundle associated over B to a vector bundle E \'ARROW\' B and \'E POT. prime\' \'ARROW\' B is other vector bundle over B. The purpose of this work is to prove this version of the Borsuk-Ulam theorem obtained by A. Dold, called parametrized version of the Borsuk-Ulam theorem. We also prove a cohomological generalization of this problem

Neste trabalho, estudamos diferentes construções e propriedades das classes características de variedades suaves e singulares. Para ilustrar a teoria, calculamos a obstrução de Euler de algumas superfícies singulares no espaço tridimensional e apresentamos uma fórmula do tipo Poincaré-Hopf para variedades singulares / In this work, we study different constructions and properties of the characteristics classes of smooth and singular manifolds. To ilustrate the theory, we compute the Euler obstructions of some singular surfaces in tridimensional space and state a Poincaré-Hopf´s formula for singular varieties

Neste trabalho, estudamos diferentes construções e propriedades das classes características de variedades suaves e singulares. Para ilustrar a teoria, calculamos a obstrução de Euler de algumas superfícies singulares no espaço tridimensional e apresentamos uma fórmula do tipo Poincaré-Hopf para variedades singulares / In this work, we study different constructions and properties of the characteristics classes of smooth and singular manifolds. To ilustrate the theory, we compute the Euler obstructions of some singular surfaces in tridimensional space and state a Poincaré-Hopf´s formula for singular varieties