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Navegando Matemática por Autor "Câmara, Leonardo Meireles"
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- ItemCaracterização geométrica do espaço moduli de conexões ASD do fibrado de Hopf quatérnio(Universidade Federal do Espírito Santo, 2017-03-06) Maroja, Aaron Aragon; Câmara, Leonardo Meireles; Bursztyn, Henrique; Batoréo, Marta JakubowiczIn the early 80’s, C.C. Taubes and K. Uhlenbeck provided the analytical foundations so that the solutions of the Yang-Mills equations, called instantons, would have a geometrical use, yet to be found in the same period. Simon Donaldson has built then a theory based on certain aspects of these solutions over 4-dimensional, oriented, closed, differentiable manifolds. In this context, one can isolate a class of connections, called anti-self-dual, that necessarily sastify the Yang-Mills equation. The collection of all such, modulo a natural equivalence relation, namely gauge equivalence, is called the moduli space M of the bundle and its study has led to astonishing insights into the structure of smooth 4- manifolds. This work is set to study in detail a particular example of Donaldson’s Theory on the Hopf bundle over the 4-dimensional manifold ?? 4 . We arrive at the BPST instantons of such bundle via the Cartan canonical 1-form on ????(2). Once these are in hand, we use the conformal invariance of anti-self-dual equations to write down a 5-parameter family of such connections. By making use of a theorem of Atiyah, Hitchin and Singer, we assert that every element of the moduli space M is uniquely represented by a connection in this family. From this we obtain a concrete realization of M as the open unit ball in R 5 . In particular, M is a 5-dimensional manifold with a natural compactification whose boundary is a copy of the base space ?? 4 .
- ItemClassificação de estruturas de Nambu lineares e p-formas singulares(Universidade Federal do Espírito Santo, 2012-08-13) Almeida, Carla Rodrigues; Alves, Magno Branco; Câmara, Leonardo Meireles; Bursztyn, Henrique; Corrêa Júnior, Maurício BarrosThe aim of this work is to study the foliations that arise from Nambu structures and present the relationship between differential forms and some of this structures. More specifically, to make a study of the Poisson geometry and of singular foliations, emphasiz-ing the case of the simplectic foliation that arises from the Poisson structure and then, to present the Nambu geometry, studying the case of the foliations that arise from the this structures of order grater than or equal to three. In this particular case, we shall show how this Nambu structures are related with differential formas and, by this relationship, classify linear Nambu structure through a result of classification of integrable differential p-forms
- ItemConexões afins e a teoria de Cartan-Einstein(Universidade Federal do Espírito Santo, 2016-07-12) Xavier, Roberta Meschese; Câmara, Leonardo Meireles; Macarini, Leonardo Magalhães; Vieira, Matheus Brioschi HerkenhoffThe Cartan-Einstein theory of gravitation is a modified version of the General Theory of Relativity. While Einstein’s theory was developed according to the hypothesis that the relativity of space-time has zero torsion, Cartan allows torsion and relate it to the angular momentum of the matter several years before the discovery of the spin of the electron. Cartan’s articles, in particular Sur les variétés the affine connexion et la théorie de la Généralisée relativité, which is the basis of this work, contains important new mathematical ideas that have influenced the development of differential geometry and, in particular, led to the general theory of affine connections. Essentially these are geometrical objects on a differentiable manifolds that connect nearby tangent spaces. In this dissertation we study the invariance of the laws of classical and relativistic mechanics in continuous media and the geometry of space-time from the standpoint of affine connections.
- ItemMétodos da teoria de calibre na gravitação newtoniana(Universidade Federal do Espírito Santo, 2023-03-30) Lima, Matheus dos Santos; Câmara, Leonardo Meireles; https://orcid.org/0000-0002-4637-8573; http://lattes.cnpq.br/9240898305551070; http://lattes.cnpq.br/6044154913017696; Martins, Gabriel Luchini; https://orcid.org/0000-0001-6627-3034; http://lattes.cnpq.br/3344738411070832; Earp, Henrique Nogueira de Sá; https://orcid.org/0000-0003-0475-4494; http://lattes.cnpq.br/4040588840128192Élie Cartan developed the concept of affine connection as a way to establish a local equivalence relation between sections of the tangent space to four-dimensional spacetime. This concept allowed for an intrinsic description of local geometric properties, independent of an external coordinate system. By adopting the method of moving frames, Cartan provided a precise geometric description of spacetime, enriching our understanding of the theory of general relativity. This dissertation aims to study Cartan’s works and highlight his contributions, through the affine connection, to the geometrization of the Newtonian physical system, as well as to identify possible connections with contemporary Gauge Theory, which offers a powerful framework for describing fundamental interactions of this nature. In this context, the main objective of this study is to propose equations that allow for finding an affine connection that is a Yang-Mills connection modeling the physical phenomena under study. By seeking this connection between Cartan’s theory and Gauge Theory, we aim to advance our understanding of fundamental interactions and possibly pave the way for future work.
- ItemO Teorema da Separação de Jordan-Brouwer em variedades diferenciáveis(Universidade Federal do Espírito Santo, 2019-05-06) Giovanelli, Joelso; Batoréo, Marta Jakubowicz; Desideri, Patricia Elaine; Câmara, Leonardo Meireles; Mandini, AlessiaThis dissertation presents a proof of the Jordan-Brouwer Theorem. This theorem states, roughly, that a hypersurface of the euclidean space R n divides the space in two sets: the “inside”and the “outside”. The proof relies on techniques from Differential Topology, e.g., transversality and winding numbers. The main reference for this work is [GP74].
- ItemSoluções de vórtice das equações de Ginzburg-Landau(Universidade Federal do Espírito Santo, 2014-12-01) Galkina, Olesya; Alves, Magno Branco; Macarini, Leonardo Magalhães; Câmara, Leonardo MeirelesIn this work we study a theorem of C.H. Taubes concerning vortex solution to the Ginzburg-Landau equations, which describe superconductivity. To prove the theorem we need to show the existence of a solution to a non-linear elliptic partial di erential equation of second order. To obtain the existence of solution we study a non-linear functional de ned on an appropriate Sobolev space. We also include two auxiliary chapters concerning complex line bundles and analytical preliminaries.
- ItemTeoria de calibre e geometria via conexões de Cartan-Ehresmann(Universidade Federal do Espírito Santo, 2012-12-07) Santos, Diego Henrique Carvalho dos; Alves, Magno Branco; Câmara, Leonardo Meireles; Macarini, Leonardo Magalhães; Bochi, Jairo da SilvaThe aim of this work is to present how works the correspondence between the gauge theory and connections in ber bundles. More precisely establishing a dictionary between gauge theory of the quantum mechanics of a charged particle under the in‡uence of an electromagnetic eld and the studies of connections in circle bundles and line bundles. Then, we analyzed two objects of studies in physics using the knowledge acquired in the study of the geometry of ber bundles. The Chern classes and the holonomy of a connection will provide a geometrical visualization of, respectively, magnetic monopoles and the Aharonov-Bohm e¤ect