Thesis (Ph.D.)--Mathematics, Georgia Institute of Technology, 2008. / Committee Chair: Prof. Jean Bellissard; Committee Member: Prof. Claude Schochet; Committee Member: Prof. Michael Loss; Committee Member: Prof. Stavros Garoufalidis; Committee Member: Prof. Thang Le.

L'homologie de contact est un invariant homologique pour variétés de contact dont la définition est basée sur l'utilisation de courbes holomorphes. Ce travail de thèse concerne l'étude de cet invariant dans le cas des produits de contact. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Orientador: Ketty Abaroa de Rezende / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-16T15:34:50Z (GMT). No. of bitstreams: 1 Lima_DahisyValadaodeSouza_M.pdf: 1595993 bytes, checksum: 49a95ad790c477c7d049695a123d9acd (MD5) Previous issue date: 2010 / Resumo: Dada uma variedade suave e fechada M, o complexo de Morse-Witten associado a uma função de Morse f : M ? R e a uma métrica Riemanniana g em M consiste de grupos de cadeia gerados pelos pontos críticos de f e um operador bordo que conta linhas de fluxos isoladas do fluxo gradiente negativo. A homologia do complexo de Morse-Witten é isomorfa à homologia singular de M. Dado um conjunto invariante isolado S, uma matriz de conexão para uma decomposição de Morse de S é uma matriz de homomorfismos entre os índices homológicos de Conley dos conjuntos de Morse. A matriz de conexão é capaz de prover informações dinâmicas sobre um fluxo. De fato, esta matriz pode detectar a existência de órbitas conectantes entre os conjuntos de Morse de S. O complexo de Morse-Witten está relacionado à teoria de matrizes de conexão. Mais precisamente, o operador bordo do complexo de Morse-Witten é um caso especial de matriz de conexão / Abstract: Given a smooth closed manifold M, the Morse-Witten complex associated to a Morse function f : M ? R and a Riemannian metric g on M consists of chain groups generated by the critical points of f and a boundary operator counting isolated flow lines of the negative gradient flow. The homology of the Morse-Witten complex is isomorphic to the singular homology of M. Give a isolated invariant set S, a connection matrix for a Morse decomposition of S is a matrix of homomorphism between the Conley homology indices of Morse sets. The connection matrix is capable of providing dynamical information of a flow. In fact, this matrix can detect the existence of connecting orbits among Morse sets of S: The Morse-Witten complex is related to connection matrices theory. More precisely, the boundary operator of the Morse-Witten complex is a special case of connection matrix / Mestrado / Matematica / Mestre em Matemática

Orientador: Ketty Abaroa de Rezende / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-23T01:07:46Z (GMT). No. of bitstreams: 1 Villapouca_MarianaGesualdi_D.pdf: 1687322 bytes, checksum: 9557d400e3eadbf12a6a305e0219b2cb (MD5) Previous issue date: 2013 / Resumo: Este trabalho tem como foco o estudo do índice de Conley discreto e do par de matrizes de conexão para difeomorfismos fitted Smale em variedades compactas. Foi estabelecido um teorema que apresenta o cálculo do índice de Conley de conjuntos básicos zero - dimensionais usando a informação dinâmica contida nas matrizes de estrutura associadas. A classificação do índice de Conley homológico reduzido de conjuntos básicos zero - dimensionais, utilizando a sua forma de Jordan real foi apresentada. Estabelecemos uma caracterização de pares de matrizes de conexão para decomposições de Morse em conjuntos básicos zero - dimensionais para uma classe de difeomorfismos fitted Smale / Abstract: Our focus in this thesis was on the further development of the discrete Conley index theory with the aim of addressing questions on the pair of connection matrices for fitted Smale diffeomorphisms on compact manifolds. A theorem was established where the computation of the discrete Conley index for zero dimensional basic sets was given with respect to the dynamical information contained in the associated structure matrices. A classification of the reduced homology Conley index of a zero dimensional basic set in terms of its Jordan real form is presented. A characterization of a pair of connection matrices for a Morse decomposition of zero dimensional basic sets of a class of fitted Smale diffeomorphisms is established / Doutorado / Matematica / Doutora em Matemática

Orientador: Luiz Antonio Barrera San Martin / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-21T00:23:49Z (GMT). No. of bitstreams: 1 Rabelo_Lonardo_D.pdf: 1560307 bytes, checksum: ad1323aadd78f3943acaf2c6e13b96d0 (MD5) Previous issue date: 2012 / Resumo: Esta tese apresenta uma abordagem para o estudo da topologia das variedades flag reais. A homologia é obtida pela determinação do operador fronteira da homologia celular. Isto se dá a partir de uma parametrização explícita das células de Shubert que fornecem a estrutura celular destas variedades. Para o anel de cohomologia de uma variedade flag maximal, encontram-se os seus geradores como classes de Stiefel-Whitney de um fibrado de linha sobre a variedade flag / Abstract: This thesis presents an approach for the study of topology of real flag manifolds. The homology is obtained by the determination of the boundary operator for the cellular homology. This follows from an explicit parametrization of the Schubert cells which gives a cellular structure for these manifolds. For the cohomology ring of a maximal flag manifold, its generators are found as Stiefel-Whitney classes of a line fiber bundle over the flag manifold / Doutorado / Matematica / Doutor em Matemática

We study the K-theory and cohomology of spaces of aperiodic and repetitive tilings with finite local complexity. Given such a tiling, we build a spectral sequence converging to its K-theory and define a new cohomology (PV cohomology) that appears naturally in the second page of this spectral sequence. This spectral sequence can be seen as a generalization of the Leray-Serre spectral sequence and the PV cohomology generalizes the cohomology of the base space of a Serre fibration with local coefficients in the K-theory of its fiber. We prove that the PV cohomology of such a tiling is isomorphic to the Cech cohomology of its hull. We give examples of explicit calculations of PV cohomology for a class of 1-dimensional tilings (obtained by cut-and-projection of a 2-dimensional lattice). We also study the groupoid of the transversal of the hull of such tilings and show that they can be recovered: 1) from inverse limit of simpler groupoids (which are quotients of free categories generated by finite graphs), and 2) from an inverse semi group that arises from PV cohomology. The underslying Delone set of punctures of such tilings modelizes the atomics positions in an aperiodic solid at zero temperature. We also present a study of (classical and harmonic) vibrational waves of low energy on such solids (acoustic phonons). We establish that the energy functional (the "matrix of spring constants" which describes the vibrations of the atoms around their equilibrium positions) behaves like a Laplacian at low energy.

Soit M une variété lisse fermée et considérons sont fibré cotangent T*M muni de la structure symplectique usuelle induite par la forme de Liouville. Une hypersurface S de T*M$ est dite étoilée fibre par fibre si pour tout point q de M, l'intersection Sq de S avec la fibre au dessus de q est le bord d'un domaine étoilé par rapport à l'origine 0q de la fibre T*qM. Un flot est naturellement associé à S, il s'agit de l'unique flot généré par le champ de Reeb le long de S, le flot de Reeb. <p><p>L'existence d'une orbite orbite fermée du flot de Reeb sur S fut annoncée par Weinstein dans sa conjecture en 1978. Indépendamment, Weinstein et Rabinowitz ont montré l'existence d'une orbite fermée sur les hypersurfaces de type étoilées dans l'espace réel de dimension 2n. Sous les hypothèses précédentes, l'existence d'une orbite fermée fut démontrée par Hofer et Viterbo. Dans le cas particulier du flot géodésique, l'existence de plusieurs orbites fermées fut notamment étudiée par Gromov, Paternain et Paternain-Petean. Dans cette thèse, ces résultats sont généralisés. <p><p>Les résultats principaux de cette thèse montrent que la structure topologique de la variété M implique, pour toute hypersurface étoilée fibre par fibre, l'existence de beaucoup d'orbites fermées du flot de Reeb. Plus précisément, une borne inférieure de la croissance du nombre d'orbites fermées du flot de Reeb en fonction de leur période est mise en évidence. /<p><p>Let M be a smooth closed manifold and denote by T*M the cotangent bundle over M endowed with its usual symplectic structure induced by the Liouville form. A hypersurface S of T*M is said to be fiberwise starshaped if for each point q in M the intersection Sq of S with the fiber at q bounds a domain starshaped with respect to the origin 0q in T*qM. There is a flow naturally associated to S, generated by the unique Reeb vector field R along S ,the Reeb flow. <p><p>The existence of one closed orbit was conjectured by Weinstein in 1978 in a more general setting. Independently, Weinstein and Rabinowitz established the existence of a closed orbit on star-like hypersurfaces in the 2n-dimensional real space. In our setting the Weinstein conjecture without the assumption was proved in 1988 by Hofer and Viterbo. The existence of many closed orbits has already been well studied in the special case of the geodesic flow, for example by Gromov, Paternain and Paternain-Petean. In this thesis we will generalize their results.<p><p>The main result of this thesis is to prove that the topological structure of $M$ forces, for all fiberwise starshaped hypersurfaces S, the existence of many closed orbits of the Reeb flow on S. More precisely, we shall give a lower bound of the growth rate of the number of closed Reeb-orbits in terms of their periods. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

In this thesis we present a slight generalisation of the Pearl complex or relative quantum homology to some non monotone Lagrangian submanifolds. First we develop the theory for the so called almost monotone Lagrangian submanifolds, We apply it to uniruling problems as well as estimates for the relative Gromov width. In the second part we develop the theory for toric fiber in toric Fano manifolds, recovering previous computaional results of Floer homology . / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Complex patterns arise in many extended nonlinear nonequilibrium systems in physics, chemistry and biology. Information extraction from these complex patterns is a challenge and has been a main subject of research for many years. We study patterns in Rayleigh-Benard convection (RBC) acquired from our laboratory experiments to develop new characterization techniques for complex spatio-temporal patterns. Computational homology, a new topological characterization technique, is applied to the experimental data to investigate dynamics by quantifying convective patterns in a unique way. The homology analysis is used to detect symmetry breakings between hot and cold flows as a function of thermal driving in experiments, where other conventional techniques, e.g., curvature and wave-number distribution, failed to reveal this asymmetry. Furthermore, quantitative information is acquired from the outputs of homology to identify different spatio-temporal states. We use this information to obtain a reduced dynamical description of spatio-temporal chaos to investigate extensivity and physical boundary effects in RBC. The results from homological analysis are also compared to other dimensionality reduction techniques such as Karhunen-Loeve decomposition and Fourier analysis.

Foot-and-mouth disease (FMD) is a highly contagious and economically important disease that affects even-toed hoofed mammals. The FMD virus (FMDV) is the causative agent of FMD, of which there are seven clinically indistinguishable serotypes. Three serotypes, namely, South African Territories (SAT)1, SAT2 and SAT3 are endemic to southern Africa and are the most antigenically diverse among the FMDV serotypes. A negative consequence of this antigenic variation is that infection or vaccination with one virus may not provide immune protection from other strains or it may only confer partial protection. The identification of B-cell epitopes is therefore key to rationally designing cross-reactive vaccines that recognize the immunologically distinct serotypes present within the population. Computational epitope prediction methods that exploit the inherent physicochemical properties of epitopes in their algorithms have been proposed as a cost and time-effective alternative to the classical experimental methods. The aim of this project is to employ in silico epitope prediction programmes to predict B-cell epitopes on the capsids of the SAT serotypes. Sequence data for 18 immunologically distinct SAT1, SAT2 and SAT3 strains from across southern Africa were collated. Since, only one SAT1 virus has had its structure elucidated by X-ray crystallography (PDB ID: 2WZR), homology models of the 18 virus capsids were built computationally using Modeller v9.12. They were then subjected to energy minimizations using the AMBER force field. The quality of the models was evaluated and validated stereochemically and energetically using the PROMOTIF and ANOLEA servers respectively. The homology models were subsequently used as input to two different epitope prediction servers, namely Discotope1.0 and Ellipro. Only those epitopes predicted by both programmes were defined as epitopes. Both previously characterised and novel epitopes were predicted on the SAT strains. Some of the novel epitopes are located on the same loops as experimentally derived epitopes, while others are located on a putative novel antigenic site, which is located close to the five-fold axis of symmetry. A consensus set of 11 epitopes that are common on at least 15 out of 18 SAT strains was collated. In future work, the epitopes predicted in this study will be experimentally validated using mutagenesis studies. Those found to be true epitopes may be used in the rational design of broadly reactive SAT vaccines / Life and Consumer Sciences / M. Sc. (Life Sciences)