This thesis is split up into two parts each revolving around Floer homology and quantum cohomology of closed monotone symplectic manifolds. In the first part we consider symplectic manifolds obtained by symplectic reduction. Our main result is that a quantum version of an abelianization formula of Martin holds, which relates the quantum cohomologies of symplectic quotients by a group and by its maximal torus. Also we show a quantum version of the Leray-Hirsch theorem for Floer homology of Lagrangian intersections in the quotient. The second part is devoted to Floer homology of a pair of monotone Lagrangian submanifolds in clean intersection. Under these assumptions the symplectic action functional is degenerated. Nevertheless Frauenfelder defines a version of Floer homology, which is in a certain sense an infinite dimensional analogon of Morse-Bott homology. Via natural filtrations on the chain level we were able to define two spectral sequences which serve as a tool to compute Floer homology. We show how these are used to obtain new intersection results for simply connected Lagrangians in the product of two complex projective spaces. The link between both parts is that in the background the same technical methods are applied; namely the theory of holomorphic strips with boundary on Lagrangians in clean intersection. Since all our constructions rely heavily on these methods we also give a detailed account of this theory although in principle many results are not new or require only straight forward generalizations.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

In 1970, Raoul Bott published The Periodicity Theorem for the Classical Groups and Some of Its Applications, in which he uses this famous result as a guideline to present some important areas and tools of Algebraic Topology. This dissertation aims to use the path Bott presented in his article as a guideline to address certain topics on Algebraic Topology. We start this incursion developing important tools used in Homotopy Theory such as spectral sequences and Eilenberg-MacLane spaces, exploring how they can be combined to aid in computation of homotopy groups. We then study important results of Morse Theory, a tool which was in the centre of Botts proof of the Periodicity Theorem. We also develop two extensions: Morse-Bott Theory, and the applications of such results to the loopspace of a manifold. We end by giving an introduction to generalised cohomology theories and K-Theory. / Em 1970, Raoul Bott publicou o artigo The Periodicity Theorem for the Classical Groups and Some of Its Applications no qual usava esse famoso resultado como um guia para apresentar importantes áreas e ferramentas da Topologia Algébrica. O presente trabalho usa o mesmo caminho traçado por Bott em seu artigo como roteiro para explorar tópicos importantes da Topologia Algébrica. Começamos esta incursão desenvolvendo ferramentas importantes da Teoria de Homotopia como sequências espectrais e espaços de Eilenberg-MacLane, explorando como estes podem ser combinados para auxiliar em cálculos de grupos de homotopia. Passamos então a estudar resultados importantes de Teoria de Morse, uma ferramenta que estava no centro da demonstração de Bott do Teorema da Periodicidade. Desenvolvemos ainda, duas extensões: Teoria de Morse-Bott e aplicações destes resultados ao espaço de laços de uma variedade. Terminamos com uma introdução a teorias de cohomologia generalizadas e K-Teoria.

Embora o fenômeno da fotodissociação (dissociação de moléculas devido à interação com um campo externo dependente do tempo) venha já há muito tempo atraindo a atenção dos pesquisadores, esta ainda longe de ser completamente compreendido. Este problema é de difícil tratamento teórico por se tratar não apenas de um problema quântico de vários corpos, mas que apresenta também dependência temporal. Este trabalho tem como alvo de estudo a evolução temporal da função de onda de uma molécula diatômica sujeita a um campo externo dependente do tempo, servindo-se para isso do potencial de Morse. Este potencial unidimensional descreve razoavelmente bem os níveis vibracionais de moléculas diatômicas e pode ter seus parâmetros ajustados de forma a representar varias moléculas. O estado fundamental do oscilador é perturbado pelo campo e a função de onda é propagada através de diferentes métodos, que são comparados entre si. É interessante notar que as partes real e imaginária da função de onda começam a oscilar, mostrando que o pacote esta ganhando energia cinética, enquanto que a densidade de probabilidade permanece inicialmente quase inalterada. E discutido um efeito similar ao Stark, devido ao fato da variação temporal do campo externo ter sido assumida proporcional a um co-seno. O princípio do processo dissociativo, com as funções de onda se estendendo para maiores valores da coordenada espacial, é observado com o prosseguimento da propagação a tempos maiores / Although the phenomenon of photodissociation (dissociation of molecules due to the interaction with an external time-dependent field) has been for a long time attracting scientists\' attention, it is yet far from being completely understood. Theoretical approach to this process is difficult not only because it is a many-body quantum problem, but also due to the time dependence of the external field. The main goal of this work is to study the time evolution of a diatomic molecule in the presence of an external time-dependent field, using the Morse potential. This unidimensional potential describes reasonably well the vibrational levels of diatomic molecules and may have its parameters adjusted in order to represent several molecules. The ground state is perturbed by the field and the wavefunction propagated through a few methods, which are compared among them. It is interesting to notice that the real and imaginary parts of the wavefunction start to oscillate, showing that the packet is gaining kinetic energy, while the probability density initially remains practically still. An effect similar to the Stark one, due to the fact that the time oscillation of the external field has been assumed proportional to a co-sine, is discussed. The beginning of the dissociation process, with the wavefunctions extending to greater values of the spatial coordinate, is observed as the propagation is continued through greater times

Orientador: Francesco Mercuri / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-07-23T06:16:08Z (GMT). No. of bitstreams: 1 Souza_FaustoMarcalde_D.pdf: 3773786 bytes, checksum: b0c8a9073cc06177fbc3a81fee13c235 (MD5) Previous issue date: 1997 / Resumo: Neste trabalho desenvolvemos a Teoria de Morse para funções de baixa diferenciabilidade (de classe C1), com segunda derivada nos pontos críticos isolados e, possivelmente degenerados. Aplicamos os resultados obtidos ao problema de geodésicas fechadas de uma métrica de Finsler, os quais permitem usar os argumentos originais do Teorema de Gromoll-Meyer para demonstrar a existência de infinitas geodésicas fechadas não constantes, geometricamente distintas, em variedades Finslerianas compactas, cuja cohomologia (real) não seja uma álgebra gerada por um só elemento. / Abstract: Not informed. / Doutorado / Geometria e Topologia / Doutor em Matemática

Orientador: Elso Drigo Filho / Banca: Sergio Henrique Monari Soares / Banca: José Roberto Ruggiero / Resumo: Neste trabalho são apresentadas as solu cões da equação de movimento clássica para uma partícula sujeita ao potencial de Morse Generalizado. Nesse potencial, utilizando parâmetros adequados recupera-se o potencial de Morse na sua forma conhecida. Essa mesma resolução também em foi feita para potenciais sim etricos de Morse. Este estudo foi realizado pensando no modelo mecânico do Acido Desoxirribonucléico (DNA), mais especi camente, na simulação das pontes de hidrogênio entre os pares de base. As equações de movimento s~ao tratadas como clássicas, pois algumas propriedades do DNA permitem a essa mol ecula um tratamento deste tipo. Manipulando-se classicamente, a resolução da equação de movimento e bastante simples e, através de t écnicas de integração, encontra-se uma solução exata para o potencial em questão. Poucos trabalhos são vistos na literatura com o potencial de Morse num estudo clássico, o que foi mais um incentivo para o desenvolvimento desse resultado. / Abstract: This study presents the solutions of classical motion equation for a particle subject to a Generalized Morse Potential. With this potential are using appropriate parameters one can recover the Morse potential as it is usually known. The same work had been done for symmetric Morse potentials. The main motivation for this work are the mechanical models for Deoxiribonucleic Acid (DNA), more speci cally, the H-bonds simulation between base pairs. The motion equations are discussed in a classical level, because some DNA properties allow these treatment. In Classical way, the resolution of motion equation is very simpler. Applying integration tecniques meets an exact solution when the Morse Potential is employed in this equation. There are few Classical Morse Potential studies founded in the literature. Due to this fact, it was an impulse to develop this result. / Mestre

Univ., Diss.--Stockholm, 2005. / Includes bibliographical references (p. [361] - 369) and index.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1959. / Typescript. Abstracted in Dissertation abstracts, v. 19 (1959) no. 11, p. 3055-3056. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 390-394).

Typescript. Includes vita and abstract. Includes bibliographical references (leaves 163-164) Also available online in Scholars' Bank; and in ProQuest, free to University of Oregon users.

x, 164 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / We study the topology of the space of metrics of positive scalar curvature on a compact manifold. The main tool we use for constructing such metrics is the surgery technique of Gromov and Lawson. We extend this technique to construct families of positive scalar curvature cobordisms and concordances which are parametrised by Morse functions and later, by generalised Morse functions. We then use these results to study concordances of positive scalar curvature metrics on simply connected manifolds of dimension at least five. In particular, we describe a subspace of the space of positive scalar curvature concordances, parametrised by generalised Morse functions. We call such concordances Gromov-Lawson concordances. One of the main results is that positive scalar curvature metrics which are Gromov-Lawson concordant are in fact isotopic. This work relies heavily on contemporary Riemannian geometry as well as on differential topology, in particular pseudo-isotopy theory. We make substantial use of the work of Eliashberg and Mishachev on wrinkled maps and of results by Hatcher and Igusa on the space of generalised Morse functions. / Committee in charge: Boris Botvinnik, Chairperson, Mathematics; James Isenberg, Member, Mathematics; Hal Sadofsky, Member, Mathematics; Christopher Phillips, Member, Mathematics; Michael Kellman, Outside Member, Chemistry