Abstract Not Provided. / Thesis / Master of Science (MSc)

The Ginibre gas is a Poisson point process dened on a space of loops related to the Feynman-Kac representation of the ideal Bose gas. Here we study thermodynamic limits of dierent ensembles via Martin-Dynkin boundary technique and show, in which way innitely long loops occur. This effect is the so-called Bose-Einstein condensation.

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 -∞.

We construct products on the homology of quotients by finite group actions of the free loop space ΛM of a compact manifold M. We compute some of the these products in the case M is as sphere. We show that there are nonnilpotent classes with respect to these products for spheres. The energy functional on ΛM associated to a Riemannian metric on M is invariant under the group actions we consider. We therefore retain information about geometrically distinct closed geodesics.

The Leray-Serre spectral sequence is a fundamental tool for studying singular homology of a fibration E->B with typical fiber F. It expresses H (E) in terms of H (B) and H (F). One of the classic examples of a fibration is given by the free loop space fibration, where the typical fiber is given by the based loop space . The first part of this thesis constructs the Leray-Serre spectral sequence in Morse homology on Hilbert manifolds under certain natural conditions, valid for instance for the free loop space fibration if the base is a closed manifold. We extend the approach of Hutchings which is restricted to closed manifolds. The spectral sequence might provide answers to questions involving closed geodesics, in particular to spectral invariants for the geodesic energy functional. Furthermore we discuss another example, the free loop space of a compact G-principal bundle, where G is a connected compact Lie group. Here we encounter an additional difficulty, namely the base manifold of the fiber bundle is infinite-dimensional. Furthermore, as H ( P) = HF (T P) and H ( Q) =HF (T Q), where HF denotes Floer homology for periodic orbits, the spectral sequence for P -> Q might provide a stepping stone towards a similar spectral sequence defined in purely Floer-theoretic terms, possibly even for more general symplectic quotients. Hutchings’ approach to the Leray-Serre spectral sequence in Morse homology couples a fiberwise negative gradient flow with a lifted negative gradient flow on the base. We study the Morse homology of a vector field that is not of gradient type. The central issue in the Hilbert manifold setting to be resolved is compactness of the involved moduli spaces. We overcome this difficulty by utilizing the special structure of the vector field. Compactness up to breaking of the corresponding moduli spaces is proved with the help of Gronwall-type estimates. Furthermore we point out and close gaps in the standard literature, see Section 1.4 for an overview. In the second part of this thesis we introduce a Lagrangian Floer homology on cotangent bundles with varying Lagrangian boundary condition. The corresponding complex allows us to obtain the Leray-Serre spectral sequence in Floer homology on the cotangent bundle of a closed manifold Q for Hamiltonians quadratic in the fiber directions. This corresponds to the free loop space fibration of a closed manifold of the first part. We expect applications to spectral invariants for the Hamiltonian action functional. The main idea is to study pairs of Morse trajectories on Q and Floer strips on T Q which are non-trivially coupled by moving Lagrangian boundary conditions. Again, compactness of the moduli spaces involved forms the central issue. A modification of the compactness proof of Abbondandolo-Schwarz along the lines of the Morse theory argument from the first part of the thesis can be utilized.

The Leray-Serre spectral sequence is a fundamental tool for studying singular homology of a fibration E->B with typical fiber F. It expresses H (E) in terms of H (B) and H (F). One of the classic examples of a fibration is given by the free loop space fibration, where the typical fiber is given by the based loop space . The first part of this thesis constructs the Leray-Serre spectral sequence in Morse homology on Hilbert manifolds under certain natural conditions, valid for instance for the free loop space fibration if the base is a closed manifold. We extend the approach of Hutchings which is restricted to closed manifolds. The spectral sequence might provide answers to questions involving closed geodesics, in particular to spectral invariants for the geodesic energy functional. Furthermore we discuss another example, the free loop space of a compact G-principal bundle, where G is a connected compact Lie group. Here we encounter an additional difficulty, namely the base manifold of the fiber bundle is infinite-dimensional. Furthermore, as H ( P) = HF (T P) and H ( Q) =HF (T Q), where HF denotes Floer homology for periodic orbits, the spectral sequence for P -> Q might provide a stepping stone towards a similar spectral sequence defined in purely Floer-theoretic terms, possibly even for more general symplectic quotients. Hutchings’ approach to the Leray-Serre spectral sequence in Morse homology couples a fiberwise negative gradient flow with a lifted negative gradient flow on the base. We study the Morse homology of a vector field that is not of gradient type. The central issue in the Hilbert manifold setting to be resolved is compactness of the involved moduli spaces. We overcome this difficulty by utilizing the special structure of the vector field. Compactness up to breaking of the corresponding moduli spaces is proved with the help of Gronwall-type estimates. Furthermore we point out and close gaps in the standard literature, see Section 1.4 for an overview. In the second part of this thesis we introduce a Lagrangian Floer homology on cotangent bundles with varying Lagrangian boundary condition. The corresponding complex allows us to obtain the Leray-Serre spectral sequence in Floer homology on the cotangent bundle of a closed manifold Q for Hamiltonians quadratic in the fiber directions. This corresponds to the free loop space fibration of a closed manifold of the first part. We expect applications to spectral invariants for the Hamiltonian action functional. The main idea is to study pairs of Morse trajectories on Q and Floer strips on T Q which are non-trivially coupled by moving Lagrangian boundary conditions. Again, compactness of the moduli spaces involved forms the central issue. A modification of the compactness proof of Abbondandolo-Schwarz along the lines of the Morse theory argument from the first part of the thesis can be utilized.

Made available in DSpace on 2015-03-13T17:10:48Z (GMT). No. of bitstreams: 1 SandraPA_DISSERT_.pdf: 6038687 bytes, checksum: 3a64b39d14e021c1abe47d90598f9430 (MD5) Previous issue date: 2012-03-22 / Our study refers to the state of Rio Grande do Norte against the deployment of oil activity in their territory. In this sense the aim of this work was to analyze the presence of the loop space of the oil production system linked to objects and actions on the Rio Grande do Norte territory. From the so-called "oil shock", an event that caused global developments in several countries, Petr?leo Brasileiro S/A (PETROBRAS) increased investments in drilling geological basins in Brazil. In the year 1973 was drilled in the sea area well which led to commercial production of oil and gas in Rio Grande do Norte. From that point on were added in some parts of the Potiguar territory, large systems of coupled objects to actions caused by several agents. In this context, geographic situations have been reorganized due to an unprecedented space circuit production accompanied by a new circle of cooperation. In the state happen all instances of the circuit: the production, distribution and consumption. In light of the theory of the geographical area seek to direct our thoughts to the operation of these bodies, and they are linked to material and immaterial flows multiscales. This perspective allows us to think the territory of Rio Grande do Norte entered into a new territorial division of labor characterized by specialization regional production. Oil activity was implemented in the territory of Rio Grande do Norte at a time marked by productive restructuring of various economic sectors. The oil sector has been acting increasingly linked the scientific and informational, with a view to increasing productivity. The presence of this circuit demanded the territory, specifically the Mossor?, an organizational structure that is different from the vast system nationally integrated private commercial corporations to small corporations, all of them relating directly or indirectly to PETROBRAS. The flows between companies whose headquarters are located in distant states and even countries have generated a continuous movement of goods, people, information and ideas, which is also causing new materialities in the territory / Nosso estudo se remete ao estado do Rio Grande do Norte frente ? implanta??o da atividade petrol?fera em seu territ?rio. Nesse sentido o objetivo geral do trabalho consistiu em analisar a presen?a do circuito espacial da produ??o petrol?fera vinculado ao sistema de objetos e de a??es presentes no territ?rio norte-rio-grandense. A partir do chamado choque do petr?leo , acontecimento mundial que causou desdobramentos em v?rios pa?ses, a Petr?leo Brasileiro S/A (PETROBRAS) aumentou os investimentos em perfura??es nas bacias geol?gicas brasileiras. No ano de 1973 foi perfurado em ?rea mar?tima o po?o que deu origem ? produ??o comercial de petr?leo e g?s no Rio Grande do Norte. Desse momento em diante foram acrescentados em algumas parcelas do territ?rio potiguar, grandes sistemas de objetos, juntamente, ?s a??es provocadas por agentes diversos. Nesse contexto, situa??es geogr?ficas foram reorganizadas em fun??o de um in?dito circuito espacial de produ??o acompanhado de um novo c?rculo de coopera??o. No estado acontecem todas as inst?ncias do circuito: a produ??o, a distribui??o e o consumo. ? luz da teoria do espa?o geogr?fico procuramos direcionar as nossas reflex?es ao funcionamento destas inst?ncias, estando elas ligadas a fluxos materiais e imateriais multiescalares. Essa perspectiva nos autoriza a pensar o territ?rio norte-rio-grandense inserido em uma nova divis?o territorial do trabalho marcada pela especializa??o regional produtiva. A atividade petrol?fera implantou-se no territ?rio norte-rio-grandense em um momento marcado pela reestrutura??o produtiva de diversos segmentos econ?micos. O setor petrol?fero passou a atuar cada vez mais atrelado ?s bases cient?ficas e informacionais, tendo em vista o aumento da produtividade. A presen?a desse circuito demandou ao territ?rio, mais especificamente a Mossor?, uma diversa estrutura organizacional que ocorre desde o vasto sistema nacionalmente integrado de corpora??es comerciais privadas at? as pequenas empresas, todas elas relacionando-se diretamente ou indiretamente com a PETROBRAS. Os fluxos entre empresas, cujas sedes localizam-se em estados e mesmo pa?ses distantes, t?m gerado um movimento cont?nuo de produtos, pessoas, informa??es e ideias, o que vem provocando tamb?m novas materialidades no territ?rio