In this thesis we use a diffeo-geometric framework based on manifolds hat are locally modeled on ``convenient'' vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold M, we construct a weak symplectic structure on each leaf I_w of a foliation of the space of compact oriented isotropic submanifolds in M equipped with top degree forms of total measure 1. These forms are called weightings and such manifolds are said to be weighted. We show that this symplectic structure on the particular leaves consisting of weighted Lagrangians is equivalent to a heuristic weak symplectic structure of Weinstein. When the weightings are positive, these symplectic spaces are symplectomorphic to reductions of a weak symplectic structure of Donaldson on the space of embeddings of a fixed compact oriented manifold into M. When M is compact, by generalizing a moment map of Weinstein we construct a symplectomorphism of each leaf I_w consisting of positive weighted isotropics onto a coadjoint orbit of the group of Hamiltonian symplectomorphisms of M equipped with the Kirillov-Kostant-Souriau symplectic structure. After defining notions of Poisson algebras and Poisson manifolds, we prove that each space I_w can also be identified with a symplectic leaf of a Poisson structure. Finally, we discuss a kinematic description of spaces of weighted submanifolds.

This thesis studies the pre-quantization of quasi-Hamiltonian group actions from a cohomological viewpoint. The compatibility of pre-quantization with symplectic reduction and the fusion product are established, and are used to understand the necessary and sufficient conditions for the pre-quantization of M(G,S), the moduli space of at flat G-bundles over a closed surface S. For a simply connected, compact, simple Lie group G, M(G,S) is known to be pre-quantizable at integer levels. For non-simply connected G, however, integrality of the level is not sufficient for pre-quantization, and this thesis determines the obstruction, namely a certain 3-dimensional cohomology class, that places further restrictions on the underlying level. The levels that admit a pre-quantization of the moduli space are determined explicitly for all non-simply connected, compact, simple Lie groups G. Partial results are obtained for the case of a surface S with marked points. Also, it is shown that via the bijective correspondence between quasi-Hamiltonian group actions and Hamiltonian loop group actions, the corresponding notions of prequantization coincide.

Given a Liouville manifold, symplectic cohomology is defined as the Hamiltonian Floer homology for the symplectic action functional on the free loop space. In this thesis, we propose two versions of periodic S^1-equivariant homology or S^1-equivariant Tate homology for the natural S^1-action on the free loop space. The first version is called periodic symplectic cohomology. We prove that there is a localization theorem or a fix point property for periodic symplectic cohomology. The second version is called the completed periodic symplectic cohomology which satisfies Goodwillie's excision isomorphism. Inspired by the work of Seidel and Solomon on the existence of dilations on symplectic cohomology, we formulate the notion of Liouville manifolds admitting higher dilations using Goodwillie's excision isomorphism on the completed periodic symplectic cohomology. As an application, we derive obstructions to existence of certain exact Lagrangian immersions in Liouville manifolds admitting higher dilations.

Bibliography: leaves 154-158.

Résumé de la thèse : ce travail porte sur la notion d'espace symétrique symplectique extrinsèque. Ces espaces sont des espaces symétriques symplectiques dont la structure est induite par le plongement dans variété symplectique ambiante munie d'une connexion. Par analogie à la théorie standard des espaces symétriques, nous démontrons un théorème d'équivalence entre les espaces symétriques symplectiques extrinsèques d'une variété qui est elle-même un espace symétrique symplectique. La définition d'un espace symétrique symplectique extrinsèque fait intervenir l'existence d'affinités globales de la variété ambiante, les ``symétries extrinsèques', qui induisent la structure symétrique de la sous-variété ; ceci mène à poser une question du type : quelles sont les variétés possédant ``beaucoup' de ces affinités~? Une question précise ainsi qu'une réponse sont fournies dans un contexte où la variété ambiante est seulement supposée munie d'une structure symplectique et d'une connexion symplectiques. Nous considérons également le cas où ces symétries commutent avec un champ $K$ d'endomorphismes symplectiques fixé de la variété, de carré $pmId$. Nous définissons une notion de courbure sectionnelle pour plans $K$-stables et montrons que les espaces à $K$-courbure sectionnelle constantes sont localement symétriques de type Ricci. Par suite nous étudions les espaces symétriques symplectiques extrinsèques dans un espace vectoriel symplectique. Nous montrons par exemple qu'un tel espace, s'ils est de dimension deux, est forcément intrinsèquement plat (c.-à-d. à courbure intrinsèque nulle), mais que son image n'est pas forcément un plan affin de l'espace vectoriel ambiant. Nous décrivons en fait explicitement tous les espaces symétriques symplectiques extrinsèques, dans un espace vectoriel, dont la courbure intrinsèque s'annule identiquement. Nous décrivons également une famille d'exemples d'espaces extrinsèques, dont nous montrons qu'elle fournit la totalité des espaces extrinsèques de codimension $2$, dans un espace vectoriel. Enfin, nous décrivons quelques exemples d'espaces symétriques symplectiques extrinsèques qui sont totalement géodésiques, dans un espace de type Ricci particulier.

The first part of this thesis investigates the Gromov width of maximal dimensional coadjoint orbits of compact simple Lie groups. An upper bound for the Gromov width is provided for all compact simple Lie groups but only for those coadjoint orbits that satisfy a certain technical assumption, whereas the lower bound is proved only for groups of type A, but without the technical restriction. The two bounds use very different techniques: the proof of the upper bound uses more analytical tools, while the proof of the lower bound is more geometric. The second part of the thesis is a short report on a joint project with my supervisor, which was concerned with the relationship between two different definitions of orbifolds: one using Lie groupoids and the other involving diffeologies. The results are summarized in Chapter 5 of this text.

Any genus $g$ surface, $\Sigma_{g,n},$ with $n$ boundary components may be given a trinion decomposition: a realization of the surface as a union of $2g-2+n$ trinions glued together along $3g-3+n$ of their boundary circles. Together with the flows of Goldman, Jeffrey and Weitsman use the trinion boundary circles in a decomposition of $\Sigma_{g,n}$ to obtain a Hamiltonian action of a compact torus $(S^1)^{3g-3+n'} $ on an open dense subset of the moduli space of certain gauge equivalence classes of flat $SU(2)-$connections on $\Sigma_{g,n}.$ Jeffrey and Weitsman also provide a complete description of the moment polytopes for these torus actions, and we make use of this description to study the cohomology of associated toric varieties. While we are able to make use of the work of Danilov to obtain the integral (rational) cohomology ring in the smooth (orbifold) case, we show that the aforementioned toric varieties almost always possess singularities worse than those of an orbifold. In these cases we use an algorithm of Bressler and Lunts to recover the intersection cohomology Betti numbers using the combinatorial information provided by the corresponding moment polytopes. The main contribution of this thesis is a computation of the intersection cohomology Betti numbers for the toric varieties associated to trinion decomposed surfaces $\Sigma_{2,0},\Sigma_{2,1},\Sigma_{3,0}, \Sigma_{3,1}, \Sigma_{4,0},$ and $\Sigma_{4,1}.$

Any genus $g$ surface, $\Sigma_{g,n},$ with $n$ boundary components may be given a trinion decomposition: a realization of the surface as a union of $2g-2+n$ trinions glued together along $3g-3+n$ of their boundary circles. Together with the flows of Goldman, Jeffrey and Weitsman use the trinion boundary circles in a decomposition of $\Sigma_{g,n}$ to obtain a Hamiltonian action of a compact torus $(S^1)^{3g-3+n'} $ on an open dense subset of the moduli space of certain gauge equivalence classes of flat $SU(2)-$connections on $\Sigma_{g,n}.$ Jeffrey and Weitsman also provide a complete description of the moment polytopes for these torus actions, and we make use of this description to study the cohomology of associated toric varieties. While we are able to make use of the work of Danilov to obtain the integral (rational) cohomology ring in the smooth (orbifold) case, we show that the aforementioned toric varieties almost always possess singularities worse than those of an orbifold. In these cases we use an algorithm of Bressler and Lunts to recover the intersection cohomology Betti numbers using the combinatorial information provided by the corresponding moment polytopes. The main contribution of this thesis is a computation of the intersection cohomology Betti numbers for the toric varieties associated to trinion decomposed surfaces $\Sigma_{2,0},\Sigma_{2,1},\Sigma_{3,0}, \Sigma_{3,1}, \Sigma_{4,0},$ and $\Sigma_{4,1}.$

The first part of this thesis investigates the Gromov width of maximal dimensional coadjoint orbits of compact simple Lie groups. An upper bound for the Gromov width is provided for all compact simple Lie groups but only for those coadjoint orbits that satisfy a certain technical assumption, whereas the lower bound is proved only for groups of type A, but without the technical restriction. The two bounds use very different techniques: the proof of the upper bound uses more analytical tools, while the proof of the lower bound is more geometric. The second part of the thesis is a short report on a joint project with my supervisor, which was concerned with the relationship between two different definitions of orbifolds: one using Lie groupoids and the other involving diffeologies. The results are summarized in Chapter 5 of this text.

In this thesis, we develop two methods for constructing Lie groupoids. The first method is a blow-up construction, corresponding to the elementary modification of a Lie algebroid along a subalgebroid over some closed hypersurface. This construction may be specialized to the Poisson groupoids and Lie bialgebroids. We then apply this method to three cases. The first is the adjoint Lie groupoid integrating the Lie algebroid of vector fields tangent to a collection of normal crossing hypersurfaces. The second is the adjoint symplectic groupoid of a log symplectic manifold. The third is the adjoint Lie groupoid integrating the tangent algebroid of a Riemann surface twisted by a divisor. The second method is a gluing construction, whereby Lie groupoids defined on the open sets of an appropriate cover may be combined to obtain global integrations. This allows us to construct and classify the Lie groupoids integrating the given Lie algebroid. We apply this method to the aforementioned cases, albeit with small differences, and characterize the category of integrations in each case.