Academic literature on the topic 'Symplectic and Poisson geometry'

This thesis explores aspects of generalized geometry, a geometric framework introduced by Hitchin and Gualtieri in the early 2000s. In the first part, we introduce a new class of submanifolds in stable generalized complex manifolds, so-called Lagrangian branes with boundary. We establish a correspondence between stable generalized complex geometry and log symplectic geometry, which allows us to prove results on local neighbourhoods and small deformations of this new type of submanifold. We further investigate Lefschetz thimbles in stable generalized complex Lefschetz fibrations and show that Lagrangian branes with boundary arise in this context. Stable generalized complex geometry provides the simplest examples of generalized complex manifolds which are neither complex nor symplectic, but it is sufficiently similar to symplectic geometry for a multitude of symplectic results to generalize. Our results on Lefschetz thimbles in stable generalized complex geometry indicate that Lagrangian branes with boundary are part of a potential generalisation of the Wrapped Fukaya category to stable generalized complex manifolds. The work presented in this thesis should be seen as a first step towards the extension of Floer theory techniques to stable generalized complex geometry, which we hope to develop in future work. The second part of this thesis studies Dorfman brackets, a generalisation of the Courant- Dorfman bracket, within the framework of double vector bundles. We prove that every Dorfman bracket can be viewed as a restriction of the Courant-Dorfman bracket on the standard VB-Courant algebroid, which is in this sense universal. Dorfman brackets have previously not been considered in this context, but the results presented here are reminiscent of similar results on Lie and Dull algebroids: All three structures seem to fit into a more general duality between subspaces of sections of the standard VB-Courant algebroid and brackets on vector bundles of the form T M ⊕ E ∗ , E → M a vector bundle. We establish a correspondence between certain properties of the brackets on one, and the subspaces on the other side.

Orientador: Paulo Regis Caron Ruffino<br>Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica<br>Made available in DSpace on 2018-08-13T23:01:19Z (GMT). No. of bitstreams: 1 Costa_PauloHenriquePereirada_M.pdf: 875708 bytes, checksum: 8862a1813f1bb85b5d0269462a80501e (MD5) Previous issue date: 2009<br>Resumo: O objetivo desse trabalho é estudar as equações de Hamilton no contexto estocástico. Sendo necessário para tal um pouco de conhecimento a cerca dos seguintes assuntos: cálculo estocástico, geometria de segunda ordem, estruturas simpléticas e de Poisson. Abordamos importantes resultados, dentre eles o teorema de Darboux (coordenadas locais) em variedades simpléticas, teorema de Lie-Weinstein que de certa forma generaliza o teorema de Darboux em variedades de Poisson. Veremos que apesar de o ambiente natural para se estudar sistemas hamiltonianos ser variedades simpléticas, no caso estocástico esses sistemas se adaptam bem em variedades de Poisson. Além disso, para atingir a nossa meta, estudaremos equações diferenciais estocásticas em variedades de dimensão finita usando o operador de Stratonovich<br>Abstract: This dissertation deals with transfering Hamilton's equations in stochastic context. This requires some knowledge about the following: stochastic calculus, second order geometry and Poisson and simplectic structures. Important results that will be discussed in this theory are Darboux's theorem (local coordinates) for simplectic manifolds, and Lie-Weintein's theorem that is in a certain way of Darboux's theorem on Poisson manifolds. We will see that although the natural environment for studying hamiltonian systems is symplectic manifolds, if we have a Poisson structure we will still be able to study them. Moreover, to achieve our goal, we will study stochastic differential equations on finite dimensional manifolds using the Stratonovich operator<br>Mestrado<br>Geometria Estocastica<br>Mestre em Matemática

In this thesis we develop a mathematically rigorous framework of the so-called ''classical limit'' of quantum systems and their semi-classical properties. Our methods are based on the theory of strict, also called C*- algebraic deformation quantization. Since this C*-algebraic approach encapsulates both quantum as classical theory in one single framework, it provides, in particular, an excellent setting for studying natural emergent phenomena like spontaneous symmetry breaking (SSB) and phase transitions typically showing up in the classical limit of quantum theories. To this end, several techniques from functional analysis and operator algebras have been exploited and specialised to the context of Schrödinger operators and quantum spin systems. Their semi-classical properties including the possible occurrence of SSB have been investigated and illustrated with various physical models. Furthermore, it has been shown that the application of perturbation theory sheds new light on symmetry breaking in Nature, i.e. in real, hence finite materials. A large number of physically relevant results have been obtained and presented by means of diverse research papers.