Esta tese estuda a geometria de problemas variacionais através da linearização e projetivização das suas equações de Euler - Lagrange. O processo de linearização fornece a passagem das equações de Euler - Lagrange para as equações de Jacobi; a minimalidade (local) de extremais está determinada pelo conceito de ponto conjugado, que tem natureza projetiva. Propriedades de minimalidade local são transformadas em propriedades de auto-interseção de uma curva na variedade de Grassmann adequada. Desenvolvemos este processo em duas aplicações: 1) O estudo da minimalidade local de extremais de problemas variacionais de ordem superior. Neste caso, encontramos uma curva não degenerada de planos isotrópicos num espaço vetorial simplético, que, após prolongamento por derivadas, fornece uma curva degenerada de planos Lagrangeanos cujas auto-interseções determinam a minimalidade. 2) No caso mais clássico de problemas de ordem um, estudamos a versão linear - projetiva do problema inverso: dada uma equação diferencial de ordem dois, quando ela é a equação de Euler - Lagrange de um problema variacional? Veremos que as condições do problema inverso linear - projetivo fornecem informações sobre os possíveis Lagrangianos, por exemplo a assinatura. / In this work we study the geometry of high order calculus of variations through the linearization and projectivization of their Euler Lagrange equations. The linearization process provides the passage from the Euler Lagrange equations to the Jacobi equations; the (local) minimality properties of the extremal is determined by conjugate points, which is a projective concept. Minimaltiy properties of the extremals are transformed into self-intersection propertie of curves in the appropriate Grassmann manifold. We develop this process in two instances: 1) The study of minimality properties of extremals of higher-order variational problems. In this case, we find a non-degenerate curve of isotropic subspaces, that, after prolongation by derivatives, gives a degenerate curve of Lagrangian planes whose self-intersections determine minimality. 2) In the classical case of order one variational problems, we study a projective-linear version of the inverse problem: given a second order differential equation, when is it the Euler-Lagrange equation of a variational problem? We will see that the conditions given by the linear projective inverse problem provides information about the possible Lagrangians, for example, its signature.

We formulate a process of quantization of classical mechanics, from a symplecticperspective. The Dirac quantization axioms are stated, and a satisfactory prequantizationmap is constructed using a complex line bundle. Using polarization, it isdetermined which prequantum states and observables can be fully quantized. Themathematical concepts of symplectic geometry, fibre bundles, and distributions are exposedto the degree to which they occur in the quantization process. Quantizationsof a cotangent bundle and a sphere are described, using real and K¨ahler polarizations,respectively.

Orientadores: Luiz Antonio Barrera San Martin, Lino Anderson da Silva Grama / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-27T10:27:44Z (GMT). No. of bitstreams: 1 Alves_LeonardoSoriani_M.pdf: 542116 bytes, checksum: b4db821b86b39eb2b221b4f63a4c9829 (MD5) Previous issue date: 2015 / Resumo: Estudamos geometria complexa generalizada, que tem como casos particulares as geometrias complexa e simplética. Começamos com os seus fundamentos algébricos num espaço vetorial e transportamos essas noções para variedades. Estudamos o colchete de Courant na soma direta dos fibrados tangente e cotangente de uma variedade, que é essencial para definir a integrabilidade das estruturas complexas generalizadas. Verificamos que em nilvariedades de dimensão 6 sempre existe estrutura complexa generalizada invariante à esquerda, ainda que algumas delas não admitam estrutura complexa ou simplética. Estudamos duas noções de T-dualidade e suas relações com geometria complexa generalizada. Por fim recapitulamos a simetria do espelho para curvas elípticas e obtemos uma manifestação de simetria do espelho através de geometria complexa generalizada / Abstract: We study generalized complex geometry, which encompasses complex and symplectic geometry as particular cases. We begin with the algebraic basics on a vector space and then we transport these concepts to manifolds. We study the Courant bracket on the direct sum of tangent and cotangent bundles of a manifold, which is essential to define the integrability of the generalized complex structures. We check that on every $6$ dimensional nilmanifolds there is a left invariant generalized complex structure, even though some of them do not admit complex or symplectic structure. We study two notions of T-dualidade and its relations to generalized complex geometry. We recall mirror symmetry for elliptic curves and derive a manifestation of mirror symmetry from generalized complex geometry / Mestrado / Matematica / Mestre em Matemática

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21537号 / 理博第4444号 / 新制||理||1639(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 小野 薫, 教授 向井 茂, 教授 望月 拓郎 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

Using Sympelctic Field Theory as a computational tool, we compute Gromov-Witten theory of target curves using gluing formulas and quantum integrable systems. In the smooth case this leads to a relation of the results of Okounkov and Pandharipande with the quantum dispersionless KdV hierarchy, while in the orbifold case we prove triple mirror symmetry between GW theory of target P^1 orbifolds of positive Euler characteristic, singularity theory of a class of polynomials in three variables and extended affine Weyl groups of type ADE.

Symplectic geometry can be traced back to Lagrange and his work on celestial mechanics and has since then been a very active field in mathematics, partly because of the applications it offers but also because of the beauty of the objects it deals with. I this thesis we begin by the simplest fact of symplectic geometry. We give the definition of a symplectic space and of the symplectic group, Sp(n). A symplectic space is the data of an even-dimensional space and of a form which satisfies a number of properties. Having done this we give a definition of the Lagrangian Grassmannian Lag(n) which consists of all n-dimensional subspaces of the symplectic space on which the symplectic form vanishes. We carefully study the topology of these spaces and their universal coverings. It is of great interest to know how the elements of the Lagrangian Grassmannian intersect each other. A lot of efforts have therefore been made to construct intersection indices for elements of Lag(n). They have gone under many names but have had a sole purpose, namely to give us a way to determine how these elements intersect. We show how these elements are constructed and extend the definition to paths of elements of Lag(n) and Sp(n). We end this thesis by extending the definition of an index defined by Conley and Zehnder bu using the properties of the Leray index. Their index plays a significant role in the theory of periodic Hamiltonian orbit.

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: Marcos Benevenuto Jardim / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-22T22:29:57Z (GMT). No. of bitstreams: 1 Branco_LucasMagalhaesPereiraCastello_M.pdf: 1981391 bytes, checksum: 7ecd7674514f634b8bb527c0bcab1a06 (MD5) Previous issue date: 2013 / Resumo: Neste trabalho os aspectos básicos da teoria de calibre são abordados, incluindo as noções de conexão e curvatura em fibrados principais e vetoriais, considerações sobre o grupo de transformações de calibre e o espaço de moduli de soluções para a equação anti-auto-dual em dimensão quatro (o espaço de moduli de instantons). Posteriormente, mapas momento e redução são introduzidos. Primeiramente, no contexto clássico de geometria simplética e depois no contexto de geometria hyperkähler. Por fim, são apresentadas aplicações da teoria de mapas momento e redução em teoria de calibre. As equações ADHM são introduzidas e mostra se que estas podem ser dadas como o conjunto de zeros de um mapa momento hyperkähler. Além disso, considerações são feitas acerca da construção ADHM de instantons, que relaciona soluções dessas equações com as soluções da equação de anti-auto-dualidade. O espaço de moduli de conexões planas é também abordado. Neste caso, a curvatura é vista como um mapa momento e os cálculos podem ser generalizados para o espaço de moduli de conexões planas sobre variedades Kähler de dimensões mais altas e para o espaço de moduli de instantons sobre variedades hyperkähler de dimensão quatro / Abstract: In this work it is developed the basic concepts of gauge theory, including the notions of connections and curvature on principal bundles and vector bundles, considerations on the group of gauge transformations and the moduli space of anti-self-dual connections in dimension four (the instanton moduli space). After, moment maps and reduction are introduced. First in the classical context of symplectic geometry, then in hyperkähler geometry. At last, applications to the theory of moment maps and reduction in gauge theory are given. The ADHM equations are introduced and it is shown that solutions to these equations can be given by the zeros of a hyperkähler moment map. Furthermore, the ADHM construction, that relates the ADHM equations to instanton solutions, is discussed. The moduli space of flat connections over a Riemann surface is also treated. In this case, the curvature is seen as a moment map and the calculations can be generalized to flat connections over higher-dimensional Kähler manifolds and to the instanton moduli space over four dimensional hyperkähler manifolds / Mestrado / Matematica / Mestre em Matemática

On étudie les propriétés de rigidité symplectique des difféomorphismes hamiltoniens en dimension finie et en dimension infinie. En dimension finie, les outils principaux qu'on utilise sont les fonctions génératrices et les capacités symplectiques. En dimension infinie on regarde les flots des équations en dérivées partielles (EDPs) hamiltoniennes et, en particulier, les flots qui peuvent être approchés uniformément par des flots hamiltoniens de dimension finie.Dans la première partie de la thèse on étudie les sélecteurs d'action définies à partir des fonctions génératrices et on construit des invariants hamiltoniens pour les sous-ensembles de $R^{2m}times T^*T^k$. Cela nous permet de démontrer un théorème non-squeezing coisotrope pour les difféomorphismes hamiltoniens à support compact de $R^{2n}$. On montre à continuation que cette propriété apparaisse dans certains cas non compacts. Finalement, on explique comment ce résultat donne aussi l'information sur le problème de rigidité symplectique en dimension intermédiaire. Encore en dimension finie, on démontre qu'on peut utiliser le théorème du chameau symplectique pour produire des sous-ensembles invariants compacts dans des surfaces d'energie.Dans la deuxième partie on étudie les propriétés de rigidité symplectique des flots des EDPs hamiltoniennes. On se place dans le contexte introduit par Kuksin et on étudie une classe particulière de EDPs semi-linéaires qui peuvent être approchées par flots hamiltoniens de dimension finie. D'abord on donne une nouvelle construction de capacité symplectique en dimension infinie à partir des capacités de Viterbo. Puis on démontre l'analogue de la rigidité intermédiaire pour certaines EDPs hamiltoniennes. Cette classe inclue l'équation d'ondes en dimension 1 avec une non-linéarité bornée, comme par exemple l'équation de Sine-Gordon. Dans la dernière partie de la thèse on s'intéresse à un analogue de la conjecture d'Arnold pour l'équation de Schrödinger périodique avec une non linéarité de convolution. / We study symplectic rigidity properties in both finite and infinite dimension. In finite dimension, the main tools that we use are generating functions and symplectic capacities. In infinite dimension we study flows of Hamiltonian partial differential equations (PDEs) and, in particular, flows which can be uniformly approximated by finite dimensional Hamiltonian diffeomorphisms.In the first part of this thesis we study the action selectors defined from generating functions and we build Hamiltonian invariants for subsets of $R^{2m}times T^*T^k$. This allows us to prove a coisotropic non-squeezing theorem for compactly supported Hamiltonian diffeomorphisms of $R^{2n}$. We then extend this result to some non-compact settings. Finally we explain how this result can give information about the middle dimensional symplectic rigidity problem. Still in finite dimensions, we show that it is possible to use the symplectic camel theorem to create energy surfaces with compact invariant subsets.In the second part of the thesis we study symplectic rigidity properties of flows of Hamiltonian PDEs. We work in the context introduced by Kuksin and study a particular class of semi-linear Hamiltonian PDEs that can be approximated by finite dimensional Hamiltonian diffeomorphisms. We first give a new construction of an infinite dimensional capacity using Viterbo's capacities. The main result of this part is the proof of the analogue of the middle dimensional rigidity for certain types of Hamiltonian PDEs. These include nonlinear string equations with bounded nonlinearity such as the Sine-Gordon equation. In the final part of this thesis we study an analogue of Arnold's conjecture for the periodic Schrödinger equations with a convolution nonlinearity.

This thesis comprises the study of two moduli spaces of piecewise J-holomorphic curves. The main scheme is to consider a subdivision of the 2-sphere into a collection of small domains and to study collections of J-holomorphic maps into a symplectic manifold. These maps are coupled by Lagrangian boundary conditions. The work can be seen as finding a 2-dimensional analogue of the finite-dimensional path space approximation by piecewise geodesics on a Riemannian manifold (Q,g). For a nice class of target manifolds we consider tangent bundles of Riemannian manifolds and symplectizations of unit tangent bundles. Via polarization they provide a rich set of Lagrangians which can be used to define appropriate boundary value problems for the J-holomorphic pieces. The work focuses on existence theory as a pre-stage to global questions such as combinatorial refinement and the quality of the approximation. The first moduli space of lifted type is defined on a triangulation of the 2-sphere and consists of disks in the tangent bundle whose boundary projects onto geodesic triangles. The second moduli space of punctured type is defined on a circle packing domain and consists of boundary punctured disks in the symplectization of the unit tangent bundle. Their boundary components map into single fibers and at punctures the disks converge to geodesics. The coupling boundary conditions are chosen such that the piecewise problem always is Fredholm of index zero and both moduli spaces only depend on discrete data. For both spaces existence results are established for the J-holomorphic pieces which hold true on a small scale. Each proof employs a version of the implicit function theorem in a different setting. Here the argument for the moduli space of punctured type is more subtle. It rests on a connection to tropical geometry discovered by T. Ekholm for 1-jet spaces. The boundary punctured disks are constructed in the vicinity of explicit Morse flow trees which correspond to the limiting objects under degeneration of the boundary condition.