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41	Non-Isotopic Symplectic Surfaces in Products of Riemann Surfaces Hays, Christopher January 2006 (has links) <html> <head> <meta http-equiv="Content-Type" content="text/html;charset=iso-8859-1"> </head> Let Σ<em><sub>g</sub></em> be a closed Riemann surface of genus <em>g</em>. Generalizing Ivan Smith's construction, for each <em>g</em> ≥ 1 and <em>h</em> ≥ 0 we construct an infinite set of infinite families of homotopic but pairwise non-isotopic symplectic surfaces inside the product symplectic manifold Σ<em><sub>g</sub></em> ×Σ<em><sub>h</sub></em>. In particular, we achieve all positive genera from these families, providing first examples of infinite families of homotopic but pairwise non-isotopic symplectic surfaces of even genera inside Σ<em><sub>g</sub></em> ×Σ<em><sub>h</sub></em>. Mathematics Symplectic Manifolds Isotopy Problem Branched Covers
42	Lagrange-d'alembert integrators Cuell, Charles Lee 08 June 2007 (has links) A Lagrange--d'Alembert integrator is a geometric numerical method for finding numerical solutions to the Lagrange--d'Alembert equations for mechanical systems with nonholonomic constraints that are linear in the velocities. The integrator is developed from geometry and principles that are analogues of the continuous theory.<p>Using discrete analogues of the symplectic form and momentum map, the resulting methods are symplectic and momentum preserving whenever the continuous system is symplectic and momentum preserving. In addition, it is possible to, in principle, generate Lagrange--d'Alembert integrators of any method order. Geometric mechanics integrators symplectic nonholonomic holonomic
43	Symplectic Rational Blow-Up and Embeddings of Rational Homology Balls Khodorovskiy, Tatyana 21 June 2013 (has links) We define the symplectic rational blow-up operation, for a family of rational homology balls \(B_n\), which appeared in Fintushel and Stern's rational blow-down construction. We do this by exhibiting a symplectic structure on a rational homology ball \(B_n\) as a standard symplectic neighborhood of a certain 2-dimensional Lagrangian cell complex. We also study the obstructions to symplectically rationally blowing up a symplectic 4-manifold, i.e. the obstructions to symplectically embedding the rational homology balls \(B_n\) into a symplectic 4-manifold. First, we present a couple of results which illustrate the relative ease with which these rational homology balls can be smoothly embedded into a smooth 4-manifold. Second, we prove a theorem and give additional examples which suggest that in order to symplectically embed the rational homology balls \(B_n\), for high \(n\), a symplectic 4-manifold must at least have a high enough \(c^2_1\) as well. / Mathematics embeddings rational homology balls symplectic topology mathematics
44	Symplectic convexity theorems and applications to the structure theory of semisimple Lie groups Otto, Michael, January 2004 (has links) Thesis (Ph. D.)--Ohio State University, 2004. / Title from first page of PDF file. Document formatted into pages; contains v, 88 p. Includes bibliographical references (p. 87-88). Available online via OhioLINK's ETD Center
45	Symplectic analysis of flexible structures by finite elements / Mao, Shenggen. January 1996 (has links) Thesis (Ph. D.)--University of Hong Kong, 1998. / Includes bibliographical references (leaf 135-143).
46	Symplektická spin geometrie / Symplectic spin geometry Holíková, Marie January 2016 (has links) The symplectic Dirac and the symplectic twistor operators are sym- plectic analogues of classical Dirac and twistor operators appearing in spin- Riemannian geometry. Our work concerns basic aspects of these two ope- rators. Namely, we determine the solution space of the symplectic twistor operator on the symplectic vector space of dimension 2n. It turns out that the solution space is a symplectic counterpart of the orthogonal situation. Moreover, we demonstrate on the example of 2n-dimensional tori the effect of dependence of the solution spaces of the symplectic Dirac and the symplectic twistor operators on the choice of the metaplectic structure. We construct a symplectic generalization of classical theta functions for the symplectic Dirac operator as well. We study several basic aspects of the symplectic version of Clifford analysis associated to the symplectic Dirac operator. Focusing mostly on the symplectic vector space of the real dimension 2, this amounts to the study of first order symmetry operators of the symplectic Dirac ope- rator, symplectic Clifford-Fourier transform and the reproducing kernel for the symplectic Fischer product including the construction of bases for the symplectic monogenics of the symplectic Dirac operator in real dimension 2 and their extension to symplectic spaces...
47	Propriedades genéricas de sistemas hamiltonianos Lemes, Ricardo Chicalé [UNESP] 05 December 2013 (has links) (PDF) Made available in DSpace on 2014-12-02T11:16:50Z (GMT). No. of bitstreams: 0 Previous issue date: 2013-12-05Bitstream added on 2014-12-02T11:21:26Z : No. of bitstreams: 1 000793711.pdf: 1081771 bytes, checksum: 9ad4a08d3ec9d6accf66ef005a138f0a (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Nosso objetivo neste trabalho é demonstrar o Teorema da Densidade Geral que é um resultado análogo ao Teorema de Kupka-Smale para campos de vetores hamiltonianos. O Teorema da Densidade Geral afirma que o conjuntos dos campos hamiltonianos em uma variedade simplética M que possuem a propriedade H2-N é residual em Xk H(M). Começamos estabelecendo as teorias simpléticas linear e não-linear básicas e depois estudamos suas conexões com os sistemas hamiltonianos, provando os principais resultados da teoria e alguns resultados relacionados. Recebem destaque o estudo das curvas genéricas de matrizes simpléticas, a noção de funções geradoras de difeomorfismos simpléticos e sua aplicação na questão da estabilidade dos pontos fixos elípticos de campos hamiltonianos, a qual é respondida parcialmente através da Forma Normal de Birkhoff. Depois de estabelecer os resultados necessários, passamos a estudar a dinâmica hamiltoniana do ponto de vista das famílias a um parâmetro de difeomorfismos simpléticos. Provamos um resultado devido a Pugh e consideramos a questão da estabilidade estrutural de certas famílias de difeomorfismos simpléticos. Finalmente, provamos o Teorema da Densidade Geral usando a noção de pseudotransversalidade dada no Apêndice C. Este trabalho é baseado nas notas de aula Lectures on Hamiltonian Systems do professor R. Clark Robinson / In this work our goal is to prove the General Density Theorem which is an analogous result for hamiltonian vector fields of the Kupka-Smale Theorem. The General Density Theorem states that the set of hamiltonian vector fields on a symplectic manifold M that has the property H2-N is a residual subset of Xk H(M). We begin by stating the basic linear and nonlinear symplectic theory and then we study its connections with hamiltonian systems, proving some of the main theorems of the theory and other related results. Here we give special attention to topics like generic curves of symplectic matrices, generating functions of symplectic diffeomorphisms and their applications in the problem of the stability of eliptic fixed points of hamiltonian systems, which is partially solved using the Birkhoff Normal Form. After stating the necessary results, we begin to study some hamiltonian dynamics using one-parameter families of symplectic diffeomorphisms. We prove a result stated by Pugh and consider the problem of structural stability of a certain type of one-parameter family. Finally we prove the General Density Theorem using the notion of pseudotransversality given in Appendix C. This work is based on the lecture notes Lectures on Hamiltonian Systems of professor R. Clark Robinson Geometria Geometria simplética Sistemas hamiltonianos Symplectic geometry
48	Propriedades genéricas de sistemas hamiltonianos / Lemes, Ricardo Chicalé. January 2013 (has links) Orientador: Vanderlei Minori Horita / Banca: Thiago Aparecido Catalan / Banca: Claudio Aguinaldo Buzzi / Resumo: Nosso objetivo neste trabalho é demonstrar o Teorema da Densidade Geral que é um resultado análogo ao Teorema de Kupka-Smale para campos de vetores hamiltonianos. O Teorema da Densidade Geral afirma que o conjuntos dos campos hamiltonianos em uma variedade simplética M que possuem a propriedade H2-N é residual em Xk H(M). Começamos estabelecendo as teorias simpléticas linear e não-linear básicas e depois estudamos suas conexões com os sistemas hamiltonianos, provando os principais resultados da teoria e alguns resultados relacionados. Recebem destaque o estudo das curvas genéricas de matrizes simpléticas, a noção de funções geradoras de difeomorfismos simpléticos e sua aplicação na questão da estabilidade dos pontos fixos elípticos de campos hamiltonianos, a qual é respondida parcialmente através da Forma Normal de Birkhoff. Depois de estabelecer os resultados necessários, passamos a estudar a dinâmica hamiltoniana do ponto de vista das famílias a um parâmetro de difeomorfismos simpléticos. Provamos um resultado devido a Pugh e consideramos a questão da estabilidade estrutural de certas famílias de difeomorfismos simpléticos. Finalmente, provamos o Teorema da Densidade Geral usando a noção de pseudotransversalidade dada no Apêndice C. Este trabalho é baseado nas notas de aula Lectures on Hamiltonian Systems do professor R. Clark Robinson / Abstract: In this work our goal is to prove the General Density Theorem which is an analogous result for hamiltonian vector fields of the Kupka-Smale Theorem. The General Density Theorem states that the set of hamiltonian vector fields on a symplectic manifold M that has the property H2-N is a residual subset of Xk H(M). We begin by stating the basic linear and nonlinear symplectic theory and then we study its connections with hamiltonian systems, proving some of the main theorems of the theory and other related results. Here we give special attention to topics like generic curves of symplectic matrices, generating functions of symplectic diffeomorphisms and their applications in the problem of the stability of eliptic fixed points of hamiltonian systems, which is partially solved using the Birkhoff Normal Form. After stating the necessary results, we begin to study some hamiltonian dynamics using one-parameter families of symplectic diffeomorphisms. We prove a result stated by Pugh and consider the problem of structural stability of a certain type of one-parameter family. Finally we prove the General Density Theorem using the notion of pseudotransversality given in Appendix C. This work is based on the lecture notes Lectures on Hamiltonian Systems of professor R. Clark Robinson / Mestre Geometria. Geometria simplética. Sistemas hamiltonianos. Symplectic geometry
49	The contact property for magnetic flows on surfaces Benedetti, Gabriele January 2015 (has links) This work investigates the dynamics of magnetic flows on closed orientable Riemannian surfaces. These flows are determined by triples (M, g, σ), where M is the surface, g is the metric and σ is a 2-form on M . Such dynamical systems are described by the Hamiltonian equations of a function E on the tangent bundle TM endowed with a symplectic form ω_σ, where E is the kinetic energy. Our main goal is to prove existence results for a) periodic orbits, and b) Poincare sections for motions on a fixed energy level Σ_m := {E = m^2/2} ⊂ T M . We tackle this problem by studying the contact geometry of the level set Σ_m . This will allow us to a) count periodic orbits using algebraic invariants such as the Symplectic Cohomology SH of the sublevels ({E ≤ m^2/2}, ω_σ ); b) find Poincare sections starting from pseudo-holomorphic foliations, using the techniques developed by Hofer, Wysocki and Zehnder in 1998. In Chapter 3 we give a proof of the invariance of SH under deformation in an abstract setting, suitable for the applications. In Chapter 4 we present some new results on the energy values of contact type. First, we give explicit examples of exact magnetic systems on T^2 which are of contact type at the strict critical value. Then, we analyse the case of non-exact systems on M different from T^2 and prove that, for large m and for small m with symplectic σ, Σ_m is of contact type. Finally, we compute SH in all cases where Σ_m is convex. On the other hand, we are also interested in non-exact examples where the contact property fails. While for surfaces of genus at least two, there is always a level not of contact type for topological reasons, this is not true anymore for S^2 . In Chapter 5, after developing the theory of magnetic flows on surfaces of revolution, we exhibit the first example on S^2 of an energy level not of contact type. We also give a numerical algorithm to check the contact property when the level has positive magnetic curvature. In Chapter 7 we restrict the attention to low energy levels on S^2 with a symplectic σ and we show that these levels are of dynamically convex contact type. Hence, we prove that, in the non-degenerate case, there exists a Poincare section of disc-type and at least an elliptic periodic orbit. In the general case, we show that there are either 2 or infinitely many periodic orbits on Σ_m and that we can divide the periodic orbits in two distinguished classes, short and long, depending on their period. Then, we look at the case of surfaces of revolution, where we give a sufficient condition for the existence of infinitely many periodic orbits. Finally, we discuss a generalisation of dynamical convexity introduced recently by Abreu and Macarini, which applies also to surfaces with genus at least two. 516.3
50	Elliptic stable envelopes and 3d mirror symmetry Kononov, Iakov January 2021 (has links) In this thesis we discuss various classical problems in enumerative geometry. We are focused on ideas and methods which can be used explicitly for practical computations. Our approach is based on studying the limits of elliptic stable envelopes with shifted equivariant or Kahler variables from elliptic cohomology to K-theory. We prove that for a variety X we can obtain K-theoretic stable envelopes for the variety of the G-fixed points of X, where G is a cyclic group acting on X preserving the symplectic form. We formalize the notion of symplectic duality, also known as 3-dimensional mirror symmetry. We obtain a factorization theorem about the limit of elliptic stable envelopes to a wall, which generalizes the result of M. Aganagic and A. Okounkov. This approach allows us to extend the action of quantum groups, quantum Weyl groups, R-matrices etc., to actions on the K-theory of the symplectic dual variety. In the case of X = Hilb, our results imply the conjectures of E. Gorsky and A. Negut. We propose a new approach to K-theoretic quantum difference equations. Mathematics Physics Geometry, Enumerative Symplectic geometry

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