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1	Comparison of Second Order Conformal Symplectic Schemes with Linear Stability Analysis Floyd, Dwayne 01 January 2014 (has links) Numerical methods for solving linearly damped Hamiltonian ordinary differential equations are analyzed and compared. The methods are constructed from the well-known Störmer-Verlet and implicit midpoint methods. The structure preservation properties of each method are shown analytically and numerically. Each method is shown to preserve a symplectic form up to a constant and are therefore conformal symplectic integrators, with each method shown to accurately preserve the rate of momentum dissipation. An analytical linear stability analysis is completed for each method, establishing thresholds between the value of the damping coefficient and the step-size that ensure stability. The methods are all second order and the preservation of the rate of energy dissipation is compared to that of a third order Runge-Kutta method that does not preserve conformal properties. Numerical experiments will include the damped harmonic oscillator and the damped nonlinear pendulum. Mathematics
2	Cohomology of the moduli space of curves of genus three with level two structure Bergvall, Olof January 2014 (has links) In this thesis we investigate the moduli space M3[2] of curves of genus 3 equipped with a symplectic level 2 structure. In particular, we are interested in the cohomology of this space. We obtain cohomological information by decomposing M3[2] into a disjoint union of two natural subspaces, Q[2] and H3[2], and then making S7- resp. S8-equivariantpoint counts of each of these spaces separately. / Målet med denna uppsats är att undersöka modulirummet M3[2] av kurvor av genus 3 med symplektisk nivå 2 struktur. Mer specifikt vill vi hitta informationom kohomologin av detta rum. För att uppnå detta delar vi först upp M[2] i en disjunkt union av två naturliga delrum, Q[2] och H3[2], och räknar därefter punkterna av dessa rum S7- respektive S8-ekvivariant. Algebraic geometry Moduli space Cohomology Symplectic structure Point count Geometry Geometri Algebra and Logic Algebra och logik
3	Cohomology of the moduli space of curves of genus three with level two structure Bergvall, Olof January 2014 (has links) In this thesis we investigate the moduli space M3[2] of curves of genus 3 equipped with a symplectic level 2 structure. In particular, we are interested in the cohomology of this space. We obtain cohomological information by decomposing M3[2] into a disjoint union of two natural subspaces, Q[2] and H3[2], and then making S7- resp. S8-equivariantpoint counts of each of these spaces separately. / Målet med denna uppsats är att undersöka modulirummet M3[2] av kurvor av genus 3 med symplektisk nivå 2 struktur. Mer specifikt vill vi hitta informationom kohomologin av detta rum. För att uppnå detta delar vi först upp M[2] i en disjunkt union av två naturliga delrum, Q[2] och H3[2], och räknar därefter punkterna av dessa rum S7- respektive S8-ekvivariant. Algebraic geometry Moduli space Cohomology Symplectic structure Point count Geometry Geometri Algebra and Logic Algebra och logik
4	The universal covers of hypertoric varieties and Bogomolov’s decomposition / 超トーリック多様体の普遍被覆とボゴモロフ分解 Nagaoka, Takahiro 23 March 2022 (has links) 京都大学 / 新制・課程博士 / 博士(理学) / 甲第23686号 / 理博第4776号 / 新制\|\|理\|\|1684(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)教授並河良典, 教授玉川安騎男, 教授望月拓郎 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM hypertoric varieties conical symplectic varieties universal cover fundamental group of regular locus Bogomolov's decomposition uniqueness of symplectic structure 400
5	Cohomologies on sympletic quotients of locally Euclidean Frolicher spaces Tshilombo, Mukinayi Hermenegilde 08 1900 (has links) This thesis deals with cohomologies on the symplectic quotient of a Frölicher space which is locally diffeomorphic to a Euclidean Frölicher subspace of Rn of constant dimension equal to n. The symplectic reduction under consideration in this thesis is an extension of the Marsden-Weinstein quotient (also called, the reduced space) well-known from the finite-dimensional smooth manifold case. That is, starting with a proper and free action of a Frölicher-Lie-group on a locally Euclidean Frölicher space of finite constant dimension, we study the smooth structure and the topology induced on a small subspace of the orbit space. It is on this topological space that we will construct selected cohomologies such as : sheaf cohomology, Alexander-Spanier cohomology, singular cohomology, ~Cech cohomology and de Rham cohomology. Some natural questions that will be investigated are for instance: the impact of the symplectic structure on these di erent cohomologies; the cohomology that will give a good description of the topology on the objects of category of Frölicher spaces; the extension of the de Rham cohomology theorem in order to establish an isomorphism between the five cohomologies. Beside the algebraic, topological and geometric study of these new objects, the thesis contains a modern formalism of Hamiltonian mechanics on the reduced space under symplectic and Poisson structures. / Mathematical Sciences / D. Phil. (Mathematics) Constant dimension Locally Euclidean Frolicher spaces Symplectic structure Symplectic geometry Symplectic quotient or reduced space Exterior algebra Ringed Frolicher space Smooth Gelfand representation Sheaf cohomology Alexander-Spanier cohomology Singular cohomology Cech cohomology and de Rham cohomology Vector fields and mechanics 514.224 Homology theory Symplectic manifolds Topological spaces Sheaf theory Geometry, Differential Hamiltonian graph theory Algebraic spaces
6	Cohomologies on sympletic quotients of locally Euclidean Frolicher spaces Tshilombo, Mukinayi Hermenegilde 08 1900 (has links) This thesis deals with cohomologies on the symplectic quotient of a Frölicher space which is locally diffeomorphic to a Euclidean Frölicher subspace of Rn of constant dimension equal to n. The symplectic reduction under consideration in this thesis is an extension of the Marsden-Weinstein quotient (also called, the reduced space) well-known from the finite-dimensional smooth manifold case. That is, starting with a proper and free action of a Frölicher-Lie-group on a locally Euclidean Frölicher space of finite constant dimension, we study the smooth structure and the topology induced on a small subspace of the orbit space. It is on this topological space that we will construct selected cohomologies such as : sheaf cohomology, Alexander-Spanier cohomology, singular cohomology, ~Cech cohomology and de Rham cohomology. Some natural questions that will be investigated are for instance: the impact of the symplectic structure on these di erent cohomologies; the cohomology that will give a good description of the topology on the objects of category of Frölicher spaces; the extension of the de Rham cohomology theorem in order to establish an isomorphism between the five cohomologies. Beside the algebraic, topological and geometric study of these new objects, the thesis contains a modern formalism of Hamiltonian mechanics on the reduced space under symplectic and Poisson structures. / Mathematical Sciences / D. Phil. (Mathematics) Constant dimension Locally Euclidean Frolicher spaces Symplectic structure Symplectic geometry Symplectic quotient or reduced space Exterior algebra Ringed Frolicher space Smooth Gelfand representation Sheaf cohomology Alexander-Spanier cohomology Singular cohomology Cech cohomology and de Rham cohomology Vector fields and mechanics 514.224 Homology theory Symplectic manifolds Topological spaces Sheaf theory Geometry, Differential Hamiltonian graph theory Algebraic spaces

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