This thesis is concerned with the 2-dimensional Toda equations and their geometric interpretation in form of r-adapted maps into flag manifolds, r-adapted maps are not only of interest due to their relation with the Toda equations, but also for their adaption to the m-synametric space structure of flag manifolds. This thesis studies the congruence question for r-adapted maps in flag manifolds. The main theorem of this thesis is a congruence theorem for г-holomorphic maps Ψ : S(^2) → G/T of constant curvature, where G can be any compact simple Lie group. It is supplemented by a congruence theorem for general r-holomorphic maps Ψ : S(^2) → G/T if G has rank 2, and a number of congruence theorems for isometric r-primitive Ψ : S(^2) → G/T of constant Kahler angle. The second group of congruence theorems is proved for the rank 2 case, as well as a selection of Lie groups with higher rank: SU(4),SU(5),F(_4),E(_6),E(_6),E(_8),Sp(n).

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.

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.

Basic concepts of differential geometry and Lie theory are introduced. Lie transformation groups are applied to linear systems of differential equations and the problem of describing rigid body orientation. Linear Hamiltonian systems are then treated as a Lie system of differential equations. This theory is applied to a particular Hamiltonian system arising from a problem in control theory, the linear state regulator problem.

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We give an overview of finite group frames and their applications to calculating summary statistics from partially ranked data, drawing upon the work of Rachel Cranfill (2009). We also provide a summary of the representation theory of compact Lie groups. We introduce both of these concepts as possible avenues beyond finite group representations, and also to suggest exploration into calculating summary statistics on Hilbert spaces using representations of Lie groups acting upon those spaces.

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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

This thesis discusses the mathematics of the Element Independent Corotational (EICR) Method and the more general Unified Small-Strain Corotational Formulation. The former was developed by Rankin, Brogan and Nour-Omid [106]. The latter, created by Felippa and Haugen [49], provides a theoretical frame work for the EICR and similar methods and its own enhanced methods. The EICR and similar corotational methods analyse non-linear deformation of a body by its discretization into finite elements, each with an orthogonal frame rotating (and translating) with the element. Such methods are well suited to deformations where non-linearity arises from rigid body deformation but local strains are small (1-4%) and so suited to linear analysis. This thesis focuses on such small-strain, non-linear deformations. The key concept in small-strain corotational methods is the separation of deformation into its rigid body and elastic components. The elastic component then can be analyzed linearly. Assuming rigid translation is removed first, this separation can be viewed as a polar decomposition (F = vR) of the deformation gradient (F) into a rigid rotation (R) followed by a small, approximately linear, stretch (v). This stretch usually causes shear as well as pure stretch. Using linear algebra, Chapter 3 explains the EICR Method and Unified Small-Strain Corotational Formulation initially without, and then with, the projector operator, reflecting their historical development. Projectors are orthogonal projections which simplify the isolation of elastic deformation and improve element strain invariance to rigid body deformation. Turning to Lie theory, Chapter 4 summarizes and applies relevant Lie theory to explore rigid and elastic deformation, finite element methods in general, and the EICR Method in particular. Rigid body deformation from a Lie perspective is well represented in the literature which is summarized. A less developed but emerging area in differential geometry (notably, Marsden/Hughes [82]), elastic deformation is discussed thoroughly followed by various Lie aspects of finite element analysis. Finally, the EICR Method is explored using Lie theory. Given the available research, complexity of the area, and level of this thesis, this exploration is less developed than the earlier linear algebraic discussion, but offers a useful alternative perspective on corotational methods. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2011-03-30 21:40:25.831

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