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Dissertations / Theses on the topic 'Complex Differential Geometry'
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1
Lam, Tsz-fung. "Nesting of 2D parts with complex geometry and material heterogeneity." Click to view the E-thesis via HKUTO, 2007. http://sunzi.lib.hku.hk/HKUTO/record/B39557005.
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Lam, Tsz-fung, and 林子峰. "Nesting of 2D parts with complex geometry and material heterogeneity." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2007. http://hub.hku.hk/bib/B39557005.
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Brown, James Ryan. "Complex and almost-complex structures on six dimensional manifolds." Diss., Columbia, Mo. : University of Missouri-Columbia, 2006. http://hdl.handle.net/10355/4466.
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Thesis (Ph.D.)--University of Missouri-Columbia, 2006.The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (February 26, 2007) Vita. Includes bibliographical references.
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Kirchhoff-Lukat, Charlotte Sophie. "Aspects of generalized geometry : branes with boundary, blow-ups, brackets and bundles." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/283007.
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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.
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Hsu, Siu-fai, and 許紹輝. "Geometric quantization of fermions and complex bosons." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2013. http://hub.hku.hk/bib/B50434500.
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Geometric quantization is a subject of finding irreducible representations of certain group or algebra and identifying those equivalent representations by geometric means. Geometric quantization of even dimensional fermionic system has been constructed based on the spinor representation of even dimensional Clifford algebras. Although geometric quantization of odd dimensional fermionic system has not been done, the existence of spinor representations in odd dimension indicates that the geometric quantization is possible.
In quantum field theory, charge conjungation can be defined on complex bosons and fermions. Without interaction, the particles and anti-particles essentially have same physical properties.
In this thesis, we will first recall the setup of geometric quantization of even dimensional fermion and bosons. Then we will show how to quantize odd dimensional fermion. After that, charge conjungation on complex fermion and boson will be defined and the remaining effort will be put on how to identify the Hilbert spaces produced by different charge conjungations.published_or_final_version
Mathematics
Master
Master of Philosophy
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Ugail, Hassan. "Time-dependent shape parameterisation of complex geometry using PDE surfaces." Nashboro Press, 2004. http://hdl.handle.net/10454/2686.
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Alves, Leonardo Soriani 1991. "Geometria complexa generalizada e tópicos relacionados." [s.n.], 2015. http://repositorio.unicamp.br/jspui/handle/REPOSIP/305829.
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Orientadores: Luiz Antonio Barrera San Martin, Lino Anderson da Silva GramaDissertação (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
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Gabella, Maxime. "The AdS/CFT correspondence and generalized geometry." Thesis, University of Oxford, 2011. http://ora.ox.ac.uk/objects/uuid:6fd2037e-d0ec-4806-b4db-631eb3693071.
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The most general AdS$_5 imes Y$ solutions of type IIB string theory that are AdS/CFT dual to superconformal field theories in four dimensions can be fruitfully described in the language of generalized geometry, a powerful hybrid of complex and symplectic geometry. We show that the cone over the compact five-manifold $Y$ is generalized Calabi-Yau and carries a generalized holomorphic Killing vector field $xi$, dual to the R-symmetry. Remarkably, this cone always admits a symplectic structure, which descends to a contact structure on $Y$, with $xi$ as Reeb vector field. Moreover, the contact volumes of $Y$, which can be computed by localization, encode essential properties of the dual CFT, such as the central charge and the conformal dimensions of BPS operators corresponding to wrapped D3-branes. We then define a notion of ``generalized Sasakian geometry'', which can be characterized by a simple differential system of three symplectic forms on a four-dimensional transverse space. The correct Reeb vector field for an AdS$_5$ solution within a given family of generalized Sasakian manifolds can be determined---without the need of the explicit metric---by a variational procedure. The relevant functional to minimize is the type IIB supergravity action restricted to the space of generalized Sasakian manifolds, which turns out to be just the contact volume. We conjecture that this contact volume is equal to the inverse of the trial central charge whose maximization determines the R-symmetry of the dual superconformal field theory. The power of this volume minimization is illustrated by the calculation of the contact volumes for a new infinite family of solutions, in perfect agreement with the results of $a$-maximization in the dual mass-deformed generalized conifold theories.
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Ma, Yilin. "Nonlinear Calderón Problem on Stein Manifolds." Thesis, The University of Sydney, 2021. https://hdl.handle.net/2123/25757.
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This thesis is devoted to the study of inverse problems for semilinear elliptic equations on Stein manifolds with Kähler metric. After developing some preliminary techniques, we will show that the Dirichlet-Neumann maps for certain semilinear elliptic equations determine the nonlinearities. We will consider two inverse problems of this kind with distinct geometric conditions imposed. The first one is the inverse problem for nonlinear Schrödinger equations on Kähler manifolds having specific Stein-like properties. The second one is the inverse problem for nonlinear magnetic Schrödinger equations on Riemann surfaces with partial data boundary measurements. In both cases, the nonlinearities involved are assumed to have certain analytic representations and vanishing lower order terms. The key observation is that, by a suitable linearisation procedure, one could transform the nonlinear problems into series of linear problems which have close connections to the techniques we develop.
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LY, KIM HA. "ON TWO APPROACHES FOR PARTIAL DIFFERENTIAL EQUATIONS IN SEVERAL COMPLEX VARIABLES." Doctoral thesis, Università degli studi di Padova, 2014. http://hdl.handle.net/11577/3423534.
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The aim of this thesis is to present influence of notations of ''type" on partial differential equations in several complex variables. The notations of "type" here include the finite and the infinite type in the sense of Hormander, and D'Angelo. In particular, in the first part, under the finite type condition, we will consider the existence and uniqueness of solutions for the initial value problem associated to the heat operator δs+□b on CR manifolds. The finite type m is the critical condition to provide pointwise estimates of the heat kernel via theory of singular integral operators developed by E. Stein and A. Nagel, D.H. Phong and E. Stein. Next, in the second part, we will introduce a new method to investigate the Cauchy-Riemann equations δu = φ. The solutions are constructed via the integral representation method. Moreover, we will show that the new method here is also applied well to the complex Monge-Ampère operator (ddc)n inCn. The main point is that our method can pass some well-known results from the case of finite type to infinite type.Lo scopo di questa tesi è quello di presentare l'influenza di notazioni di " tipo'' su equazioni differenziali alle derivate parziali in più variabili complesse. Le notazioni di "tipo" qui includono il finito e il tipo di infinito, nel senso di Hormander, e D'Angelo. In particolare, nella prima parte, a condizione tipo finito, prenderemo in considerazione l'esistenza e l'unicità delle soluzioni per il problema del valore iniziale associato ai operatore calore δs+□b su varietà CR. Il tipo finito m è la condizione fondamentale per fornire stime puntuali del nucleo del calore attraverso la teoria degli operatori integrali singolari sviluppate da E. Stein e A. Nagel, D.H. Phong e E. Stein. Prossimo, nella seconda parte, introdurremo un nuovo metodo per indagare la equazioni Cauchy-Riemann δu = φ. Le soluzioni sono costruite con via metodo rappresentazione integrale. Inoltre, mostreremo che il nuovo metodo qui viene applicato anche ben al complesso operatore Monge-Ampère (ddc)n inCn. Il punto principale è che il nostro metodo può passare alcuni risultati noti dal caso di tipo finito al tipo di infinito.
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Shaddad, Amna. "The classification and dynamics of the momentum polytopes of the SU(3) action on points in the complex projective plane with an application to point vortices." Thesis, University of Manchester, 2018. https://www.research.manchester.ac.uk/portal/en/theses/the-classification-and-dynamics-of-the-momentum-polytopes-of-the-su3-action-on-points-in-the-complex-projective-plane-with-an-application-to-point-vortices(456a7a49-ef1b-4660-a8e6-8d4cd0791d9d).html.
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We have fully classified the momentum polytopes of the SU(3) action on CP(2)xCP(2) and CP(2)xCP(2) xCP(2), both actions with weighted symplectic forms, and their corresponding transition momentum polytopes. For CP(2)xCP(2) the momentum polytopes are distinct line segments. The action on CP(2)xCP(2) xCP(2), has 9 different momentum polytopes. The vertices of the momentum polytopes of the SU(3) action on CP(2)xCP(2) xCP(2), fall into two categories: definite and indefinite vertices. The reduced space corresponding to momentum map image values at definite vertices is isomorphic to the 2-sphere. We show that these results can be applied to assess the dynamics by introducing and computing the space of allowed velocity vectors for the different configurations of two-vortex systems.
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Bäcklund, Pierre. "Studies on boundary values of eigenfunctions on spaces of constant negative curvature." Doctoral thesis, Uppsala University, Department of Mathematics, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-8920.
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This thesis consists of two papers on the spectral geometry of locally symmetric spaces of Riemannian and Lorentzian signature. Both works are concerned with the idea of relating analysis on such spaces to structures on their boundaries.
The first paper is motivated by a conjecture of Patterson on the Selberg zeta function of Kleinian groups. We consider geometrically finite hyperbolic cylinders with non-compact Riemann surfaces of finite area as cross sections. For these cylinders, we present a detailed investigation of the Bunke-Olbrich extension operator under the assumption that the cross section of the cylinder has one cusp. We establish the meromorphic continuation of the extension of Eisenstein series and incomplete theta series through the limit set. Furthermore, we derive explicit formulas for the residues of the extension operator in terms of boundary values of automorphic eigenfunctions.
The motivation for the second paper comes from conformal geometry in Lorentzian signature. We prove the existence and uniqueness of a sequence of differential intertwining operators for spherical principal series representations, which are realized on boundaries of anti de Sitter spaces. Algebraically, these operators correspond to homomorphisms of generalized Verma modules. We relate these families to the asymptotics of eigenfunctions on anti de Sitter spaces.
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Souici, Zobida. "Transformations holomorphiquement projectives des espaces symétriques complexes." Lyon 1, 1988. http://www.theses.fr/1988LYO11758.
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Ce travail est consacre a l'etude des proprietes h-projectives des espaces symetriques complexes. Dans le chapitre i (preliminaire) on rappelle quelques definitions et resultats sur les courbes holomorphiquement planaites, les transformations h-projectives, le tenseur de weyl h-projectif et les espaces symetriques complexes. On construit et on etudie dans le chapitre ii, deux familles de domaines symetriques de l'espace projectif complexe et on demontre que ce sont les seuls domaines symetriques. La construction effectuee au chapitre ii est utilisee, dans le chapitre iii pour obtenir une classification algebrique des espaces symetriques complexes h-projectivement plats. Dans le cas simplement connexe, on ramene la recherche des classes d'isomorphisme a la determination effectuee au chapitre ii d'orbites du groupe projectif complexe. Le cas non simplement connexe est egalement etudie. Le chapitre iv est consacre a la recherche des espaces symetriques complexes admettant des transformations h-projectives non affines. On demontre, tout d'abord, que ces espaces sont necessairement h-projectivement plats, et, en exploitant la classification du chapitre iii, on determine, a isomorphisme pres, tous les espaces symetriques complexes dont le groupe des transformations h-projectives ne se reduit pas a un groupe de transformations affines
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Gonzalez, Castro Gabriela, and Hassan Ugail. "Shape morphing of complex geometries using partial differential equations." Academy Publisher, 2007. http://hdl.handle.net/10454/2643.
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An alternative technique for shape morphing
using a surface generating method using partial differential
equations is outlined throughout this work. The boundaryvalue
nature that is inherent to this surface generation
technique together with its mathematical properties are
hereby exploited for creating intermediate shapes between
an initial shape and a final one. Four alternative shape
morphing techniques are proposed here. The first one is
based on the use of a linear combination of the boundary
conditions associated with the initial and final surfaces,
the second one consists of varying the Fourier mode for
which the PDE is solved whilst the third results from a
combination of the first two. The fourth of these alternatives
is based on the manipulation of the spine of the surfaces,
which is computed as a by-product of the solution. Results
of morphing sequences between two topologically nonequivalent
surfaces are presented. Thus, it is shown that the
PDE based approach for morphing is capable of obtaining
smooth intermediate surfaces automatically in most of the
methodologies presented in this work and the spine has been
revealed as a powerful tool for morphing surfaces arising
from the method proposed here.
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Cekić, Mihajlo. "The Calderón problem for connections." Thesis, University of Cambridge, 2017. https://www.repository.cam.ac.uk/handle/1810/267829.
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This thesis is concerned with the inverse problem of determining a unitary connection $A$ on a Hermitian vector bundle $E$ of rank $m$ over a compact Riemannian manifold $(M, g)$ from the Dirichlet-to-Neumann (DN) map $\Lambda_A$ of the associated connection Laplacian $d_A^*d_A$. The connection is to be determined up to a unitary gauge equivalence equal to the identity at the boundary. In our first approach to the problem, we restrict our attention to conformally transversally anisotropic (cylindrical) manifolds $M \Subset \mathbb{R}\times M_0$. Our strategy can be described as follows: we construct the special Complex Geometric Optics solutions oscillating in the vertical direction, that concentrate near geodesics and use their density in an integral identity to reduce the problem to a suitable $X$-ray transform on $M_0$. The construction is based on our proof of existence of Gaussian Beams on $M_0$, which are a family of smooth approximate solutions to $d_A^*d_Au = 0$ depending on a parameter $\tau \in \mathbb{R}$, bounded in $L^2$ norm and concentrating in measure along geodesics when $\tau \to \infty$, whereas the small remainder (that makes the solution exact) can be shown to exist by using suitable Carleman estimates. In the case $m = 1$, we prove the recovery of the connection given the injectivity of the $X$-ray transform on $0$ and $1$-forms on $M_0$. For $m > 1$ and $M_0$ simple we reduce the problem to a certain two dimensional $\textit{new non-abelian ray transform}$. In our second approach, we assume that the connection $A$ is a $\textit{Yang-Mills connection}$ and no additional assumption on $M$. We construct a global gauge for $A$ (possibly singular at some points) that ties well with the DN map and in which the Yang-Mills equations become elliptic. By using the unique continuation property for elliptic systems and the fact that the singular set is suitably small, we are able to propagate the gauges globally. For the case $m = 1$ we are able to reconstruct the connection, whereas for $m > 1$ we are forced to make the technical assumption that $(M, g)$ is analytic in order to prove the recovery. Finally, in both approaches we are using the vital fact that is proved in this work: $\Lambda_A$ is a pseudodifferential operator of order $1$ acting on sections of $E|_{\partial M}$, whose full symbol determines the full Taylor expansion of $A$ at the boundary.
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Angoshtari, Arzhang. "Geometric discretization schemes and differential complexes for elasticity." Diss., Georgia Institute of Technology, 2013. http://hdl.handle.net/1853/49026.
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In this research, we study two different geometric approaches, namely, the discrete exterior calculus and differential complexes, for developing numerical schemes for linear and nonlinear elasticity. Using some ideas from discrete exterior calculus (DEC), we present a geometric discretization scheme for incompressible linearized
elasticity. After characterizing the configuration manifold of volume- preserving discrete deformations, we use Hamilton’s principle on this configuration manifold. The discrete Euler-Lagrange equations are obtained without using Lagrange multipliers.
The main difference between our approach and the mixed finite element formulations is that we simultaneously use three different discrete spaces for the displacement field. We test the efficiency and robustness of this geometric scheme using some numerical examples. In particular, we do not see any volume locking and/or checkerboarding of pressure in our numerical examples. This suggests that our choice of discrete solution
spaces is compatible. On the other hand, it has been observed that the linear elastostatics complex can be used to find very efficient numerical schemes. We use some geometric techniques to obtain differential complexes for nonlinear elastostatics.
In particular, by introducing stress functions for the Cauchy and the second Piola-Kirchhoff stress tensors, we show that 2D and 3D nonlinear elastostatics admit separate kinematic and kinetic complexes. We show that stress functions corresponding to the first Piola-Kirchhoff stress tensor allow us to write a complex for 3D nonlinear
elastostatics that similar to the complex of 3D linear elastostatics contains both the kinematics an kinetics of motion. We study linear and nonlinear compatibility equations
for curved ambient spaces and motions of surfaces in R3. We also study the relationship between the linear elastostatics complex and the de Rham complex. The geometric approach presented in this research is crucial for understanding connections
between linear and nonlinear elastostatics and the Hodge Laplacian, which can enable one to convert numerical schemes of the Hodge Laplacian to those for linear and possibly nonlinear elastostatics.
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Stienon, Mathieu. "A propos d'une structure complexe sur un espace de twisteurs pour certaines variété symplectiques." Doctoral thesis, Universite Libre de Bruxelles, 2004. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/211176.
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Kuang, Shilong. "Analysis of conjugate heat equation on complete non-compact Riemannian manifolds under Ricci flow." Diss., UC access only, 2009. http://proquest.umi.com/pqdweb?index=7&did=1907270831&SrchMode=2&sid=2&Fmt=2&VInst=PROD&VType=PQD&RQT=309&VName=PQD&TS=1270053784&clientId=48051.
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Thesis (Ph. D.)--University of California, Riverside, 2009.Includes abstract. Includes bibliographical references (leaves 74-76). Issued in print and online. Available via ProQuest Digital Dissertations.
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Jbilou, Asma. "Equations hessiennes complexes sur des variétés kählériennes compactes." Phd thesis, Université de Nice Sophia-Antipolis, 2010. http://tel.archives-ouvertes.fr/tel-00463111.
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Sur une variété kählérienne compacte connexe de dimension 2m, ! étant la forme de Kähler, une forme volume donnée dans [!]m et k un entier 1 < k < m, on cherche à résoudre de façon unique dans [!] l'équation ˜ !k ^!m−k = en utilisant une notion de k-positivité pour ˜ ! 2 [!] (les cas extrêmes sont résolus : k = m par Yau, k = 1 trivialement). Nous résolvons par la méthode de continuité l'équation hessienne d'ordre k complexe elliptique correspondante sous l'hypothèse que la variété est à courbure bisectionelle holomorphe non-négative, ici requise seulement pour établir un pincement a priori de valeurs propres.
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Lytle, George H. "APPROXIMATIONS IN RECONSTRUCTING DISCONTINUOUS CONDUCTIVITIES IN THE CALDERÓN PROBLEM." UKnowledge, 2019. https://uknowledge.uky.edu/math_etds/61.
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In 2014, Astala, Päivärinta, Reyes, and Siltanen conducted numerical experiments reconstructing a piecewise continuous conductivity. The algorithm of the shortcut method is based on the reconstruction algorithm due to Nachman, which assumes a priori that the conductivity is Hölder continuous. In this dissertation, we prove that, in the presence of infinite-precision data, this shortcut procedure accurately recovers the scattering transform of an essentially bounded conductivity, provided it is constant in a neighborhood of the boundary. In this setting, Nachman’s integral equations have a meaning and are still uniquely solvable.
To regularize the reconstruction, Astala et al. employ a high frequency cutoff of the scattering transform. We show that such scattering transforms correspond to Beltrami coefficients that are not compactly supported, but exhibit certain decay at infinity. For this class of Beltrami coefficients, we establish that the complex geometric optics solutions to the Beltrami equation exist and exhibit the same subexponential decay as described in the 2006 work of Astala and Päivärinta. This is a first step toward extending the inverse scattering map of Astala and Päivärinta to non-compactly supported conductivities.
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Tipler, Carl. "Constructions de métriques extrémales : résolutions de singularités, déformations complexes." Phd thesis, Université de Nantes, 2011. http://tel.archives-ouvertes.fr/tel-00676452.
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Le problème abordé dans cette thèse est celui de l'existence de métriques extrémales. Si (M, J, g) est une variété kahlérienne compacte, une métrique extrémale est une métrique kählérienne dont la norme L2 de la courbure scalaire est minimale pour les métriques représentant la même classe de Kähler. On propose de nouvelles constructions de métriques extrémales utilisant des méthodes perturbatives. Dans un premier temps, on montre que si (M, J, g) est une surface orbifold extrémale qui ne possède que des singularités isolées de type Hirzebruch-Jung, alors une résolution de (M, J) admet une métrique extrémale. On donne des applications de ce résultat sur l'existence de métriques extrémales sur les éclatements de surfaces réglées paraboliques. Dans une seconde partie, on etudie la stabilié des métriques extrémales sous déformations complexes. Ceci est un travail réalisé en collaboration avec Yann Rollin et Santiago Simanca. On donne un critère suffisant pour assurer la stabilité d'une métrique extrémale lors d'une déformation complexe munie d'une action holomorphe d'un groupe compact. On généralise ainsi des résultats de S.Simanca et C.Lebrun. Ceci nous permet également de retrouver un résultat de S.Donaldson, a savoir une métrique Kähler-Einstein sur une déformation de la variété de Mukai et Umemura.
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Auvray, Hugues. "Équation de Monge-Ampère complexe, métriques kählériennes de type Poincaré et instantons gravitationnels ALF." Phd thesis, Université Pierre et Marie Curie - Paris VI, 2012. http://tel.archives-ouvertes.fr/tel-00750891.
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Ce travail de thèse s'intéresse à la résolution d'équations de Monge-Ampère complexes et à ses applications sur certains types de variétés non compactes. Ce mémoire décrit plus précisément deux situations distinctes dans lesquelles on résout des équations de Monge-Ampère, avant de tirer les conséquences de ces résolutions. Dans une première partie, on travaille sur le complémentaire d'un diviseur à croisements normaux dans une variété kählérienne compacte. On fixe sur le complémentaire du diviseur une classe de métriques kählériennes à singularités cusp le long du diviseur. Pour construire des géodésiques entre métriques de cette classe, on résout une équation de Monge-Ampère homogène, sur le produit de notre ouvert de Zariski par une surface de Riemann à bord. On applique cette construction à un résultat d'unicité de métriques à courbure scalaire constante dans la classe considérée ; on résout encore pour cela une équation de Monge-Ampère avec second membre sur le complémentaire du diviseur. On exhibe enfin des obstructions topologiques à l'existence de métriques à courbure scalaire constante au sein des classes de métriques kählériennes singulières envisagées. La seconde partie du mémoire traite d'une construction analytique d'instantons gravitationnels ALF, ou variétés complètes de dimension 4, hyperkählériennes, à croissance cubique du volume. On donne la construction d'instantons diédraux ; on considère plus exactement des résolutions de singularités kleiniennes diédrales. Le traitement d'une équation de Monge-Ampère, donné pour des variétés kählériennes ALF assez générales, nous permet sur nos exemples de corriger un prototype simple pour obtenir la métrique hyperkählérienne recherchée.
23
Spinaci, Marco. "Déformations des applications harmoniques tordues." Phd thesis, Grenoble, 2013. http://tel.archives-ouvertes.fr/tel-00877310.
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On étudie les déformations des applications harmoniques $f$ tordues par rapport à une représentation. Après avoir construit une application harmonique tordue "universelle", on donne une construction de toute déformations du premier ordre de $f$ en termes de la théorie de Hodge ; on applique ce résultat à l'espace de modules des représentations réductives d'un groupe de Kähler, pour démontrer que les points critiques de la fonctionnelle de l'énergie $E$ coïncident avec les représentations de monodromie des variations complexes de structures de Hodge. Ensuite, on procède aux déformations du second ordre, où des obstructions surviennent ; on enquête sur l'existence de ces déformations et on donne une méthode pour les construire. En appliquant ce résultat à la fonctionnelle de l'énergie comme ci-dessus, on démontre (pour n'importe quel groupe de présentation finie) que la fonctionnelle de l'énergie est strictement pluri sous-harmonique sur l'espace des modules des représentations. En assumant de plus que le groupe soit de Kähler, on étudie les valeurs propres de la matrice hessienne de $E$ aux points critiques.
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Gutt, Jean. "On the minimal number of periodic Reeb orbits on a contact manifold." Phd thesis, Université de Strasbourg, 2014. http://tel.archives-ouvertes.fr/tel-01016954.
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Le sujet de cette thèse est la question du nombre minimal d'orbites de Reeb distinctes sur une variété de contact qui est le bord d'une variété symplectique compacte. L'homologie symplectique $S^1$-équivariante positive est un des outils principaux de cette thèse; elle est construite à partir d'orbites périodiques de champs de vecteurs hamiltoniens sur une variété symplectique dont le bord est la variété de contact considérée. Nous analysons la relation entre les différentes variantes d'homologie symplectique d'une variété symplectique exacte compacte (domaine de Liouville) et les orbites de Reeb de son bord. Nous démontrons certaines propriétés de ces homologies. Pour un domaine de Liouville plongé dans un autre, nous construisons un morphisme entre leurs homologies. Nous étudions ensuite l'invariance de ces homologies par rapport au choix de la forme de contact sur le bord. Nous utilisons l'homologie symplectique $S^1$-équivariante positive pour donner une nouvelle preuve d'un théorème de Ekeland et Lasry sur le nombre minimal d'orbites de Reeb distinctes sur certaines hypersurfaces dans $\R^{2n}$. Nous indiquons comment étendre au cas de certaines hypersurfaces dans certains fibrés en droites complexes négatifs. Nous donnons une caractérisation et une nouvelle façon de calculer l'indice de Conley-Zehnder généralisé, défini par Robbin et Salamon pour tout chemin de matrices symplectiques. Ceci nous a mené à développer de nouvelles formes normales de matrices symplectiques.
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Aribi, Amine. "Le spectre du sous-laplacien sur les variétés CR strictement pseudoconvexes." Phd thesis, Université François Rabelais - Tours, 2012. http://tel.archives-ouvertes.fr/tel-00960234.
Full textAbstract:
Le but de cette thèse est d'étudier le spectre du sous-laplacien sur les variétés CR strictement peusdoconvexes. Nous prouvons que le spectre du sous-laplacien $\Delta_b$ est discret sur un domaine borné $\Omega \subset M$ d'une variété CR strictement pseudoconvexe qui satisfait l'inégalité de Poincaré, sous les conditions de Dirichlet au bord. Nous étudions le comportement des valeurs propres du sous-laplacien $\Delta_b$ sur une variété CR strictement pseudoconvexe compacte $M$, en tant que fonctionnelle sur l'espace ${\mathcal P}_+$ de formes de contact positivement orientées sur $M$ en dotant ${\mathcal P}_+$ d'une topologie métrique naturelle. Nous établissons des inégalités pour les valeurs propres de $\Delta_b$ sur des variétés CR strictement pseudoconvexes ( éventuellement à bord non vide). Nos estimations prolongent les résultats obtenus par P-C. Niu \& H. Zhang \cite{NiZh} pour les valeurs propres du sous-laplacien avec conditions de Dirichlet au bord sur un domaine borné du groupe de Heisenberg, et sont dans l'esprit des inégalités de Payne-P\'lya-Weinberger et Yang. Nous obtenons une nouvelle borne inférieure sur la première valeur propre non nulle $\lambda_1 (\theta )$ du sous-laplacien $\Delta_b$ sur une variété CR strictement pseudoconvexe compacte $M$ munie d'une forme de contact $\theta$ dont la connexion de Tanaka-Webster est à courbure de Ricci minorée.
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"Topics in complex differential geometry." Chinese University of Hong Kong, 1987. http://library.cuhk.edu.hk/record=b5885783.
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Bailey, Michael. "On the Local and Global Classification of Generalized Complex Structures." Thesis, 2012. http://hdl.handle.net/1807/32657.
Full textAbstract:
We study a number of local and global classification problems in generalized complex geometry. Generalized complex geometry is a relatively new type of geometry which has applications to string theory and mirror symmetry. Symplectic and complex geometry are special cases.
In the first topic, we characterize the local structure of generalized complex manifolds by proving that a generalized complex structure near a complex point arises from a holomorphic Poisson structure. In the proof we use a smoothed Newton’s method along the lines of Nash, Moser and Conn.
In the second topic, we consider whether a given regular Poisson structure and transverse complex structure come from a generalized complex structure. We give cohomological criteria, and we find some counterexamples and some unexpected examples, including a compact, regular generalized complex manifold for which nearby symplectic leaves are not symplectomorphic.
In the third topic, we consider generalized complex structures with nondegenerate type change; we describe a generalized Calabi-Yau structure induced on the type change locus, and prove a local normal form theorem near this locus. Finally, in the fourth topic, we give a classification of generalized complex principal bundles satisfying a certain transversality condition; in this case, there is a generalized flat connection, and the classification involves a monodromy map to the Courant automorphism group.
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(6597026), Hongshan Li. "Vanishing Theorems for the logarithmic de Rham complex of unitary local system." Thesis, 2019.
Find full textAbstract:
This work includes various proofs of cohomology vanishing for logarithmic de Rham complex of unitary local system defined on an open algebraic complex manifold, which has a projective compactification by normal crossing divisor
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Sun, Jian. "Kähler-Einstein metrics and Sobolev inequality /." 2000. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:9965165.
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McBride, Matthew Scott. "D-bar and Dirac Type Operators on Classical and Quantum Domains." 2012. http://hdl.handle.net/1805/2931.
Full textAbstract:
Indiana University-Purdue University Indianapolis (IUPUI)I study d-bar and Dirac operators on classical and quantum domains subject to the APS boundary conditions, APS like boundary conditions, and other types of global boundary conditions. Moreover, the inverse or inverse modulo compact operators to these operators are computed. These inverses/parametrices are also shown to be bounded and are also shown to be compact, if possible. Also the index of some of the d-bar operators are computed when it doesn't have trivial index. Finally a certain type of limit statement can be said between the classical and quantum d-bar operators on specialized complex domains.
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Philip, Eliza. "Function Theory On Non-Compact Riemann Surfaces." Thesis, 2012. https://etd.iisc.ac.in/handle/2005/2330.
Full textAbstract:
The theory of Riemann surfaces is quite old, consequently it is well developed. Riemann surfaces originated in complex analysis as a means of dealing with the problem of multi-valued functions. Such multi-valued functions occur because the analytic continuation of a given holomorphic function element along different paths leads in general to different branches of that function. The theory splits in two parts; the compact and the non-compact case. The function theory developed on these cases are quite dissimilar. The main difficulty one encounters in the compact case is the scarcity of global holomorphic functions, which limits one’s study to meromorphic functions. This however is not an issue in non-compact Riemann surfaces, where one enjoys a vast variety of global holomorphic functions. While the function theory of compact Riemann surfaces is centered around the Riemann-Roch theorem, which essentially tells us how many linearly independent meromorphic functions there are having certain restrictions on their poles, the function theory developed on non-compact Riemann surface engages tools for approximation of functions on certain subsets by holomorphic maps on larger domains. The most powerful tool in this regard is the Runge’s approximation theorem. An intriguing application of this is the Gunning-Narasimhan theorem, which says that every connected open Riemann surface has an immersion into the complex plane. The main goal of this project is to prove Runge’s approximation theorem and illustrate its effectiveness in proving the Gunning-Narasimhan theorem. Finally we look at an analogue of Gunning-Narasimhan theorem in the case of a compact Riemann surface.
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Philip, Eliza. "Function Theory On Non-Compact Riemann Surfaces." Thesis, 2012. http://etd.iisc.ernet.in/handle/2005/2330.
Full textAbstract:
The theory of Riemann surfaces is quite old, consequently it is well developed. Riemann surfaces originated in complex analysis as a means of dealing with the problem of multi-valued functions. Such multi-valued functions occur because the analytic continuation of a given holomorphic function element along different paths leads in general to different branches of that function. The theory splits in two parts; the compact and the non-compact case. The function theory developed on these cases are quite dissimilar. The main difficulty one encounters in the compact case is the scarcity of global holomorphic functions, which limits one’s study to meromorphic functions. This however is not an issue in non-compact Riemann surfaces, where one enjoys a vast variety of global holomorphic functions. While the function theory of compact Riemann surfaces is centered around the Riemann-Roch theorem, which essentially tells us how many linearly independent meromorphic functions there are having certain restrictions on their poles, the function theory developed on non-compact Riemann surface engages tools for approximation of functions on certain subsets by holomorphic maps on larger domains. The most powerful tool in this regard is the Runge’s approximation theorem. An intriguing application of this is the Gunning-Narasimhan theorem, which says that every connected open Riemann surface has an immersion into the complex plane. The main goal of this project is to prove Runge’s approximation theorem and illustrate its effectiveness in proving the Gunning-Narasimhan theorem. Finally we look at an analogue of Gunning-Narasimhan theorem in the case of a compact Riemann surface.
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Herrera, Andrea Cecilia. "Estructuras Killing-Yano invariantes en variedades homogéneas." Bachelor's thesis, 2018. http://hdl.handle.net/11086/6554.
Full textAbstract:
Tesis (Doctor en Matemática)--Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación, 2018.Estudiamos 2-formas de Killing-Yano invariantes a izquierda sobre grupos de Lie, o equivalentemente, sobre álgebras de Lie. Más particularmente, clasificamos las álgebras de Lie de dimensión 4 que admitan tales estructuras, y para cada una de tales álgebras, listamos todas las posibles estructuras de Killing-Yano (salvo equivalencia). Realizando extensiones centrales de estas álgebras obtenemos ejemplos de 2-formas de Killing-Yano Conformes en dimensión 5. Las álgebras de Lie obtenidas resultan ser aquellas álgebras de Lie sasakianas (de dimensión 5) con centro no trivial. Analizamos tensores de Killing-Yano invariantes para variedades homogéneas G/K que admitan una métrica G-invariante. Más específicamente estudiamos la ecuación de Killing-Yano en variedades bandera generalizadas y damos ejemplos concretos en variedades bandera maximales de dimensiones 6, 8 y 12.
We study left invariant Killing-Yano 2-forms on Lie groups. This is equivalent to work at the Lie algebra level. In particular, we classify four dimensional Lie algebras that admit such structures, and for each of them we list all the possibles Killing-Yano structures (up to equivalence). Doing central extensions on the obtained Lie algebras, we find examples of Conformal Killing-Yano tensors in dimension five. Further, the obtained central extensions are Sasakian Lie algebras of dimension five with no trivial center. We analyse G-invariant Killing-Yano tensors on homogeneous spaces G/K that admits a G-invariant metric. As an application we study the Killing-Yano equation on Generalized flag manifolds and we find examples of invariant Killing Yano tensors on full flag manifolds of dimension six, eight and twelve.
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Ghosh, Kartick. "On some canonical metrics on holomorphic vector bundles over Kahler manifolds." Thesis, 2023. https://etd.iisc.ac.in/handle/2005/6152.
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This thesis consists of two parts. In the first part, we introduce coupled Kähler-
Einstein and Hermitian-Yang-Mills equations. It is shown that these equations have
an interpretation in terms of a moment map. We identify a Futaki-type invariant as an
obstruction to the existence of solutions of these equations. We also prove a Matsushima-
Lichnerowicz-type theorem as another obstruction. Using the Calabi ansatz, we produce
nontrivial examples of solutions of these equations on some projective bundles. Another
class of nontrivial examples is produced using deformation. In the second part, we prove
a priori estimates for vortex-type equations. We then apply these a priori estimates
in some situations. One important application is the existence and uniqueness result
concerning solutions of the Calabi-Yang-Mills equations. We recover a priori estimates
of the J-vortex equation and the Monge-Ampère vortex equation. We establish a corre-
spondence result between Gieseker stability and the existence of almost Hermitian-Yang-
Mills metric in a particular case. We also investigate the Kählerity of the symplectic form
which arises in the moment map interpretation of the Calabi-Yang-Mills equations.
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(9132815), Kuang-Ru Wu. "Hermitian-Yang-Mills Metrics on Hilbert Bundles and in the Space of Kahler Potentials." Thesis, 2020.
Find full textAbstract:
The two main results in this thesis have a common point: Hermitian--Yang--Mills (HYM) metrics. In the first result, we address a Dirichlet problem for the HYM equations in bundles of infinite rank over Riemann surfaces. The solvability has been known since the work of Donaldson \cite{Donaldson92} and Coifman--Semmes \cite{CoifmanSemmes93}, but only for bundles of finite rank. So the novelty of our first result is to show how to deal with infinite rank bundles. The key is an a priori estimate obtained from special feature of the HYM equation.
In the second result, we take on the topic of the so-called ``geometric quantization." This is a vast subject. In one of its instances the aim is to approximate the space of K\"ahler potentials by a sequence of finite dimensional spaces. The approximation of a point or a geodesic in the space of K\"ahler potentials is well-known, and it has many applications in K\"ahler geometry. Our second result concerns the approximation of a Wess--Zumino--Witten type equation in the space of K\"ahler potentials via HYM equations, and it is an extension of the point/geodesic approximation.
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"The geometry of complete positively curved Kähler manifolds." 2003. http://library.cuhk.edu.hk/record=b6073552.
Full textAbstract:
by Chen Bing-Long."August 2003."
Thesis (Ph.D.)--Chinese University of Hong Kong, 2003.
Includes bibliographical references (p. 68-71).
Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Mode of access: World Wide Web.
Abstracts in English and Chinese.
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(6983546), Stefano Silvestri. "The Dynamics of Semigroups of Contraction Similarities on the Plane." Thesis, 2019.
Find full textAbstract:
Given a parametrized family of Iterated Function System (IFS) we give sufficient conditions for a parameter on the boundary of the connectedness locus, M, to be accessible from the complement of M.
Moreover, we provide a few examples of such parameters and describe how they are connected to Misiurewicz parameter in the Mandelbrot set, i.e. the connectedness locus of the quadratic family z^2+c.
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"On the structure of complete Kähler manifolds with positive bisectional curvature." 2005. http://library.cuhk.edu.hk/record=b5896407.
Full textAbstract:
Yu Chengjie.Thesis (M.Phil.)--Chinese University of Hong Kong, 2005.
Includes bibliographical references (leaves 65-67).
Abstracts in English and Chinese.
Abstract --- p.i
Acknowledgments --- p.ii
Chapter 1 --- Introduction --- p.1
Chapter 2 --- A multiplicity estimate and applications --- p.5
Chapter 2.1 --- A multiplicity estimate --- p.6
Chapter 2.2 --- Sharp bounds for the dimensions of the spaces of holomorphic functions of polynomial growth --- p.14
Chapter 2.3 --- Siegel's theorem on the fields of rational functions --- p.15
Chapter 3 --- Quasi-embedding of complete Kahler manifolds --- p.21
Chapter 3.1 --- The original map F0 --- p.21
Chapter 3.2 --- Almost injectivity of F0 --- p.26
Chapter 3.3 --- Almost surjectivity of F0 --- p.28
Chapter 3.4 --- Weaker conditions for almost surjectivity --- p.41
Chapter 3.5 --- Existence of quasi-embedding --- p.48
Chapter 4 --- Desingularization of quasi-embeddings --- p.51
Chapter 4.1 --- Normalization of a map with polynomial growth --- p.51
Chapter 4.2 --- The method to desingularize a quasi-embedding --- p.54
Chapter 4.3 --- The case of dimension two --- p.55
Chapter 4.4 --- A uniformization theorem --- p.63
Bibliography --- p.65
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Kinzebulatov, Damir. "Geometric Analysis on Solutions of Some Differential Inequalities and within Restricted Classes of Holomorphic Functions." Thesis, 2012. http://hdl.handle.net/1807/32309.
Full textAbstract:
Pars 1 and 2 are devoted to study of solutions of certain differential inequalities.
Namely, in Part 1 we show that a germ of an analytic set (real or complex) admits
a Gagliardo-Nirenberg type inequality with a certain exponent s>=1. At a regular point
s=1, and the inequality becomes classical. As our examples show, s can be strictly greater than 1 even for an isolated singularity.
In Part 2 we prove the property of unique continuation for solutions of differential inequality |\Delta u|<=|Vu| for a large class of potentials V. This result can be applied to the problem of absence of positive eigenvalues for
self-adjoint Schroedinger operator -\Delta+V defined in the sense of the form sum.
The results of Part 2 are joint with Leonid Shartser.
In Parts 3 and 4 we derive the basic elements of complex function theory within
some subalgebras of holomorphic functions (including extension from submanifolds, corona type theorem, properties of divisors, approximation property). Our key instruments and results are the analogues of Cartan theorems A and B for the `coherent sheaves' on the maximal ideal spaces of these subalgebras, and of Oka-Cartan theorem on coherence of the sheaves of ideals of the corresponding complex analytic subsets.
More precisely, in Part 3 we consider the algebras of holomorphic functions on regular
coverings of complex manifolds whose restrictions to each fiber belong to a translation-invariant Banach subalgebra of bounded functions endowed with sup-norm.
The model examples of such subalgebras are Bohr's holomorphic almost periodic functions on tube domains, and all fibrewise bounded holomorphic functions on regular coverings of complex manifolds.
In Part 4 the primary object of study is the subalgebra of bounded holomorphic functions on the unit disk whose moduli can have only boundary discontinuities of the first kind.
The results of Parts 3 and 4 are joint with Alexander Brudnyi.
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"Isometric immersions of complete surfaces with non-positive curvature." 2000. http://library.cuhk.edu.hk/record=b5890388.
Full textAbstract:
by Fan Xuqian.Thesis (M.Phil.)--Chinese University of Hong Kong, 2000.
Includes bibliographical references (leaves 99-100).
Abstracts in English and Chinese.
Chapter 1 --- Introduction --- p.5
Chapter 2 --- The Theorem of Efimov --- p.7
Chapter 2.1 --- The Idea of the Proof of the Efimov's Theorem --- p.8
Chapter 2.2 --- Proof of the Efimov's Main Lemma --- p.12
Chapter 2.3 --- Proof of Lemma 2.3 --- p.48
Chapter 2.4 --- Proof of Lemma 2.4 --- p.52
Chapter 3 --- Isometric Immersion into R3 of Complete Surfaces with Negative Curvature --- p.62
Chapter 3.1 --- The Sketch of the Proof of Theorem 3.1 --- p.66
Chapter 3.2 --- Proof of Lemma 3.4 --- p.75
Chapter 3.3 --- Proof of Lemma 3.5 --- p.76
Chapter 3.4 --- Proof of Lemma 3.6 --- p.86
Chapter 3.5 --- Proof of Lemma 3.7 --- p.89
Chapter 3.6 --- The Geometric Properties of the Immersion --- p.95
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Ochoa, Mayorga Victor Manuel. "Geometric approach to multi-scale 3D gesture comparison." Phd thesis, 2010. http://hdl.handle.net/10048/1530.
Full textAbstract:
The present dissertation develops an invariant framework for 3D gesture comparison studies. 3D gesture comparison without Lagrangian models is challenging not only because of the lack of prediction provided by physics, but
also because of a dual geometry representation, spatial dimensionality and non-linearity associated to 3D-kinematics.
In 3D spaces, it is difficult to compare curves without an alignment operator since it is likely that discrete curves are not synchronized and do not share a common point in space. One has to assume that each and every single trajectory in the space is unique. The common answer is to assert the similitude between two or more trajectories as estimating an average distance error from the aligned curves, provided that the alignment operator is found.
In order to avoid the alignment problem, the method uses differential geometry for position and orientation curves. Differential geometry not only reduces the spatial dimensionality but also achieves view invariance. However,
the nonlinear signatures may be unbounded or singular. Yet, it is shown that pattern recognition between intrinsic signatures using correlations is robust for position and orientation alike.
A new mapping for orientation sequences is introduced in order to treat quaternion and Euclidean intrinsic signatures alike. The new mapping projects a 4D-hyper-sphere for orientations onto a 3D-Euclidean volume. The projection uses the quaternion invariant distance to map rotation sequences into 3D-Euclidean curves. However, quaternion spaces are sectional discrete spaces.
The significance is that continuous rotation functions can be only approximated for small angles. Rotation sequences with large angle variations can only be interpolated in discrete sections.
The current dissertation introduces two multi-scale approaches that improve numerical stability and bound the signal energy content of the intrinsic signatures. The first is a multilevel least squares curve fitting method similar to Haar wavelet. The second is a geodesic distance anisotropic kernel filter.
The methodology testing is carried out on 3D-gestures for obstetrics training. The study quantitatively assess the process of skill acquisition and transfer of manipulating obstetric forceps gestures. The results show that the multi-scale correlations with intrinsic signatures track and evaluate gesture differences between experts and trainees.
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Mühle, Steffen. "Nanoscale Brownian Dynamics of Semiflexible Biopolymers." Doctoral thesis, 2020. http://hdl.handle.net/21.11130/00-1735-0000-0005-1433-B.
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ARRIGHETTI, Walter. "Mathematical models and methods for Electromagnetism on fractal geometries." Doctoral thesis, 2007. http://hdl.handle.net/11573/1656600.
Full textAbstract:
This work summarizes the research path done by Walter Arrighetti during his three years of Doctorate of Research in Electromagnetism at Università degli Studi di Roma “La Sapienza,” Rome, under the guidance of Professor Giorgio Gerosa. This work was mainly motivated by the struggle to find simpler and simpler models to introduce complex geometries (like fractal ones, for example, which are complicated but far from being ‘irregular’) in physical field theories like the Classical Electrodynamics, and which stand at the base of most contemporary applied research activities: from antennas (of any sizes, bandwidths and operational distances) to waveguides & resonators (for devices ranging from IC motherboards , to high-speed fibre channel links), to magnetic resonance (RMI) devices (for both diagnostic and research purposes), all the way up to particle accelerators. All of these models need not only a solid physical base, but also a specifically crafted ensemble of mathematical methods, in order to tackle with problems which “standard-geometry” models (both in the continuum and the discrete cases) are not best-suited for. During his previous years of study towards the Laurea degree in Electronic Engineering, the author used different approaches toward Fractal Electrodynamics, form purely-analytical, to computer-assisted numerical simulations of applied electromagnetic structures (both radiating and wave-guiding), down to algebraic-topological ones. The latter approaches, more often than not, proved to be the best way to start with, because the author found out that self-similarity (a property which many complicated geometries —even non-fractal ones— seem to, at least, tend to possess) can be easily interpreted as a topological symmetry, wonderfully described using “ad hoc” nontrivial algebraic languages. Whatever can be successfully described in the language of Algebra (either via numbers, symmetry groups, graphs, polynomials, etc.) is then always simplified (or “quotiented” — so to speak in a more strict mathematical language) and, when numerical computation takes the way towards the solution of a specific applied problem, those simplifications turn in handy to reduce the complexity of it. For example, the strict self-similarity possessed by some fractals (like those generated via an Iterated Function System — or IFS) allows to numerically store the geometrical data for a fractal object in a sequence of simpler and simpler data which are, for example, instantly recovered by a computer starting from the simplest data (like simplices, squares/cubes, circles/spheres and regular polygons/polytopes). For the same reason, all the physical properties that depend on the geometry (or the topology — i.e. basically the number of “holes” or inner connections) of the domain can be reduced, estimated or be even completely known a priori, even before a numerical simulation is performed. In this work, several of these methods (coming from apparently different branches of pure and applied Mathematics) are presented and finally joined with Electromagnetism equations to solve some more or less applied problems. Since many of the mathematical tools used to build the studied models and methods are advanced and generally not sufficiently known to experts in either such different fields, the first two Chapters are devoted to a brief introduction of some purely mathematical topics. In that context, the author found that the best way to accomplish this was to re-write all those different results from different branches of both pure and applied Mathematics in a formalism as more solid and unified as possible, with continuous links back and forth to different topics (and to the next more applied Chapters). That approach is seldom found in most graduate-level texts. For example, very similar mathematical objects may be even called or classified in different ways, according to the different mathematical contexts they are introduced in, which is exactly the opposite philosophy which has guided underneath in writing these first Chapters. On the other end, simpler and more trivial mathematical definitions, formalisms or electromagnetic problems, when not elsewhere referenced to, can be found in [9], Arrighetti W., Analisi di Strutture Elettromagnetiche Frattali, the author’s Laurea degree dissertation (currently only in Italian language). The most original part of the work is in the last three Chapters where —always using the same “language” and helping with cross-links, as well as to the Bibliography— methods are introduced and then applied to model some electromagnetic problems (previously either unsolved — or already-known, but here solved with a different, usually simpler, or at least more elegant approach).
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Anarella, Mateo. "Control de líneas con movimiento infinitesimalmente helicoidal de paso fijo." Bachelor's thesis, 2020. http://hdl.handle.net/11086/15371.
Full textAbstract:
Tesis (Lic. en Matemática)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía, Física y Computación, 2020.Sea L el espacio de líneas orientadas de R^3 o del espacio hiperbólico H^3. Se estudia la controlabilidad del sistema de control en L dado por la condición de que una curva de líneas orientadas describa en cada instante, a nivel infinitesimal, un helicoide de paso prefijado. El sistema se describe con precisión como cierto subfibrado de TL sobre L (que no es topológicamente trivial) y resulta controlable salvo en el caso euclídeo con helicoide plano (paso infinito).
Let L be the space of oriented lines of R^3 or hyperbolic space H^3. We study the controllability of the control system in L given by the condition that a curve of oriented lines describes at each instant, at the infinitesimal level, a helicoid with prescribed pitch. The system is defined precisely as a certain subbundle of TL over L (which is not topologically trivial) and turns out to be controllable except in the Euclidean case with flat helicoid (infinite pitch).
Fil: Anarella, Mateo. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina.
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Moas, Ruth Paola. "La energía de las secciones unitarias normales de la grassmanniana asociadas a productos cruz." Doctoral thesis, 2020. http://hdl.handle.net/11086/19767.
Full textAbstract:
Tesis (Doctor en Matemática)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía, Física y Computación, 2020.Sea G(k,n) la grassmanniana de subespacios orientados de Rn de dimensión k con su métrica riemanniana canónica. Estudiamos la energía de funciones que asignan a cada P en G(k,n) un vector unitario normal a P. Son secciones de un fibrado esférico E(k,n) sobre G(k,n). Los productos cruz doble y triple octoniónicos inducen de manera natural secciones de este tipo para k=2, n=7 y k=3, n=8, respectivamente. Probamos que son aplicaciones armónicas en E(k,n) munido de la métrica de Sasaki. Esto, junto con el resultado bien conocido de que los campos vectoriales de Hopf en esferas de dimensión impar son aplicaciones armónicas en su fibrado tangente unitario, nos permite concluir que todas las secciones normales unitarias de las grassmannianas asociadas a productos cruz son aplicaciones armónicas. También mostramos que estos fibrados esféricos no poseen secciones paralelas, que trivialmente habrían tenido energía mínima. En una segunda instancia analizamos la energía de aplicaciones que asignan a cada P en G(2,8) una estructura compleja ortogonal J(P) en el subespacio ortogonal a P. Estas asignaciones son secciones del subfibrado esférico unitario del fibrado vectorial sobre P en G(2,8) cuya fibra en cada P consiste esencialmente de las transformaciones antisimétricas del subespacio ortogonal a P. Probamos que la sección naturalmente inducida por el producto cruz triple octoniónico es una aplicación armónica. Comentamos la relación con la armonicidad de la estructura casi compleja canónica de la esfera de dimensión 6.
Let G(k,n) be the Grassmannian of oriented subspaces of Rn of dimension k with its canonical symmetric Riemannian metric. We study the energy of maps assigning a unit vector normal to P to each P in G(k,n) . They are sections of a sphere bundle E(k,n) over G(k,n). The octonionic double and triple cross products induce in a natural way such sections for k=2, n=7 and k=3, n=8, respectively. We prove that they are harmonic maps into E(k,n) endowed with the Sasaki metric. This, together with the well-known result that Hopf vector fields on odd dimensional spheres are harmonic maps into their unit tangent bundles, allows us to conclude that all unit normal sections of the Grassmannians associated with cross products are harmonic. We also show that these sphere bundles do not have parallel sections, which trivially would have had minimum energy. In a second instance we analyze the energy of maps assigning an orthogonal complex structure J(P) on P to each P in G(2,8). They are sections of the unit sphere bundle over G(2,8) whose fiber at each P consists essentially of the skewsymmetric transformations on P?. We prove that the section naturally induced by the octonionic triple product is a harmonic map. We comment on the relationship with the harmonicity of the canonical almost complex structure of the sphere of dimension 6.
publishedVersion
Fil: Moas, Ruth Paola. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina.