Made available in DSpace on 2015-05-14T13:21:10Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 422678 bytes, checksum: a7a607df8d67afa93aa6137919ecb1f5 (MD5) Previous issue date: 2011-08-17 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The present work discusses a study of extactic curves in the projective plane, providing a method that guarantees the existence of first integrals for certain vector fields. To achieve this goal, this study covers the following topics: vector fields, first integrals (with the main result presented in Jouanolou's Theorem), holomorphic foliations (in particular, foliations on the projective plane) and algebraic solutions (where the main result is the well-known theorem of Darboux, which guarantees the existence of rational first integrals for algebraic foliations on the projective plane). / O presente trabalho aborda um estudo das curvas estáticas no plano projetivo, proporcionando um método que garante a existência de integrais primeiras para certos campos vetorias. Para atingir tal objetivo, o presente estudo abrange os seguintes tópicos: Campos Vetoriais, Integrais Primeiras (tendo como principal resultado apresentado o Teorema de Jouanolou), Folheações Holomorfas (em particular, folheações no plano projetivo) e as Soluções Algébricas (onde o principal resultado é o conhecido teorema de Darboux, que garante a existência de integrais primeiras racionais para folheações algébricas no plano projetivo).

We generalize the notions of structural stability and hyperbolicity for the family of (multivalued) complex maps Hc(z) = zr + c; where r > 1 is rational and zr = exp r log z: We discovered that Hc is structurally stable at every hyperbolic parameter satisfying the escaping condition. Surprisingly, there may be infinitely many attracting periodic points for Hc. The set of such points gives rise to the dual Julia set, which is a Cantor set coming from a Conformal Iterated Funcion System. Both the Julia set and its dual are projections of holomorphic motions of dynamical systems (single valued maps) defined on compact subsets of Banach spaces, denoted by Xc and Wc, respectively. For c close to zero: (1) we show that Jc is a union of quasiconformal arcs around the unit circle; (2) the set Xc is an holomorphic motion of the solenoid X0; (3) using the formalism of Gibbs states we exhibit an upper bound for the Hausdorff dimension of Jc; which implies that Jc has zero Lebesgue measure. / Generalizamos as noções de estabilidade estrutural e hiperbolicidade para a família de correspondências holomorfas Hc(z) = zr + c; onde r > 1 é racional e zr = exp r log z: Descobrimos que Hc é estruturalmente estável em todos os parâmetros hiperbólicos satisfazendo a condição de fuga. Tipicamente Hc possui infinitos pontos periódicos atratores, fato totalmente inesperado, uma vez que este número é sempre finito para aplicações racionais. O conjunto de tais pontos dá origem ao chamado conjunto de Julia dual, que é um conjunto de Cantor proveniente de um Conformal Iterated Function System. Tanto o conjunto de Julia e quanto seu dual são projeções de movimentos holomorfos de sistemas definidos em subconjuntos compactos denotados por Xc e Wc; respectivamente de um espaço de Banach. Para todo c próximo de zero: (1) mostramos que Jc é reunião de arcos quase-conformes próximos do círculo unitário; (2) o conjunto Xc é um movimento holomorfo do solenóide X0; (3) utilizando o formalismo dos estados de Gibbs, exibimos um limitante superior para a dimensão de Hausdorff de Jc. Consequentemente, Jc possui medida de Lebesgue nula.

L'objet de cette thèse est la construction d'objets géométriques sur une variété C paramétrant des courbes rationnelles dans l'espace des twisteurs d'une variété hyperkählérienne. On établira une correspondance entre la géométrie complexe de l'espace des twisteurs et des propriétés différentielles sur C (opérateurs différentiels et courbure de la structure riemanienne complexe héritée de la variété hyperkählérienne). Les premiers chapitres précisent le cadre et les résultats connus. Dans les chapitres 4, 5 et 6 on établit une équivalence de catégories entre fibrés triviaux en restriction à chaque droite de l'espace des twisteurs et les fibrés à connexion sur C satisfaisant une condition de courbure. Le chapitre 7 prolonge cette correspondance sur le plan cohomologique tandis que le chapitre 8 en fait l'étude infinitésimale en reliant la courbure de la connexion avec les épaississements infinitésimaux des fibrés le long des droites. / The purpose of this thesis is to construct geometric objects on a manifold C parametrizing rational curves in the twistor space of a hyperkähler manifold. We shall establish a correspondence between the complex geometry of the twistor space and some differential properties of C (differential operators and curvature of a complex riemannian structure inherited from the base hyperkähler manifold). The first chapters gather some classical results of the theory of hyperkähler manifolds and their twistor spaces. In the chapters 4, 5 and 6, we construct an equivalence of categories between bundles on the twistor space which are trivial on each line and bundles with a connexion of C satisfying certain curvature conditions. The chapter 7 extends this correspondence on the cohomological level whereas the chapter 8 explores its infinitesimal version ; it links curvature of the connexion with thickening (in the sense of LeBrun) of the bundle along the lines.

We give a proof of the Oka-Weil theorem which states that on compact, polynomially convex subsets of Cn, holomorphic functions can be approximated uniformly by holomorphic polynomials. / Vi ger ett bevis av Oka-Weil sats som säger att på kompakta och polynomkonvexa delmängder av Cn kan holomorfa funktioner approximeras likformigt med holomorfa polynom.

Durante o desenvolvimento de holometálobos alguns tecidos são eliminados/remodelados durante a metamorfose. A autofagia age nesse processo degradando componentes citoplasmáticos, inicialmente isolando-os em dupla membrana, estrutura chamada autofagossomo e esses conteúdos são degradados por hidrolases lisossomais. Porém, aspectos apoptóticos podem estar presentes nesse processo, como o envolvimento de caspases e a fragmentação nuclear. Alterações morfológicas na glândula salivar e no corpo gorduroso, que são bons exemplos de órgãos que sofrem morte celular programada (MCP) no desenvolvimento de R. americana, foram analisados por microscopia de luz e eletrônica. Durante a remoção desses órgãos, núcleos apresentam morfologia condensada e com fragmentação confirmada por TUNEL. Ambos tecidos mostraram formação de autofagossomos, mas a glândula salivar completa o processo de MCP durante a metamorfose. Genes antiapoptóticos e autofágicos que têm importante papel na MCP foram caracterizados. MCP em R. americana apresenta cooperação de aspetos da autofagia e da apoptose. / In the development of holometabolous insects, some tissues are eliminated/remodelated during metamorphosis. Autophagy acts in this process by degrading cytoplasm contents, initially by surrounding them within a double membrane, a structure called autophagosome and its contents are degraded by lysosomal hydrolases. However, some features of apoptotic cell death may be present in this process, such as the involvement of caspases and nuclear fragmentation. Morphological changes of salivary gland and fat body, good examples of organs that suffer programmed cell death (PCD) during R. americana development, were analyzed by light and electron microscopy. During the removal of these organs, nuclei present fragmented and condensed morphology, confirmed by TUNEL assay. Both tissues show the formation of autophagosomes, but the salivary gland completes the process of PCD during metamorphosis. Antiapoptotic and autophagic genes that play important function in the PCD, were characterized. R. americana PCD occurs with the cooperation of autophagy and apoptosis features.

Cette thèse est consacrée à l'étude de quelques problèmes en dynamique holomorphe discrete et continue à l'aide de la Théorie du Pluripotentiel. Le premier problème présenté concerne la description des paires d'endomorphismes holomorphes permutables du plan projectif complexe qui ne partagent pas une itérée. Nous nous intéressons au cas où les degrés des deux applications coïncident après un certain nombre d'itérations. Nous montrons que telles applications sont des exemples de Lattès ou bien des relèvements des exemples de Lattès unidimensionnels. Combiné avec un théorème de T.-C. Dinh et N. Sibony ce résultat complète la classification des paires permutables en dimension deux. Ensuite, nous nous intéressons à la dynamique des laminations par variétés complexes. Nous montrons que, dans une variété kählérienne compacte, le carré de la classe de cohomologie d'un cycle feuilleté dirigé par une lamination transversalement Lipschitz est toujours zéro. Parmi les conséquences nous montrons que l'espace projectif complexe $\pr^{n}$ n'admet pas de cycle feuilleté transversalement Lipschitz de dimension $q \leq \frac{n}{2}$. Cela généralise un résultat de J.E. Forn\ae ss et N. Sibony. Dans la dernière partie nous étudions les mesures de Monge-Ampère à potentiel höldérien. Nous montrons que ces mesures satisfont un analogue d'un théorème de H. Skoda concernant l'intégrabilité exponentielle d'une fonction plurisousharmonique en termes de ses nombres de Lelong. Ce résultat peut être vu comme une très forte compacité pour les fonctions plurisousharmoniques qui sont eux-mêmes un outil fondamental en dynamique holomorphe. / This thesis is devoted to the study of some problems in discrete and continuous holomorphic dynamics with the tools of Pluripotential Theory. The first problem we consider involves the description of commuting pairs of holomorphic endomorphisms of the complex projective plane that do not share an iterate. We consider the case when their degrees coincide after some number of iterations. We show that these maps are either Lattès maps or lifts of one-dimensional Lattès maps. Together with a theorem of T.-C. Dinh and N. Sibony this result completes the classification of commuting pairs in dimension two. Later on, we turn our attention to the dynamics of laminations by complex manifolds. We show that, on a compact Kähler manifold, the square of the cohomology class of a foliated cycle directed by a transversally Lipschitz lamination is always zero. As a corollary we show that the complex projective space $\pr^n$ do not carry any transversally Lipschitz foliated cycle of dimension $q \leq \frac{n}{2}$, generalizing a result by J.E. Forn\ae ss and N. Sibony. In the last part we study Monge-Ampère measures with Hölder continuous potential. We show that these measures satisfy an analogue of a theorem of H. Skoda concerning the exponential integrability of plurisubharmonic functions in terms of its Lelong numbers. This result can be viewed as a strong compactness property of plurisubharmonic functions, a class of functions of fundamental importance in holomorphic dynamics.

We study the equivariant cohomology of moduli spaces of objects in the derived category of elliptic surfaces in order to find new examples of infinite dimensional quantum integrable systems and their geometric representation theoretic interpretation in enumerative geometry. This problem is related to a program to understand the cohomological and K-theoretic Hall algebras of holomorphic symplectic surfaces and to understand how it related to the Donaldson-Thomas theory of threefolds fibered in those surfaces. We use the theory of noncommutative deformations of Poisson surfaces and especially van den Berg’s noncommutative P1 bundles as well as Rains’s analysis of moduli theory for quasi-ruled noncommutative surfaces as well as the theory of Bridgeland stability conditions and their relative versions to understand equivariant deformations and birational transformations of Hilbert schemes of points on equivariant elliptic surfaces. The moduli spaces are closely related to elliptic versions of classical integrable systems. We also use these moduli spaces to construct vertex algebra representations of extensions of toroidal extended affine algebras on their equivariant cohomology, building on work of Eswara-Rao–Moody–Yokonuma, of Billig, and of Chen–Li–Tan on vertex representations of toroidal algebras, full toroidal algebras, and toroidal extended affine algebras. Using Fourier-Mukai transforms and their relative action on families of dg-categories we study the relationship between automorphisms of toroidal extended affine algebras and families of derived equivalences on dg categories, in particular finding a relativistic (difference) generalization of the Laumon-Rothstein deformation of the Fourier-Mukai duality. Finally, using the above analysis we extend the construction of Maulik–Okounkov’s stable envelopes to moduli of framed torsionfree sheaves on noncommutative surfaces in some cases and use this to study coproducts on associated algebras assigned to elliptic surfaces with applications to understanding new representation theoretic structures in the Donaldson-Thomas theory of local curves.

Dans cette thèse, nous étudions des problèmes aux limites pour les systèmes elliptiques sous forme divergence avec coefficients complexes dans L^{infty}. Nous prouvons des estimations a priori, discutons de la solvabilité et d'extrapolation de la solvabilité. Nous utilisons une transformation via des équations Cauchy-Riemann généralisées due à P. Auscher, A. Axelsson et A. McIntosh. On peut résoudre les équations Cauchy-Riemann généralisées via la semi-groupe engendré par un opérateur différentiel perturbé d'ordre un de type Dirac. A l'aide du semi-groupe, nous étudions la théorie L^{p} avec une discussion sur la bisectorialité, le calcul fonctionnel holomorphe et les estimations hors-diagonales pour des opérateurs dans le calcul fonctionnel. En particulier, nous développons une théorie L^{p}-L^{q} pour des opérateurs dans le calcul fonctionnel d'opérateur de type Dirac perturbé. Les problèmes de Neumann, Régularité et Dirichlet se formulent avec des estimations quadratiques et des estimations pour la fonction maximale nontangentielle. Cela conduit à à démontrer de telles estimations pour le semi-groupe d'opérateur de Dirac Pour cela, nous utilisons les espaces Hardy associés et les identifions dans certains cas avec des sous-espaces des espaces de Hardy et Lebesgue classiques. Nous obtenons enfin des estimations a priori pour les problème aux limites via une extension utilisant des espaces de Sobolev associés. Nous utilisons les estimations a priori pour une discussion sur la solvabilité des problèmes aux limites et montrer un théorème d'extrapolation de la solvabilité. / In this this thesis we study boundary value problems for elliptic systems in divergence form with complex coefficients in L^{\infty}. We prove a priori estimates, discuss solvability and extrapolation of solvability. We use a transformation to generalized Cauchy-Riemann equations due to P. Auscher, A. Axelsson, and A. McIntosh. The generalized Cauchy-Riemann equations can be solved by the semi-group generated by a perturbed first order Dirac/differential operator. In relation to semi-group theory we setup the L^p theory by a discussion of bisectoriality, holomorphic functional calculus and off-diagonal estimates for operators in the functional calculus. In particular, we develop an L^p-L^q theory for operators in the functional calculus of the first order perturbed Dirac/differential operators. The formulation of Neumann, Regularity and Dirichlet problems involve square function estimates and nontangential maximal function estimates. This leads us to discuss square function estimates and nontangential maximal function estimates involving operators in the functional calculus of the perturbed first order Dirac/differential operator. We discuss the related Hardy spaces associated to operators and prove identifications by subspaces of classical Hardy and Lebesgue spaces. We obtain the a priori estimates by an extension of the square function estimates and nontangential maximal function estimates to Sobolev spaces associated to operators. We use the a priori estimates for a discussion of solvability and extrapolation of solvability.

Consideramos o fluxo de Reeb associado a uma forma de contato em S¹ x S² que induz a estrutura de contato tight. Assumimos que o fluxo admite um par de órbitas periódicas L0 e L1 cujo link L = L0 L1 é transversalmente isotópico a ( S¹ x )( S¹ x ), em que n = (0,0,1) e s = (0,0,1) são os pólos norte e sul de S², respectivamente. O objetivo é provar que, nestas condições, existem infinitas órbitas periódicas no complementar desse link cujas classes de homotopia no complementar do link são prescritas de acordo com os números de rotação de L0 e L1. / We consider the Reeb flow associated to a contact form on S¹ x S² which induces a tight contact structure. We assume that the flow admits a pair of closed orbits L0 and L1 whose link L = L0 L1 is transversely isotopic to (S¹ x)(S¹ x), where n = (0,0,1) and s =(0,0,1) are the north and south poles of S², respectively. The main goal is to prove that, under these conditions, there exit infinitely many closed orbits in the complement of this link whose homotopy classes in the complement of this link are prescribed according to the rotation numbers of L0 and L1.

Neste trabalho, estudamos a dinâmica de Reeb associada a uma forma de contato $\\lambda$ definida numa 3-variedade compacta e conexa M. Assumimos que $\\lambda$ é tight e a primeira classe de Chern da estrutura de contato $\\xi=\\ker\\lambda$ se anula sobre $\\pi_2(M)$. No nosso primeiro resultado, supomos que M é fechada e existe uma órbita fechada L do fluxo de Reeb que é um p-nó trivial com número de auto-enlaçamento $-1/p$. Supomos, além disso, que o número de rotação transversal da p-ésima iterada de L é estritamente menor do que 1. Nestas condições, provamos que existe uma órbita fechada (de Reeb) contrátil geometricamente distinta de L e não-enlaçada em L cujo número de rotação transversal é 1. Apresentamos também uma versão deste resultado para o caso em que M é uma 3-variedade cujo bordo é difeomorfo a um toro e invariante pelo fluxo de Reeb e não existem órbitas fechadas contidas no bordo. Nosso segundo resultado é uma caracterização dinâmica do toro sólido. Seja $\\lambda$ uma forma de contato não-degenerada definida em uma 3-variedade M cujo bordo é difeomorfo a um toro e invariante pelo fluxo de Reeb. Se o fluxo de Reeb satisfaz certas hipóteses de torção sobre o bordo, então ou existe uma órbita fechada contrátil com índice de Conley-Zehnder 2 ou M é folheada por discos transversais ao campo de Reeb. Neste último caso, M é difeomorfa a um toro sólido e existe uma órbita fechada não-contrátil em M que é ponto fixo da aplicação de retorno induzida pela folheação. / In this work, we study the Reeb dynamics associated to a tight contact form $\\lambda$ defined on a compact, connected 3-manifold M. Suppose that the first Chern class of $\\xi=\\ker\\lambda$ vanish on $\\pi_2(M)$. In our first result, we assume that M is closed and there exists a closed Reeb orbit L which is a p-unknotted, has self-linking number $-1/p$ and the transverse rotation number of the p-th iterate of L is less than 1. Under these conditions, we verify that there exists a contractible closed Reeb orbit which is geometrically distinct from L and not linked to L with transverse rotation number 1. We also prove a version of this result when M is a compact 3-manifold M whose boundary is diffeomorphic to a torus and invariant by the flow and, moreover, there does not exist closed Reeb orbits on the boundary. Our second result is a dynamical characterization of the solid torus. We assume that $\\lambda$ is a contact form on a compact 3-manifold M whose boundary is diffeomorphic to a torus. Under the hypothesis of $\\lambda$ being non-degenerate, if the flow is tangent to $\\partial M$ and satisfies some twist conditions on the boundary, then either there exists a contractible closed Reeb orbit which has Conley-Zehnder index 2 or M is foliated by disks transverse to the Reeb flow. In this last case, we see that M is diffeomorphic to a solid torus and there exists a non-contractible closed Reeb orbit M which is a fixed point of the return map induced by the foliation.