The influence of Chow groups on singular cohomology is motivated by classical results by Mumford and Roitman and has been investigated extensively. On the other hand, the converse influence is rather conjectural and it takes place in the framework of the ``philosophy of mixed motives'', which is mainly due to Grothendieck, Bloch and Beilinson. In the spirit of exploring this influence, Voisin formulated in 1996 a conjecture on 0--cycles on the self--product of surfaces of geometric genus one. There are few examples in which Voisin's conjecture has been verified, but it is still open for a general $K3$ surface. Our aim is to present a new example in which Voisin's conjecture is true, a family of Todorov surfaces. We give an explicit description of the family as quotient of complete intersection of four quadrics in $mathbb{P}^{6}$. We verify Voisin's conjecture for the family of Todorov surfaces of type $(2,12)$. Our main tool is Voisin's ``spreading of cycles'', we use it to establish a relation between 0--cycles on the Todorov surface and on the associated K3 surface. We give a motivic version of this result and some interesting motivic applications.

In the present work, we analyse the categories of mixed Hodge complexes and mixed Hodge diagrams of differential graded algebras in these two directions: we prove the existence of both a Cartan-Eilenberg structure, via the construction of cofibrant minimal models, and a cohomological descent structure. This allows to interpret the results of Deligne, Beilinson, Morgan and Navarro within a common homotopical framework. In the additive context of mixed Hodge complexes we recover Beilinson's results. In our study we go a little further and show that the homotopy category of mixed Hodge complexes, and the derived category of mixed Hodge structures are equivalent to a third category whose objects are graded mixed Hodge structures and whose morphisms are certain homotopy classes, which are easier to manipulate. In particular, we obtain a description of the morphisms in the homotopy category in terms of morphisms and extensions of mixed Hodge structures, and recover the results of Carlson [Car80] in this area. As for the multiplicative analogue, we show that every mixed Hodge diagram can be represented by a mixed Hodge algebra which is Sullivan minimal, and establish a multiplicative version of Beilinson's Theorem. This provides an alternative to Morgan's construction. The main difference between the two approaches is that Morgan uses ad hoc constructions of models à la Sullivan, specially designed for mixed Hodge theory, while we follow the line of Quillen's model categories or Cartan-Eilenberg categories, in which the main results are expressed in terms of equivalences of homotopy categories, and the existence of certain derived functors. In particular, we obtain not only a description of mixed Hodge diagrams in terms of Sullivan minimal algebras, but we also have a description of the morphisms in the homotopy category in terms of certain homotopy classes, parallel to the additive case. In addition, our approach generalizes to broader settings, such as the study of compactificable analytic spaces, for which the Hodge and weight filtrations can be defined, but do not satisfy the properties of mixed Hodge theory. Combining these results with Navarro's functorial construction of mixed Hodge diagrams, and using the cohomological descent structure defined via the Thom-Whitney simple, we obtain a more precise and alternative proof of that the rational homotopy type, and the rational homotopy groups of every simply connected complex algebraic variety inherit functorial mixed Hodge structures. As an application, and extending the Formality Theorem of Deligne-Griffiths-Morgan-Sullivan for compact Kähler varieties and the results of Morgan for open smooth varieties, we prove that every simply connected complex algebraic variety (possibly open and singular) and every morphism between such varieties is filtered formal: its rational homotopy type is entirely determined by the first term of the spectral sequence associated with the multiplicative weight filtration. / En aquest treball, analitzem les categories de complexos de Hodge mixtos i de diagrames de Hodge d'àlgebres diferencials graduades en aquestes dues direccions: provem l'existència d'una estructura de Cartan-Eilenberg, via la construcció de models cofibrants minimals, i d'una estructura de descens cohomològic. Aquest estudi permet interpretar els resultats de Deligne, Beilinson, Morgan i Navarro en un marc homotòpic comú.

霍奇排名是在近幾年才運用在排名的一種方法。在大多數現在的資料庫 中,資料庫很龐大,有些甚至會需要網路連結,而且很多會有資料不完整或 是資料不平衡的狀況。我們選擇用霍奇排名這種排名方法來處理可能會遇到 的這些困擾。 這篇論文主要目的是想用運用基本的線性代數來研究霍奇排名和推導組合霍奇理論。 / HodgeRank is a method of ranking that is new in recent years. In most of modern datasets, the amount of data is very large, some also need the internet connection, and plenty of them have the feature that incomplete or imbalanced. We use the method of HodgeRank to deal with these difficulties. This thesis is primary using elementary linear algebra to survey HodgeRank and deduce the combinatorial Hodge Theorem.

Cette thèse présente une nouvelle classe de schémas de discrétisation spatiale sur maillages polyédriques, nommée Compatible Discrete Operator (CDO) et en étudie l'application aux équations elliptiques et de Stokes. La préservation au niveau discret des caractéristiques essentielles du système continu sert de fil conducteur à la construction des opérateurs. Les opérateurs de de Rham définissent les degrés de liberté en accord avec la nature physique des champs à discrétiser. Les équations sont décomposées de manière à différencier les relations topologiques (lois de conservation) des relations constitutives (lois de fermeture).Les relations topologiques sont associées à des opérateurs différentiels discrets et les relations constitutives à des opérateurs de Hodge discrets. Une particularité de l'approche CDO est l'utilisation explicite d'un second maillage, dit dual, pour bâtir l'opérateur de Hodge discret. Deux familles de schémas CDO sont ainsi considérées : les schémas vertex-based lorsque le potentiel est discrétisé aux sommets du maillage (primal), et les schémas cell-based lorsque le potentiel est discrétisé aux sommets du maillage dual (les sommets duaux étant en bijection avec les cellules primales).Les schémas CDO associés à ces deux familles sont présentés et leur convergence est analysée. Une première analyse s'appuie sur une définition algébrique de l'opérateur de Hodge discret et permet d'identifier trois propriétés clés : symétrie, stabilité et $mathbb{P}_0$-consistance. Une seconde analyse s'appuie sur une définition de l'opérateur de Hodge discret à l'aide d'opérateurs de reconstruction pour lesquels sont identifiées les propriétés à satisfaire. Par ailleurs, les schémas CDO fournissent une vision unifiée d'une large gamme de schémas de la littérature (éléments finis, volumes finis, schémas mimétiques…).Enfin, la validité et l'efficacité de l'approche CDO sont illustrées sur divers cas tests et plusieurs maillages polyédriques / This thesis presents a new class of spatial discretization schemes on polyhedral meshes, called Compatible Discrete Operator (CDO) schemes and their application to elliptic and Stokes equations. In CDO schemes, preserving the structural properties of the continuous equations is the leading principle to design the discrete operators. De Rham maps define the degrees of freedom according to the physical nature of fields to discretize. CDO schemes operate a clear separation between topological relations (balance equations) and constitutive relations (closure laws).Topological relations are related to discrete differential operators, and constitutive relations to discrete Hodge operators. A feature of CDO schemes is the explicit use of a second mesh, called dual mesh, to build the discrete Hodge operator. Two families of CDO schemes are considered: vertex-based schemes where the potential is located at (primal) mesh vertices, and cell-based schemes where the potential is located at dual mesh vertices (dual vertices being in one-to-one correspondence with primal cells).The CDO schemes related to these two families are presented and their convergence is analyzed. A first analysis hinges on an algebraic definition of the discrete Hodge operator and allows one to identify three key properties: symmetry, stability, and $mathbb{P}_0$-consistency. A second analysis hinges on a definition of the discrete Hodge operator using reconstruction operators, and the requirements on these reconstruction operators are identified. In addition, CDO schemes provide a unified vision on a broad class of schemes proposed in the literature (finite element, finite element, mimetic schemes...).Finally, the reliability and the efficiency of CDO schemes are assessed on various test cases and several polyhedral meshes

Dans cette thèse, nous étudions la géométrie analytique p-adique et la cohomologie l-adique de certains espaces de Rapoport-Zink, en utilisant la théorie des filtrations de Harder-Narasimhan des schémas en groupes finis et plats élaborée par Fargues.Cette thèse se compose de trois parties. La première partie traite de certains espaces de Rapoport-Zink non-basiques, qui satisfont à la condition que leur polygone de Newton et polygone de Hodge ont un point de contact non-trivial, qui est un point de rupture pour le polygone de Newton. Sous cette hypothèse, nous prouvons que ces espaces de Rapoport-Zink peuvent être décomposés en une somme directe d'espaces de modules des types de Rapoport-Zink associés à certains sous-groupes paraboliques appropriés, donc leurs cohomologie l-adique sont des induites paraboliques et en particulier ne contiennent pas de représentations supercuspidales. Nous prouvons ces faits en démontrant d'abord un théorème sur la filtration de Hodge-Newton pour les groupes p-divisibles avec des structures additionelles sur des anneaux de valuation complets de rang un et de caractéristique mixte (0,p).Dans la deuxième partie, nous considérons les espaces de Rapoport-Zink basiques de signature (1,n-1) pour les groupes unitaires associés à l'extension quadratique non ramifiée de Qp. On étudie l'action de Hecke sur ces espaces en détails. En utilisant la théorie des filtrations de Harder-Narasimhan des schémas en groupes finis et plats, et la stratification de Bruhat-Tits de la fibre spéciale réduite Mred étudié par Vollaard-Wedhorn, on trouve un certain domaine analytique compact DK telle que ses itérés dans le groupe G(Qp)×Jb(Qp) forme un recouvrement localement fini de tout l'espace MK. Nous appelons un tel phénomène une décomposition cellulaire localement finie.Dans la troisième partie, nous démontrons une formule de Lefschetz pour ces espaces pour l'action des éléments semi-simples réguliers elliptiques, en tenant compte de l'action de ces éléments sur les cellules et en appliquant le théorème principal de Mieda. De la même manière, nous pouvons aussi reprouver la formule de Lefschetz pour les espaces de Lubin-Tate précédemment obtenue par Strauch et Mieda. Cette formule de Lefschetz devrait caractériser la réalisation de correspondances de Jacquet-Langlands locales pour les groupes unitaires dans la cohomologie l-adique de ces espaces de Rapoport-Zink, dès que certains problèmes correspondants de théorie des représentations auront été résolus. / In this thesis we study p-adic analytic geometry and l-adic cohomology of some Rapoport-Zink spaces, using the theory of Harder-Narasimhan filtration of finite flat group schemes developed by Fargues .This thesis consists of three parts. The first part deals with some non-basic Rapoport-Zink spaces, which satisfy the condition that their Newton polygon and Hodge polygon have a non-trivial contact point, which is a breakpoint for the Newton polygon. Under this hypothesis, we prove these Rapoport-Zink spaces can be decomposed as a direct sum of smaller Rapoport-Zink spaces associated to some suitable parabolic subgroups, thus their l-adic cohomology is parabolically induced and in particular contain no supercuspidal representations. We prove these facts by first proving a theorem about the Hodge-Newton filtration for p-divisible groups with additional structures over complete valuation rings of rank one and mixed characteristic (0,p).In the second part, we consider the basic Rapoport-Zink spaces with signature (1,n-1) for the unitary groups associated to the unramified quadratic extension of Qp. We study the Hecke action on these spaces in details. By using the theory of Harder-Narasimhan filtrations of finite flat group schemes, and the Bruhat-Tits stratification of the reduced special fiber Mred studied by Vollaard-Wedhorn, we find some compact analytic domain DK such that its translates under the group G(Qp)×Jb(Qp) form a locally finite cover of the whole space MK. We call such a phenomenon a locally finite cell decomposition.In the third part we prove a Lefschetz trace formula for these spaces for the action of regular semi-simple elliptic elements, by considering the action of these elements on the cells and applying Mieda's main theorem. In the same way we can also reprove the Lefschetz trace formula for Lubin-Tate spaces as previously obtained by Strauch and by Mieda. This Lefschetz trace formula should characterize the realization of local Jacquet-Langlands correspondences for unitary groups in the l-adic cohomology of these Rapoport-Zink spaces, as soon as some corresponding representation theoretic problems are solved.

[pt] Campos vetoriais representam um papel principal em diversas aplicações científicas. Eles são comumente gerados via simulações computacionais. Essas simulações podem ser um processo custoso, dado que em muitas vezes elas requerem alto tempo computacional. Quando pesquisadores desejam quantificar a incerteza relacionada a esse tipo de aplicação, costuma-se gerar um conjunto de realizações de campos vetoriais, o que torna o processo ainda mais custoso. A Decomposição de Helmholtz-Hodge é uma ferramenta útil para a interpretação de campos vetoriais uma vez que ela distingue componentes conservativos (livre de rotação) de componentes que preservam massa (livre de divergente). No presente trabalho, vamos explorar a aplicabilidade de tal técnica na análise de incerteza de campos vetoriais 2D. Primeiramente, apresentaremos uma abordagem utilizando a Decomposição de Helmholtz-Hodge como uma ferramenta básica na análise de conjuntos de campos vetoriais. Dado um conjunto de campos vetoriais epsilon, obtemos os conjuntos formados pelos componentes livre de rotação, livre de divergente e harmônico, aplicando a Decomposição Natural de Helmholtz- Hodge em cada campo vetorial em epsilon. Com esses conjuntos em mãos, nossa proposta não somente quantifica, por meio de análise estatística, como cada componente é pontualmente correlacionado ao conjunto de campos vetoriais original, como também permite a investigação independente da incerteza relacionado aos campos livre de rotação, livre de divergente e harmônico. Em sequência, propomos duas técnicas que em conjunto com a Decomposição de Helmholtz-Hodge geram, de forma estocástica, campos vetoriais a partir de uma única realização. Por fim, propomos também um método para sintetizar campos vetoriais a partir de um conjunto, utilizando técnicas de Redução de Dimensionalidade e Projeção Inversa. Testamos os métodos propostos tanto em campos sintéticos quanto em campos numericamente simulados. / [en] Vector field plays an essential role in a large range of scientific applications. They are commonly generated through computer simulations. Such simulations may be a costly process because they usually require high computational time. When researchers want to quantify the uncertainty in such kind of applications, usually an ensemble of vector fields realizations are generated, making the process much more expensive. The Helmholtz-Hodge Decomposition is a very useful instrument for vector field interpretation because it traditionally distinguishes conservative (rotational-free) components from mass-preserving (divergence-free) components. In this work, we are going to explore the applicability of such technique on the uncertainty analysis of 2-dimensional vector fields. First, we will present an approach of the use of the Helmholtz-Hodge Decomposition as a basic tool for the analysis of a vector field ensemble. Given a vector field ensemble epsilon, we firstly obtain the corresponding rotational-free, divergence-free and harmonic component ensembles by applying the Natural Helmholtz-Hodge Decomposition to each1 vector field in epsilon. With these ensembles in hand, our proposal not only quantifies, via a statistical analysis, how much each component ensemble is point-wisely correlated to the original vector field ensemble, but it also allows to investigate the uncertainty of rotational-free, divergence-free and harmonic components separately. Then, we propose two techniques that jointly with the Helmholtz-Hodge Decomposition stochastically generate vector fields from a single realization. Finally, we propose a method to synthesize vector fields from an ensemble, using both the Dimension Reduction and Inverse Projection techniques. We test the proposed methods with synthetic vector fields as well as with simulated vector fields.

[pt] A equação diferencial parcial de Poisson é de fundamental importância em várias áreas de pesquisa, dentre elas: matemática, física e engenharia. Para resolvê-la numericamente utilizam-se vários métodos, tais como os já tradicionais métodos das diferenças finitas e dos elementos finitos. Este trabalho propõe um método para resolver a equação de Poisson, utilizando uma abordagem de sistema de partículas conhecido como SPH, do inglês Smoothed Particles Hydrodynamics. O método proposto para a solução da equação de Poisson e os operadores diferenciais discretos definidos no método SPH, chamados de operadores SPH, são utilizados neste trabalho em duas aplicações: na decomposição de campos vetoriais; e na simulação numérica de escoamentos de fluidos monofásicos e bifásicos utilizando a equação de Navier-Stokes. / [en] Poisson`s equation is of fundamental importance in many research areas in engineering and the mathematical and physical sciences. Its numerical solution uses several approaches among them finite differences and finite elements. In this work we propose a method to solve Poisson`s equation using the particle method known as SPH (Smoothed Particle Hydrodynamics). The proposed method together with an accurate analysis of the discrete differential operators defined by SPH are applied in two related situations: the Hodge-Helmholtz vector field decomposition and the numerical simulation of the Navier-Stokes equations.

Orientador: Marcos Benevenuto Jardim / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T16:48:31Z (GMT). No. of bitstreams: 1 Cardoso_NunoFilipedeAndrade_M.pdf: 1858794 bytes, checksum: bbe47182338feb3de60b480df87b52a7 (MD5) Previous issue date: 2014 / Resumo: Os teoremas de degenerescência de Hodge e de anulamento de Kodaira, Akizuki e Nakano são de suma importância na teoria de variedades complexas. Usando o teorema de comparação de Serre, ambos podem ser traduzidos para o contexto de esquemas projetivos e suaves sobre um corpo de característica zero. Para corpos de característica positiva, no entanto, os dois deixam de valer sem hipóteses adicionais, sendo que os primeiros contra-exemplos foram encontrados por Mumford e Raynaud. O objetivo desta dissertação é apresentar um teorema devido a Deligne e Illusie que assegura a degenerescência da seqüência espectral de Hodge-de Rham e uma versão do teorema de Kodaira, Akizuki e Nakano para certos esquemas projetivos e suaves sobre um corpo perfeito de característica positiva. Nos propusemos a dar um tratamento, na medida do possível, auto-suficiente / Abstract: The Hodge degeneration theorem and the Kodaira, Akizuki and Nakano's vanishing theorem are of paramount importance in the theory of complex manifolds. Using Serre's comparison theorem, both can be translated to the context of smooth projective schemes over a field of characteristic zero. For fields of positive characteristic, however, both fail to hold without additional hypothesis, and the first counterexamples were found by Mumford and Raynaud. Our goal in this dissertation is to present a theorem due to Deligne and Illusie that ensures the degeneration of the Hodge-de Rham spectral sequence and a version of the theorem of Kodaira, Akizuki and Nakano for certain smooth projective schemes over a perfect field of positive characteristic. We tried to keep the treatment as self-contained as possible / Mestrado / Matematica / Mestre em Matemática

L’objectif de cette thèse est l’étude de la fonction zêta des hauteurs motivique associée à un problème de comptage de courbes sur les compactifications équivariantes d’espaces affines, résolvant au chapitre 6 l’analogue motivique de la conjecture de Manin pour celles-ci. La fonction zêta des hauteurs provenant du problème de comptage considéré est récrite convenablement à l’aide d'une formule de Poisson motivique démontrée au cinquième chapitre, qui généralise celle de Hrushovski-Kazhdan. Chaque terme est alors décomposé sous la forme d'un produit eulérien motivique, dont la définition et les propriétés sont établies au chapitre 3. La convergence de ces produits eulériens doit être comprise pour une topologie des poids que nous introduisons au quatrième chapitre et qui repose d'une part sur la théorie des modules de Hodge de Saito, et d'autre part sur une mesure motivique sur l’anneau de Grothendieck des variétés avec exponentielles, construite dans le chapitre 2 à l’aide de la notion de cycles évanescents motiviques. On en déduit ainsi une description de l'asymptotique d'une proportion positive des coefficients du polynôme de Hodge-Deligne des espaces de modules des courbes sur la compactification équivariante donnée, lorsque le degré tend vers l'infini. / The goal of this thesis is the study of the motivic height zeta function associated to the problem of counting curves on equivariant compactifications of vector groups, solving in chapter 6 the motivic analogue of Manin's conjecture for such varieties.The motivic height zeta function coming from this counting problem is rewritten in a convenient way using a Poisson summation formula proved in chapter 5, and which generalises Hrushovski and Kazhdan's motivic Poisson formula. Each term is then expressed as a motivic Euler product, the definition and properties of the latter being established in chapter 3. The convergence of these Euler products must be understood for a weight topology which we introduce in the fourth chapter and which relies both on Saito's theory of mixed Hodge modules and on a motivic measure on the Grothendieck ring of varieties with exponentials, constructed in chapter 2 using the notion of motivic vanishing cycles. We deduce from this a description of the asymptotic of a positive proportion of the coefficients of the Hodge-Deligne polynomial of the moduli spaces of curves on the given equivariant compactification, when the degree goes to infinity.

In dieser Arbeit wird die de-Rham-Kohomologie für die Quantengruppen zu den vier klassischen Serien von Lie-Gruppen bestimmt und es wird der Hodgeschen Zerlegungssatz gezeigt. Als entscheidendes Mittel wurde der Laplace-Beltrami-Operator L für Woronowicz’ äußere Algebren entwickelt. Für transzendente Werte von q und reguläre Kalkülparameter z ist L diagonalisierbar. Für die obigen Quantengruppen bestimmen wir die Eigenwerte von L, die neben q und z von zwei integralen dominanten Gewichten abhängen. Wie im klassischen Fall wird die de-Rham-Kohomologie durch harmonische Formen repräsentiert. Jedoch entspricht nur im Fall der A-Serie jeder harmonischen Form auch eine de-Rham-Kohomologieklasse. Im Falle der B-, C- und D-Serien sind biinvariante Formen nicht notwendig geschlossen. Es gilt aber, dass jede biinvariante Form harmonisch ist. Das zweite Hauptresultat ist die Hodge-Zerlegung für die Quantengruppen GLq(N) und SLq(N): Ist der Kalkülparameter z regulär, so lässt sich jede Form eindeutig zerlegen in die Summe aus einem Rand, einem Korand und einem Kohomologierepräsentanten. Ferner gilt, analog zum klassischen Fall, dass die folgenden drei Formenräume übereinstimmen: die biinvarianten Formen, die harmonischen Formen und die de-Rham-Kohomologie. Für die orthogonalen und symplektischen Quantengruppen gibt es keine vollständige Hodge-Zerlegung. Nur für die Elemente, die im Bild des Laplace-Beltrami-Operators liegen, gibt es eine eindeutige Zerlegung in Rand und Korand. Für die Standardkalküle auf den Quantengruppen GLq(N) und SLq(N) wird die Größe von Woronowicz’ äußerer Algebra bestimmt. Es wird gezeigt, dass der Raum der linksinvarianten k-Formen (N² über k)-dimensional ist. Die Algebra der biinvarianten Formen ist graduiert kommutativ. Ihre Poincaré-Reihe ist (1+t)(1+t³) ... (1+t^(2N-1)). Biinvariante Formen sind geschlossen. / Consider one of the standard bicovariant first order differential calculi for the quantum groups GLq(N), SLq(N), SOq(N), or SPq(N), where q is a transcendental complex number. It is shown that the de Rham cohomology of Woronowicz'' external algebra coincides with the de Rham cohomologies of its left-invariant, its right-invariant and its bi-invariant subcomplexes. In the cases GLq(N) and SLq(N), the cohomology ring is isomorphic to the left-invariant external algebra and to the vector space of harmonic forms. We prove a Hodge decomposition theorem in these cases. The main technical tool is the spectral decomposition of the quantum Laplace-Beltrami operator. As in the classical case all three spaces of differential forms coincide: bi- invariant forms, harmonic forms and the de-Rham-cohomology. For orthog- onal and symplectic quantum groups there is no complete Hodge decompo- sition. In case of the standard calculi on the quantum groups GLq(N) and SLq(N), the size of exterior algebra is computed. The space of left-invariant k-forms has dimension C(N², k) (binomial coefficient). The algebra of bi-invariant forms is graded commutative with Poincaré series (1+t)(1+t³) ... (1+t^(2N-1)). Bi-invariant forms are closed.