Cette thèse comporte trois chapitres indépendants portant sur l’itération des applicationsrationnelles sur des variétés projectives et plus spécifiquement sur l’étude du comportement dela suite des degrés des itérés de telles applications.Dans le premier chapitre, nous donnons une construction des invariants fondamentaux quesont les degrés dynamiques dans un cadre très général, et ce sans hypothèse ni sur la caractéristique ni sur les singularités de l’espace ambiant. Cette construction repose sur des propriétésde positivité des cycles algébriques, et propose une alternative aux approches analytiques deDinh et Sibony ou algébriques de Truong.Le second chapitre est issu d’un article écrit en commun avec Jian Xiao. Notre contributionporte sur des objets centraux en géométrie convexe appelés valuations. Nous transférons à l’espace des valuations des notions de positivité des cycles algébriques récemment introduites parLehmann et Xiao, ce qui nous permet d’étendre l’opération de convolution originellement définie par Bernig et Fu à une sous-classe de valuations suffisamment positives.Le troisième chapitre constitue le coeur de la thèse, et porte sur des estimations des degrésdynamiques des automorphismes dit modérés de la quadrique affine de dimension 3. Nos arguments sont de nature variée, et s’appuient sur l’action du groupe modéré sur un complexe carréCAT(0) et Gromov hyperbolique récemment introduite par Bisi, Furter et Lamy.Nous avons finalement collecté dans un dernier et court chapitre quelques pistes de recherchedirectement inspirées des travaux présentés ici. / This thesis is divided into three independent chapters on the iterates of rational maps on projective varieties and more specifically on the study of the growth of the degree sequences of the iterates of such maps. In the first chapter, we give a construction of the fundamental invariants called dynamical degrees. Our method holds in a very general setting, without any conditions on the characteristic of the field or on the singularities of the ambient space.This construction is based on the study of positivity properties of algebraic cycles and gives an alternative approach to the analytical technics of Dinh and Sibony or to the algebraic arguments of Truong.The second chapter is taken from an article written in joint work with Jian Xiao. Our paper focuses on central objects in convex geometry called valuations. We transfer some positivity notions of algebraic cycles recently introduced by Lehmann and Xiao, this allows us to extend the convolution operation defined by Bernig and Fu to a subspace of sufficiently positive valuations.The third chapter is the core of this thesis and focuses on the dynamical degrees of the so-called tame automorphisms of an affine quadric threefold. Our arguments are of various nature and rely on the action of the tame group on a CAT(0), Gromov hyperbolic square complex recently introduced by Bisi, Furter and Lamy. Finally, we have collected in the last chapter a few perpectives directly inspired by this work.

Adinkras and arithmetical graphs have divergent origins. In the spirit of Feynman diagrams, adinkras encode representations of supersymmetry algebras as graphs with additional structures. Arithmetical graphs, on the other hand, arise in algebraic geometry, and give an arithmetical structure to a graph. In this thesis, we will interpret adinkras as arithmetical graphs and see what can be learned. Our work consists of three main strands. First, we investigate arithmetical structures on the underlying graph of an adinkra in the specific case where the underlying graph is a hypercube. We classify all such arithmetical structures and compute some of the corresponding volumes and linear ranks. Second, we consider the case of a reduced arithmetical graph structure on the hypercube and explore the wealth of relationships that exist between its linear rank and several notions of genus that appear in the literature on graph theory and adinkras. Third, we study modifications of the definition of an arithmetical graph that incorporate some of the properties of an adinkra, such as the vertex height assignment or the edge dashing. To this end, we introduce the directed arithmetical graph and the dashed arithmetical graph. We then explore properties of these modifications in an attempt to see if our definitions make sense, answering questions such as whether the volume is still an integer and whether there are still only finitely many arithmetical structures on a given graph.

Made available in DSpace on 2016-04-27T16:57:17Z (GMT). No. of bitstreams: 1 Antonia Zulmira da Silva.pdf: 1733925 bytes, checksum: 3460611fce63b2ca5913c2f811ada3ac (MD5) Previous issue date: 2012-05-14 / Secretaria da Educação do Estado de São Paulo / The purpose of this investigation was to find evidence of indicators of algebraic thinking development for the topic 'First-degree algebraic equations' from the mathematics Teacher's Manual adopted by public middle schools in São Paulo state, Brazil, and thus provide a written contribution to this teaching material. The investigation sought to answer the following research questions: Do the activities proposed in the topic 'First-degree algebraic equations' from the mathematics Teacher s Manual for the third quarter of the eighth grade enable teachers to foster the development of algebraic thinking among students? If so, which indicators are most evident? The definition used for indicators of algebraic thinking development drew on Fiorentini, Miorim, and Miguel (1993) and Fiorentini, Fernandes and Cristóvão (2005) with regard to aspects of algebraic thinking and on Ursini et al. (2005) concerning use of variables. Concurrently, the so-called multimeanings of equations, as defined by Ribeiro and Machado (2009), were taken into account. Desk research, as defined by Lüdke and André (1986), was the method selected for the study. Of the twelve indicators of algebraic thinking development investigated, nine were detected in the activities examined. The results obtained showed that these activities enable teachers to foster the development of algebraic thinking among students. A final, stand-alone section summarizes the theoretical framework adopted and includes a chart of the algebraic thinking indicators investigated, in addition to a synthetic view of the analyses providing evidence of these indicators in the activities. This summarized section is also available in CD-ROM format / O presente estudo teve por objetivo evidenciar indicadores de desenvolvimento do pensamento algébrico no tópico Equações algébricas de primeiro grau do Caderno do professor de Matemática adotado na docência do Ensino Fundamental da rede pública do Estado de São Paulo, com a finalidade de escrever um produto que contribuísse com esse material. O objetivo se desdobrou nas seguintes questões de pesquisa: As atividades presentes no tópico Equações algébricas de primeiro grau do Caderno do professor de Matemática do terceiro bimestre do oitavo ano do Ensino Fundamental possibilitam que o professor conduza os alunos ao desenvolvimento do pensamento algébrico? Em caso afirmativo, que indicadores são priorizados? Para definir os indicadores de desenvolvimento do pensamento algébrico, tomamos como referências sobre o pensamento algébrico Fiorentini, Miorim e Miguel (1993) e Fiorentini, Fernandes e Cristóvão (2005) e, a respeito do uso das variáveis, Ursini et al. (2005). Ao mesmo tempo, investigamos os multissignificados das equações, segundo Ribeiro e Machado (2009). Para a condução da pesquisa, utilizamos o método de análise documental, conforme Lüdke e André (1986). Dentre os doze indicadores de desenvolvimento do pensamento algébrico considerados, nove foram evidenciados nas atividades analisadas. Os resultados permitiram concluir que as atividades analisadas possibilitam que o professor conduza os alunos a desenvolver o pensamento algébrico. O produto deste trabalho contém referências aos elementos teóricos do trabalho, um quadro com os indicadores do pensamento algébrico utilizados nas análises e a síntese das análises das atividades, evidenciando os indicadores do pensamento algébrico. Esse produto está anexado a esta dissertação e também encontra-se disponível em CD-ROM

Made available in DSpace on 2014-06-11T19:22:18Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-02-25Bitstream added on 2014-06-13T20:28:17Z : No. of bitstreams: 1 arruda_rl_me_sjrp.pdf: 624072 bytes, checksum: 23ddd00e27d1ad781e2d1cec2cb65dee (MD5) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / O objetivo principal deste trabalho é estudar o Teorema de Riemann-Roch, um dos resultados fundamentais na teoria de curvas algébricas, e apresentar algumas de suas aplicações. Este teorema é uma importante ferramenta para a classificação das curvas algébricas, pois relaciona propriedades algébricas e topológicas. Daremos uma descrição das curvas algébricas de gênero g, 1≤ g ≤ 5, e faremos um breve estudo dos pontos de inflexão de um sistema linear sobre uma curva algébrica / The main purpose of this work is to discuss The Riemann-Roch Theorem, wich is one of the most important results of the theory algebraic curves, and to present some applications. This theorem is an important tool of the classification of algebraic curves, sinces relates algebraic and topological properties. We will describle the algebraic curves of genus g, 1≤ g ≤ 5, and also study inflection points of a linear system on an algebraic curve

This thesis is devoted to the proof of a theorem showing the existence of a closed model category structure for weakly enriched categories. It requires first of all the definitions of weakly enriched categories and equivalences of weakly enriched categories such that these definitions recover some existing notions of higher order weak categories, for example Segal categories, Tamsamani n-categories and strict n-categories. In order to prove our theorem, we elaborate a theory of plans for cell addition following the approach of the small object argument <i>à la</i> Quillen. We conclude this work with the proof that our theorem recovers the case of Segal categories. This last result requires a fundamental groupoid-geometric realization adjunction between Segal groupoids and topological spaces.

This thesis consists of two parts, a first part on computations in algebraic geometry, and a second part on deformation quantization. More specifically, it is a collection of four papers. In the papers I, II and III, we present algorithms and an implementation for the computation of degrees of characteristic classes in algebraic geometry. Paper IV is a contribution to the field of deformation quantization and actions of the Grothendieck-Teichmüller group. In Paper I, we present an algorithm for the computation of degrees of Segre classes of closed subschemes of complex projective space. The algorithm is based on the residual intersection theorem and can be implemented both symbolically and numerically. In Paper II, we describe an algorithm for the computation of the degrees of Chern-Schwartz-MacPherson classes and the topological Euler characteristic of closed subschemes of complex projective space, provided an algorithm for the computation of degrees of Segre classes. We also explain in detail how the algorithm in Paper I can be implemented numerically. Together this yields a symbolical and a numerical version of the algorithm. Paper III describes the Macaulay2 package CharacteristicClasses. It implements the algorithms from papers I and II, as well as an algorithm for the computation of degrees of Chern classes. In Paper IV, we show that L-infinity-automorphisms of the Schouten algebra T_poly(R^d) of polyvector fields on affine space R^d which satisfy certain conditions can be globalized. This means that from a given L-infinity-automorphism of T_poly(R^d) an L-infinity-automorphism of T_poly(M) can be constructed, for a general smooth manifold M. It follows that Willwacher's action of the Grothendieck-Teichmüller group on T_poly(R^d) can be globalized, i.e., the Grothendieck-Teichmüller group acts on the Schouten algebra T_poly(M) of polyvector fields on a general manifold M. / <p>At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 2: Manuscript. Paper 3: Manuscript. Paper 4: Accepted.</p>

Nous allons étudier les propriétés des courbes algébriques sur des surfaces K3 spéciales, du point de vue de la théorie de Brill-Noether.La démonstration de Lazarsfeld du théorème de Gieseker-Petri a mis en lumière l'importance de la théorie de Brill-Noether des courbes admettant un plongement dans une surface K3. Nous allons donner une démonstration détaillée de ce résultat classique, inspirée par les idées de Pareschi. En suite, nous allons décrire le théorème de Green et Lazarsfeld, fondamental pour tout notre travail, qui établit le comportement de l'indice de Clifford des courbes sur les surfaces K3.Watanabe a montré que l'indice de Clifford de courbes sur certaines surfaces K3, admettant un recouvrement double des surfaces de del Pezzo, est calculé en utilisant les involutions non-symplectiques. Nous étudions une situation similaire pour des surfaces K3 avec un réseau de Picard isomorphe à U(m), avec m>0 un entier quelconque. Nous montrons que la gonalité et l'indice de Clifford de toute courbe lisse sur ces surfaces, avec une seule exception déterminée explicitement, sont obtenus par restriction des fibrations elliptiques de la surface. Ce travail est basé sur l'article suivant :M. Ramponi, Gonality and Clifford index of curves on elliptic K3 surfaces with Picard number two, Archiv der Mathematik, 106(4), p. 355–362, 2016.Knutsen et Lopez ont étudié en détail la théorie de Brill-Noether des courbes sur les surfaces d'Enriques. En appliquant leurs résultats, nous allons pouvoir calculer la gonalité et l'indice de Clifford de toute courbe lisse sur les surfaces K3 qui sont des recouvrements universels d'une surface d'Enriques. Ce travail est basé sur l'article suivant :M. Ramponi, Special divisors on curves on K3 surfaces carrying an Enriques involution, Manuscripta Mathematica, 153(1), p. 315–322, 2017. / We study the properties of algebraic curves lying on special K3 surfaces, from the viewpoint of Brill-Noether theory.Lazarsfeld's proof of the Gieseker-Petri theorem has revealed the importance of the Brill-Noether theory of curves which admit an embedding in a K3 surface. We give a proof of this classical result, inspired by the ideas of Pareschi. We then describe the theorem of Green and Lazarsfeld, a key result for our work, which establishes the behaviour of the Clifford index of curves on K3 surfaces.Watanabe showed that the Clifford index of curves lying on certain special K3 surfaces, realizable as a double covering of a smooth del Pezzo surface, can be determined by a direct use of the non-simplectic involution carried by these surfaces. We study a similar situation for some K3 surfaces having a Picard lattice isomorphic to U(m), with m>0 any integer. We show that the gonality and the Clifford index of all smooth curves on these surfaces, with a single, explicitly determined exception, are obtained by restriction of the elliptic fibrations of the surface. This work is based on the following article:M. Ramponi, Gonality and Clifford index of curves on elliptic K3 surfaces with Picard number two, Archiv der Mathematik, 106(4), p. 355-362, 2016.Knutsen and Lopez have studied in detail the Brill-Noether theory of curves lying on Enriques surfaces. Applying their results, we are able to determine and compute the gonality and Clifford index of any smooth curve lying on the general K3 surface which is the universal covering of an Enriques surface. This work is based on the following article:M. Ramponi, Special divisors on curves on K3 surfaces carrying an Enriques involution, Manuscripta Mathematica, 153(1), p. 315-322, 2017.

Made available in DSpace on 2015-05-15T11:45:58Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 634504 bytes, checksum: ce035cc957832598c53dda96372e7cb7 (MD5) Previous issue date: 2011-02-24 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, based on algebraic function fields, we give constructions of disjoint linear codes. In addition,we study the asymptotic behavior of disjoint linear codes from our constructions. / Neste trabalho, baseados em corpos de funções algébricas, forneceremos construções de códigos lineares disjuntos. Além disso, nós estudaremos comportamentos assintóticos de códigos lineares disjuntos a partir da nossa construção.

In this paper introduce new approach for unified theory for continuous and discrete time (optimal) control problems based on the generalized Cayley transformation. We also relate the associated discrete and continuous generalized algebraic Riccati equations. We demonstrate the potential of this new approach proving new result for discrete algebraic Riccati equations. But we also discuss where this new approach as well as all other approaches still is non-satisfactory. We explain a discrepancy observed between the discrete and continuous cse and show that this discrepancy is partly due to the consideration of the wrong analogues. We also present an idea for a metatheorem that relates general theorems for discrete and continuous control problems.

Tropical geometry is an area of mathematics that interfaces algebraic geometry and combinatorics. The main object of study in tropical geometry is the tropical variety, which is the combinatorial counterpart of a classical variety. A classical variety is converted into a tropical variety by a process called tropicalization, thus reducing the problems of algebraic geometry to problems of combinatorics. This new tropical variety encodes several useful information about the original variety, for example an algebraic variety and its tropical counterpart have the same dimension. In this thesis, we look at the some of the computational aspects of tropical algebraic geometry. We study a generalization of Grobner basis theory of modules which unlike the standard Grobner basis also takes the valuation of coefficients into account. This was rst introduced in (Maclagan & Sturmfels, 2009) in the settings of polynomial rings and its computational aspects were first studied in (Chan & Maclagan, 2013) for the polynomial ring case. The motivation for this comes from tropical geometry as it can be used to compute tropicalization of varieties. We further generalize this to the case of modules. But apart from that it has many other computational advantages. For example, in the standard case the size of the initial submodule generally grows with the increase in degree of the generators. But in this case, we give an example of a family of submodules where the size of the initial submodule remains constant. We also develop an algorithm for computation of Grobner basis of submodules of modules over Z=p`Z[x1; : : : ; xn] that works for any weight vector. We also look at some of the important applications of this new theory. We show how this can be useful in efficiently solving the submodule membership problem. We also study the computation of Hilbert polynomials, syzygies and free resolutions.