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Dans ce travail, nous étudions les surfaces dont les composantes irréductibles sont des surfaces toriques. En particulier, nous donnons une formule pour calculer l'obstruction d'Euler locale de ces surfaces. Comme application de cette formule, nous calculons l'obstruction d'Euler locale pour certaines familles de surfaces déterminantales. De plus, nous définissons et donnons une formule pour calculer la caractéristique d'Euler évanescente d'une surface torique normale $X_{sigma}$. Nous montrons que ce nombre est relié à la seconde multiplicité polaire de $X_{sigma}$. Nous présentons aussi une formule pour l'obstruction d'Euler d'une fonction $f: X_{sigma} to mathbb{C}$ et pour le nombre de Brasselet d'une telle fonction. Comme application de ce résultat nous calculons l'obstruction d'Euler d'un type de polynôme sur une famille de surfaces déterminantales. / In this work we study surfaces with the property that their irreducible components are toric surfaces. In particular, we present a formula to compute the local Euler obstruction of such surfaces. As an application of this formula we compute the local Euler obstruction for some families of determinantal surfaces. Furthermore, we define the vanishing Euler characteristic of a normal toric surface $X_{sigma}$, we give a formula to compute it, and we relate this number with the second polar multiplicity of $X_{sigma}$. We also present a formula for the Euler obstruction of a function $f: X_{sigma} to mathbb{C}$ and for the Brasselet number of it. As an application of this result we compute the Euler obstruction of a type of polynomial on a family of determinantal surfaces.

Neste trabalho, nós abordamos alguns resultados de T. Fukuda e de N. Dutertre e T. Fukui sobre a topologia das singularidades de Morin. Em particular, apresentamos uma nova prova para o Teorema de Dutertre-Fukui [2, Theorem 6.2], para o caso em que N = Rn, usando a Teoria de Morse para variedades com bordo. Baseados nas propriedades de um n-campo de vetores gradiente (∇ f1; : : : ∇fn) de uma aplicação de Morin f : M → Rn, com dim M ≥ n, na segunda parte deste trabalho, nós introduzimos o conceito de n-campos de Morin para n-campos de vetores que não são necessariamente gradientes. Nós também generalizamos o resultado de T. Fukuda [3, Theorem 1], que estabelece uma equivalência módulo 2 entre a característica de Euler de uma variedade diferenciável M e a característica de Euler dos conjuntos singulares de uma aplicação de Morin definida sobre M, para o contexto dos n-campos de Morin. / In this work, we revisit results of T. Fukuda and N. Dutertre and T. Fukui on the topology of Morin maps. In particular, we give a new proof for Dutertre-Fukui\'s Theorem [2, Theorem 6.2] when N = Rn, using Morse Theory for manifolds with boundary. Based on the properties of a gradient n-vector field (∇ f1; : : : ∇ fn) of a Morin map f : M → Rn, where dim M ≥ n, in the second part of this work, we introduce the concept of Morin n-vector field for n-vector fields V = (V1; : : : ; Vn) that are not necessarily gradients. We also generalize the result of T. Fukuda [3, Theorem 1], which establishes a module 2 equivalence between Euler\'s characteristic of a manifold M and Euler\'s characteristic of the singular sets of a Morin map defined on M, to the context of Morin n-vector fields.

Neste trabalho, nós abordamos alguns resultados de T. Fukuda e de N. Dutertre e T. Fukui sobre a topologia das singularidades de Morin. Em particular, apresentamos uma nova prova para o Teorema de Dutertre-Fukui [2, Theorem 6.2], para o caso em que N = Rn, usando a Teoria de Morse para variedades com bordo. Baseados nas propriedades de um n-campo de vetores gradiente (∇ f1; : : : ∇fn) de uma aplicação de Morin f : M → Rn, com dim M ≥ n, na segunda parte deste trabalho, nós introduzimos o conceito de n-campos de Morin para n-campos de vetores que não são necessariamente gradientes. Nós também generalizamos o resultado de T. Fukuda [3, Theorem 1], que estabelece uma equivalência módulo 2 entre a característica de Euler de uma variedade diferenciável M e a característica de Euler dos conjuntos singulares de uma aplicação de Morin definida sobre M, para o contexto dos n-campos de Morin. / In this work, we revisit results of T. Fukuda and N. Dutertre and T. Fukui on the topology of Morin maps. In particular, we give a new proof for Dutertre-Fukui\'s Theorem [2, Theorem 6.2] when N = Rn, using Morse Theory for manifolds with boundary. Based on the properties of a gradient n-vector field (∇ f1; : : : ∇ fn) of a Morin map f : M → Rn, where dim M ≥ n, in the second part of this work, we introduce the concept of Morin n-vector field for n-vector fields V = (V1; : : : ; Vn) that are not necessarily gradients. We also generalize the result of T. Fukuda [3, Theorem 1], which establishes a module 2 equivalence between Euler\'s characteristic of a manifold M and Euler\'s characteristic of the singular sets of a Morin map defined on M, to the context of Morin n-vector fields.

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>

Orientador: Prof. Dr. Rafael de Mattos Grisi / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional, 2015. / Esse trabalho tem por objetivo apresentar um pouco da teoria de grafos no ensino Básico. Nele serão abordados conceitos básicos da teoria de grafos com maior enfoque sobre os grafos eulerianos e semieulerianos e o teorema das quatro cores. Apresentamos e discutimos também algumas propostas de atividades que foram e poderão ser desenvolvidas no Ensino Fundamental e Médio, possibilitando ao aluno o desenvolvimento de algumas habilidades como investigar, analisar, modelar, dentre outras. A prática dessas atividades foi realizada em uma escola da rede estadual do Estado de São Paulo com uma turma do 9o ano do Ensino Fundamental e com uma turma do 3o ano do Ensino Médio, no ano de 2014. / This work aims to present some of the so called graph theory in the Basic education. It will address the basic concepts of graph theory with greater focus on the Euler graphs and the four color theorem. We also discuss some proposals for activities that have been developed in primary and secondary education, enabling the student to develop some skills to investigate, analyze and model problems using graphs. The practice of these activities took place in a state school of São Paulo with a class of 9th graders of the elementary school and a group of the 3rd year of high school, in 2014.

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<p>This dissertation establishes limit theory for topological functionals of geometric complexes from a stochastic process viewpoint. Standard filtrations of geometric complexes, such as the Čech and Vietoris-Rips complexes, have a natural parameter <i>r </i>which governs the formation of simplices: this is the basis for persistent homology. However, the parameter <i>r</i> may also be considered the time parameter of an appropriate stochastic process which summarizes the evolution of the filtration.</p><p>Here we examine the stochastic behavior of two of the foremost classes of topological functionals of such filtrations: the Betti numbers and the Euler characteristic. There are also two distinct setups in which the points underlying the complexes are generated, where the points are distributed randomly in <i>R^d</i> according to a general density (the traditional setup) and where the points lie in the tail of a heavy-tailed or exponentially-decaying “noise” distribution (the extreme-value theory (EVT) setup).<br></p><p>These results constitute some of the first results combining topological data analysis (TDA) and stochastic process theory. The first collection of results establishes stochastic process limits for Betti numbers of Čech complexes of Poisson and binomial point processes for two specific regimes in the traditional setup: the sparse regime—when the parameter <i>r </i>governing the formation of simplices causes the Betti numbers to concentrate on components of the lowest order; and the critical regime—when the parameter <i>r</i> is of the order <i>n^-1/d</i> and the geometric complex becomes highly connected with topological holes of every dimension. The second collection of results establishes a functional strong law of large numbers and a functional central limit theorem for the Euler characteristic of a random geometric complex for the critical regime in the traditional setup. The final collection of results establishes functional strong laws of large numbers for geometric complexes in the EVT setup for the two classes of “noise” densities mentioned above.<br></p>

Les données provenant de l'échantillonnage fin d'un processus continu (champ aléatoire) peuvent être représentées sous forme d'images. Un test statistique permettant de détecter une différence entre deux images peut être vu comme un ensemble de tests où chaque pixel est comparé au pixel correspondant de l'autre image. On utilise alors une méthode de contrôle de l'erreur de type I au niveau de l'ensemble de tests, comme la correction de Bonferroni ou le contrôle du taux de faux-positifs (FDR). Des méthodes d'analyse de données ont été développées en imagerie médicale, principalement par Keith Worsley, utilisant la géométrie des champs aléatoires afin de construire un test statistique global sur une image entière. Il s'agit d'utiliser l'espérance de la caractéristique d'Euler de l'ensemble d'excursion du champ aléatoire sous-jacent à l'échantillon au-delà d'un seuil donné, pour déterminer la probabilité que le champ aléatoire dépasse ce même seuil sous l'hypothèse nulle (inférence topologique). Nous exposons quelques notions portant sur les champs aléatoires, en particulier l'isotropie (la fonction de covariance entre deux points du champ dépend seulement de la distance qui les sépare). Nous discutons de deux méthodes pour l'analyse des champs anisotropes. La première consiste à déformer le champ puis à utiliser les volumes intrinsèques et les compacités de la caractéristique d'Euler. La seconde utilise plutôt les courbures de Lipschitz-Killing. Nous faisons ensuite une étude de niveau et de puissance de l'inférence topologique en comparaison avec la correction de Bonferroni. Finalement, nous utilisons l'inférence topologique pour décrire l'évolution du changement climatique sur le territoire du Québec entre 1991 et 2100, en utilisant des données de température simulées et publiées par l'Équipe Simulations climatiques d'Ouranos selon le modèle régional canadien du climat. / Data coming from a fine sampling of a continuous process (random field) can be represented as images. A statistical test aiming at detecting a difference between two images can be seen as a group of tests in which each pixel is compared to the corresponding pixel in the other image. We then use a method to control the type I error over all the tests, such as the Bonferroni correction or the control of the false discovery rate (FDR). Methods of data analysis have been developped in the field of medical imaging, mainly by Keith Worsley, using the geometry of random fields in order to build a global statistical test over the whole image. The expected Euler characteristic of the excursion set of the random field underlying the sample over a given threshold is used in order to determine the probability that the random field exceeds this same threshold under the null hypothesis (topological inference). We present some notions relevant to random fields, in particular isotropy (the covariance function between two given points of a field depends only on the distance between them). We discuss two methods for the analysis of non\-isotropic random fields. The first one consists in deforming the field and then using the intrinsic volumes and the Euler characteristic densities. The second one uses the Lipschitz-Killing curvatures. We then perform a study of sensitivity and power of the topological inference technique comparing it to the Bonferonni correction. Finally, we use topological inference in order to describe the evolution of climate change over Quebec territory between 1991 and 2100 using temperature data simulated and published by the Climate Simulation Team at Ouranos, with the Canadian Regional Climate Model CRCM4.2.

Let G be a finite graph with vertex set V and edge set E. A graph complex on G is an abstract simplicial complex consisting of subsets of E. In particular, we may interpret such a complex as a family of subgraphs of G. The subject of this thesis is the topology of graph complexes, the emphasis being placed on homology, homotopy type, connectivity degree, Cohen-Macaulayness, and Euler characteristic. We are particularly interested in the case that G is the complete graph on V. Monotone graph properties are complexes on such a graph satisfying the additional condition that they are invariant under permutations of V. Some well-studied monotone graph properties that we discuss in this thesis are complexes of matchings, forests, bipartite graphs, disconnected graphs, and not 2-connected graphs. We present new results about several other monotone graph properties, including complexes of not 3-connected graphs and graphs not coverable by p vertices. Imagining the vertices as the corners of a regular polygon, we obtain another important class consisting of those graph complexes that are invariant under the natural action of the dihedral group on this polygon. The most famous example is the associahedron, whose faces are graphs without crossings inside the polygon. Restricting to matchings, forests, or bipartite graphs, we obtain other interesting complexes of noncrossing graphs. We also examine a certain "dihedral" variant of connectivity. The third class to be examined is the class of digraph complexes. Some well-studied examples are complexes of acyclic digraphs and not strongly connected digraphs. We present new results about a few other digraph complexes, including complexes of graded digraphs and non-spanning digraphs. Many of our proofs are based on Robin Forman's discrete version of Morse theory. As a byproduct, this thesis provides a loosely defined toolbox for attacking problems in topological combinatorics via discrete Morse theory. In terms of simplicity and power, arguably the most efficient tool is Forman's divide and conquer approach via decision trees, which we successfully apply to a large number of graph and digraph complexes. / QC 20100622