Dans la présente thèse, on établit des bornes supérieures sur les nombres de Betti des ensembles définis à l'aide de fonctions pfaffiennes, en fonction de la complexité pfaffienne (ou format) de ces ensembles. Les fonctions pfaffiennes ont été définies par Khovanskii, comme solutions au comportement quasi-polynomial de certains systèmes polynomiaux d'équations différentielles. Les ensembles semi-pfaffiens satisfont une condition de signe booléene sur des fonctions pfaffiennes, et les ensembles sous-pfaffiens sont projections de semi-pfaffiens. Wilkie a démontré que les fonctions pfaffiennes engendrent une structure o-minimale, et Gabrielov a montré que cette structure pouvait etre efficacement décrite par des ensembles pfaffiens limites. A l'aide de la théorie de Morse, de déformations, de recurrences sur le niveau combinatoire et de suites spectrales, on donne dans cette thèse des bornes effectives pourtoutes les catégories d'ensembles pré-citées.

In this thesis, we consider semi-algebraic sets over a real closed field R defined by quadratic polynomials. Semi-algebraic sets of R^k are defined as the smallest family of sets in R^k that contains the algebraic sets as well as the sets defined by polynomial inequalities, and which is also closed under the boolean operations (complementation, finite unions and finite intersections). We prove new bounds on the topological complexity of semi-algebraic sets over a real closed field R defined by quadratic polynomials, in terms of the parameters of the system of polynomials defining them, which improve the known results. We conclude the thesis with presenting two new algorithms along with their implementations.

Let e > 3 and mo,... ,me_i be positive integers with gcd(m0,... ,me_i) = 1, which form an almost arithmetic sequence, i.e., some e - 1 of these form an arithmetic progression. We further assume that m0,... ,mc_1 generate F := Σ e-1 I=0 Nmi minimally. Note that any three integers and also any arithmetic progression form an almost arithmetic sequence. We assume that 0 < m0 < • • • < me-2 form an arithmetic progression and n := mc-i is arbitrary Put p := e - 2. Let K be a field and XQ) ... ,Xj>, Y,T be indeterminates. Let p denote the kernel of the if-algebra homomorphism η: K[XQ, ..., XV) Y) -* K^T], defined by r){Xi) = Tm\.. .η{Xp) = Tmp, η](Y) = Tn. Then, p is the defining ideal for the affine monomial curve C in A^, defined parametrically by Xo = Trr^)...)Xv = T^}Y = T*. Furthermore, p is a homogeneous ideal with respect to the gradation on K[X0)... ,XP,F], given by wt(Z0) = mo, • • •, wt(Xp) = mp, wt(Y) = n. Let 4 := K[XQ> ...,XP) Y)/p denote the coordinate ring of C. With the assumption ch(K) = 0, in Chapter 1 we have derived an explicit formula for μ(DerK(A)), the minimal number of generators for the A-module DerK(A), the derivation module of A. Furthermore, since type(A) = μ(DerK(A)) — 1 and the last Betti number of A is equal to type(A), we therefore obtain an explicit formula for the last Betti number of A as well A minimal set of binomial generatorsG for the ideal p had been explicitly constructed by PatiL In Chapter 2, we show that the set G is a Grobner basis with respect to grevlex monomial ordering on K[X0)..., Xp, Y]. As an application of this observation, in Chapter 3 we obtain an explicit minimal free resolution for affine monomial curves in A4K defined by four coprime positive integers mo,.. m3, which form a minimal arithmetic progression. (Please refer the pdf file forformulas)

Made available in DSpace on 2014-06-11T19:31:43Z (GMT). No. of bitstreams: 0 Previous issue date: 2012-04-26Bitstream added on 2014-06-13T19:02:12Z : No. of bitstreams: 1 martins_crp_dr_rcla.pdf: 3528391 bytes, checksum: 6600de4077c7912391d57dccd5f0ff47 (MD5) / No presente trabalho apresentamos uma análise da a “Parte Primeira” da obra “Sulla Risoluzione Delle Equazioni Algebriche” (1852), de Enrico Betti (1823–1892). Com foco na Teoria das Equações Algébricas, ela pode também ser elencada como as obras que pertencem à fase embrionária da Teoria de Grupos, uma vez que a parte citada contempla a então chamada Teoria das Substituições, que envolve os conceitos de permutação, grupo de permutação, sub-grupo e sub-grupo normal, entre outros. Iremos apresentar seu conteúdo matemático, relacionando-o com a forma com que hoje é estudado, como também, os fatos históricos e os resultados matemáticos anteriores aos abordados pela obra citada, principalmente aos que se referem à vida e obra de Evariste Galois (1811-1832) / We present an analysis of the “First Part” of the work “Sulla Risoluzione Delle Equazioni Algebriche” (1852), Enrico Betti (1823–1892). Focusing on the Theory of Algebraic Equations, it can also be classified as works belonging to the early stage of the Theory of Groups, since the portion cited includes the so-called Theory of substitutions, which involves the concepts of permutation, permutation group, sub-group or sub-normal group, among others. We will present its mathematical content, linking it with the way today is studied, as well as the historical facts and mathematical results covered by the previous work cited, especially those that relate to the life and work of Evariste Galois (1811 -1832)

Simplicial complexes are mathematical objects whose importance stretches from topology to commutative algebra and combinatorics. In this thesis we focus on the family of balanced simplicial complexes. A d-dimensional simplicial complex is balanced if its 1-skeleton can be properly (d+1)-colored, as in the classical graph theoretic sense. Equivalently, a d-dimensional complex is balanced iff it admits a non-degenerate simplicial projection to the d-simplex. We present results on these complexes from a number of different points of view. After two introductory chapters, we exhibit in chapter 3 an infinite family of balanced counterexamples to Stanley's partitionability conjecture. These complexes, which are in addition constructible, answer a question of Duval et al. in the negative. Next we shift to combinatorial topology, and study cross-flips, i.e., local moves on balanced manifolds introduced by Izmestiev, Klee and Novik, which preserve both the coloring and the topological type. In chapter 4 we provide an explicit description and enumeration of an interesting subset of these moves and use it to prove a Pachner-type theorem. Indeed, we show that any two balanced combinatorial manifolds with boundary which are PL-homeomorphic can be transformed one into the other by a sequence of shellings and inverse shellings which preserve both the coloring and the topological type at each step. This solves a problem proposed by Izmestiev, Klee and Novik. Chapter 5 is devoted to the study of certain algebraic invariants of simplicial complexes in the balanced case. Here upper bounds for the graded Betti numbers of the Stanley-Reisner ring of balanced simplicial complexes are investigated in several level of generalities, and we show that they are sharper than in the general case. First, we employ Hochster formula to obtain inequalities for the case of arbitrary balanced complexes. Next, we focus on the balanced Cohen-Macaulay case and we obtain two upper bounds via two different strategies. Using similar ideas we also bound the Betti numbers in the linear strand of balanced normal d-pseudomanifolds, for d>2. Finally, we explicitly compute graded Betti numbers of the class of stacked cross-polytopal spheres, and conjecture that they provide a sharp upper bound for those of all balanced pseudomanifolds with the same dimension and number of vertices. In the last chapter, we implement cross-flips on balanced surfaces and 3-manifolds, and use this computer program to search for balanced manifolds on few vertices, possibly vertex-minimal. Reducing the barycentric subdivision of vertex minimal triangulations, we find a long list of balanced triangulations of interesting spaces on few vertices. Among those stand out a balanced vertex-minimal triangulation of the dunce hat (11-vertices) and of the 2- and 3-dimensional real projective space (9 and 16 vertices respectively). Using obstructions from knot theory and a careful choice of flips we find a balanced non-shellable 3-sphere and a balanced shellable non-vertex-decomposable 3-sphere on 28 and 22 vertices respectively. These are the smallest instances known in the literature.

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京都大学 / 新制・課程博士 / 博士(理学) / 甲第25104号 / 理博第5011号 / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)准教授 森前 智行, 教授 高橋 義朗, 准教授 戸塚 圭介 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

A complexidade de um módulo M, sobre uma álgebra de dimensão finita R, é a medida do crescimento da dimensão de suas sizigias. No nosso trabalho, estudamos esse conceito, nos concentrando muito mais no caso das álgebras autoinjetiva. Relacionamos esse crescimento com o comportamento da componente do carcás de Auslander-Reiten, a qual o módulo M pertence. Em particular, estudamos, com bastante cuidado, o caso em que a complexidade é 1, o que significa que a dimensão das sizigias são eventualmente constante. Surpreendentemente, o comportamento de todos os módulos numa mesma componente é muito parecido. / The complexity of a module M under a finite dimensional algebra R is the measure of the growth of its syzygies\' dimension. In our work, we study this concept concentrating on the case of the selfinjective algebras. We relate this growth with the behavior of the Auslander-Reiten component containing this module. In particular, we study, carefully, the case in which the complexity is 1. Surprisingly, the behavior of every module in the same component as M is very similar.

In dieser Arbeit studieren wir Lorentz-Mannigfaltigkeiten mit spezieller Holonomie, d.h. ihre Holonomiedarstellung wirkt schwach-irreduzibel aber nicht irreduzibel. Aufgrund der schwachen Irreduzibilität lässt die Darstellung einen ausgearteten Unterraum invariant und damit also auch eine lichtartige Linie. Geometrisch hat dies zur Folge, dass wir zwei parallele Unterbündel (die Linie und ihr orthogonales Komplement) des Tangentialbündels erhalten. Diese Arbeit nutzt diese und weitere Objekte um zu beweisen, dass kompakte Lorentzmannigfaltigkeiten mit Abelscher Holonomie geodätisch vollständig sind. Zudem werden Lorentzmannigfaltigkeiten mit spezieller Holonomie und nicht-negativer Ricci-Krümung auf den Blättern der Blätterung, induziert durch das orthogonale Komplement der parellelen Linie, und maximaler erster Bettizahl untersucht. Schließlich werden vollständige Ricci-flache Lorentzmannigfaltigkeiten mit vorgegebener voller Holonomie konstruiert. / In the present thesis we study dimensional Lorentzian manifolds with special holonomy, i.e. such that their holonomy representation acts indecomposably but non-irreducibly. Being indecomposable, their holonomy group leaves invariant a degenerate subspace and thus a light-like line. Geometrically, this means that, since being holonomy invariant, this line gives rise to parallel subbundles of the tangent bundle. The thesis uses these and other objects to prove that Lorentian manifolds with Abelian holonomy are geodesically complete. Moreover, we study Lorentzian manifolds with special holonomy and non-negative Ricci curvature on the leaves of the foliation induced by the orthogonal complement of the parallel light-like line whose first Betti number is maximal. Finally, we provide examples of geodesically complete and Ricci-flat Lorentzian manifolds with special holonomy and prescribed full holonomy group.

Recently computational topology had an important development in data analysis giving birth to the field of Topological Data Analysis. Persistent homology appears as a fundamental tool based on the topology of data that can be represented as points in metric space. In this work, we apply techniques of Topological Data Analysis, more precisely, we use persistent homology to calculate topological features more persistent in data. In this sense, the persistence diagrams are processed as feature vectors for applying Machine Learning algorithms. In order to classification, we used the following classifiers: Partial Least Squares-Discriminant Analysis, Support Vector Machine, and Naive Bayes. For regression, we used Support Vector Regression and KNeighbors. Finally, we will give a certain statistical approach to analyze the accuracy of each classifier and regressor. / Recentemente a topologia computacional teve um importante desenvolvimento na análise de dados dando origem ao campo da Análise Topológica de Dados. A homologia persistente aparece como uma ferramenta fundamental baseada na topologia de dados que possam ser representados como pontos num espaço métrico. Neste trabalho, aplicamos técnicas da Análise Topológica de Dados, mais precisamente, usamos homologia persistente para calcular características topológicas mais persistentes em dados. Nesse sentido, os diagramas de persistencia são processados como vetores de características para posteriormente aplicar algoritmos de Aprendizado de Máquina. Para classificação, foram utilizados os seguintes classificadores: Análise de Discriminantes de Minimos Quadrados Parciais, Máquina de Vetores de Suporte, e Naive Bayes. Para a regressão, usamos a Regressão de Vetores de Suporte e KNeighbors. Finalmente, daremos uma certa abordagem estatística para analisar a precisão de cada classificador e regressor.