An abelian arrangement is a finite set of codimension one abelian subvarieties (possibly translated) in a complex abelian variety. We are interested in the topology of the complement of an arrangement. If the arrangement is unimodular, we provide a combinatorial presentation for a differential graded algebra (DGA) that is a model for the complement, in the sense of rational homotopy theory. Moreover, this DGA has a bi-grading that allows us to compute the mixed Hodge numbers. If the arrangement is chordal, then this model is a Koszul algebra. In this case, studying its quadratic dual gives a combinatorial description of the Q-nilpotent completion of the fundamental group and the minimal model of the complement of the arrangement. This dissertation includes previously unpublished co-authored material.

This dissertation uses methods from homological algebra and computational commutative algebra to study four problems. We use Hilbert function computations and classical homology theory and combinatorics to answer questions with a more applied mathematics content: splines approximation, hyperplane arrangements, configuration spaces and coding theory. In Chapter II we study a problem in approximation theory. Alfeld and Schumaker give a formula for the dimension of the space of piecewise polynomial functions (splines) of degree d and smoothness r. Schenck and Stiller conjectured that this formula holds for all d 2r + 1. In this chapter we show that there exists a simplicial complex such that for any r, the dimension of the spline space in degree d = 2r is not given by this formula. Chapter III is dedicated to formal hyperplane arrangements. This notion was introduced by Falk and Randell and generalized to formality by Brandt and Terao. In this chapter we prove a criteria for formal arrangements, using a complex constructed from vector spaces introduced by Brandt and Terao. As an application, we give a simple description of formality of graphic arrangements in terms of the homology of the flag complex of the graph. Chapter IV approaches the problem of studying configuration of smooth rational curves in P2. Since an irreducible conic in P2 is a P1 (so a line) it is natural to ask if classical results about line arrangements in P2, such as addition-deletion type theorem, Yoshinaga criterion or Terao's conjecture verify for such configurations. In this chapter we answer these questions. The addition-deletion theorem that we find takes in consideration the fine local geometry of singularities. The results of this chapter are joint work with H. Schenck. In Chapter V we study a problem in algebraic coding theory. Gold, Little and Schenck find a lower bound for the minimal distance of a complete intersection evaluation codes. Since complete intersections are Gorenstein, we show a similar bound for the minimal distance depending on the socle degree of the reduced zero-dimensional Gorenstein scheme. The results of this chapter are a work in progress.

We are studying the Tutte Polynomial of hyperplane arrangements. We discuss some previous work done to compute these polynomials. Then we explain our method to calculate the Tutte Polynomial of some arrangements more efficiently. We next discuss the details of the program used to do the calculation. We use this program and present the actual Tutte Polynomials calculated for the arrangements E6, E7, and E8. / text

A recent result of Salvetti and Settepanella gives, for a complexified real arrangement, an explicit description of a minimal CW decomposition as well as an explicit algebraic complex which computes local system homology. We apply their techniques to discriminantal arrangements in two dimensional complex space and calculate the boundary maps which will give local system homology groups given any choice of local system. This calculation generalizes several known results; examples are given related to Milnor fibrations, solutions of KZ equations, and the LKB representation of the braid group. Another algebraic object associated to a hyperplane arrangement is the module of derivations. We analyze the behavior of the derivation module for an affine arrangement over an infinite field and relate its derivation module to that of its cone. In the case of an arrangement fibration, we analyze the relationship between the derivation module of the total space arrangement and those of the base and fiber arrangements. In particular, subject to certain restrictions, we establish freeness of the total space arrangement given freeness of the base and fiber arrangements.

Let f be a non-constant, homogeneous, complex polynomial in n variables. We may associate to f a fibration with typical fiber F known as the Milnor fiber. For regular and isolated singular points of f at the origin, the topology of the Milnor fiber is well-understood. However, much less is known about the topology in the case of non-isolated singular points. In this thesis we analyze the Milnor fiber associated to a hyperplane arrangement, ie, f is a reduced, homogeneous polynomial with degree one irreducible components in n variables. If n > 2then the origin will be a non-isolated singular point. In particular, we use the fundamental group of the complement of the arrangement in order to construct a regular CW-complex that is homotopy equivalent to the Milnor fiber. Combining this construction with some local combinatorics of the arrangement, we generalize some known results on the upper bounds for the first betti number of the Milnor fiber. For several classes of arrangements we show that the first homology group of the Milnor fiber is torsion free. In the final section, we use methods that depend on the embedding of the arrangement in the complex projective plane (ie not necessarily combinatorial data) in order to analyze arrangements to which the known results on arrangements do not apply.

<p> This thesis deals with combinatorial properties of hyperplane arrangements. In particular, we address a conjecture of Deza, Terlaky and Zinchenko stating that the largest possible average diameter of a bounded cell of a simple hyperplane arrangement is not greater than the dimension. We prove that this conjecture is asymptotically tight in fixed dimension by constructing a family of hyperplane arrangements containing mostly cubical cells. The relationship with a result of Dedieu, Malajovich and Shub, the conjecture of Hirsch, and a result of Haimovich are presented.</p> <p> We give the exact value of the largest possible average diameter for all simple arrangements in dimension two, for arrangements having at most the dimension plus two hyperplanes, and for arrangements having six hyperplanes in dimension three. In dimension three, we strengthen the lower and upper bounds for the largest possible average diameter of a bounded cell of a simple hyperplane arrangements.</p> <p> Namely, let ΔA(n, d) denote the largest possible average diameter of a bounded cell of a simple arrangement defined by n hyperplanes in dimension d. We show that • ΔA(n, 2) = 2[n/2] / (n-1)(n-2) for n ≥ 3, • ΔA(d + 2, d) = 2d/d+1, • ΔA(6, 3) = 2, • 3 - 6/n-1 + 6([n/2]-2) / (n-1)(n-2)(n-3) ≤ ΔA(n, 3) ≤ 3 + 4(2n^2-16n+21) / 3(n-1)(n-2)(n-3) • ΔA (n, d) ≥ 1 + (d-1)(n-d d)+(n-d)(n-d-1) for n ≥ 2d. We also address another conjecture of Deza, Terlaky and Zinchenko stating that the minimum number Φ0A~(n, d) of facets belonging to exactly one bounded cell of a simple arrangement defined by n hyperplanes in dimension d is at least d (n-2 d-1). We show that • Φ0A(n, 2) = 2(n - 1) for n ≥ 4, • Φ0A~(n, 3) ≥ n(n-2)/3 +2 for n ≥ 5. We present theoretical frameworks, including oriented matroids, and computational tools to check by complete enumeration the open conjectures for small instances. Preliminary computational results are given.</p> / Thesis / Master of Science (MSc)

In this thesis, we study the relationship between radial projections, and orthogonal and minimal projections in l_4^3. Specifically, we calculate the norm of the maximum radial projection and we prove that the hyperplane constant, with respect to the radial projection, is not achieved by a minimal projection in this space. We will also show our numerical results, obtained using computer software, and use them to approximate the norms of the radial, orthogonal, and minimal projections in l_4^3. Specifically, we show, numerically, that the maximum minimal projection is attained for ker{1,1,1} as well as compute the norms for the maximum radial and orthogonal projections.

Ce rapport de thèse présente une analyse détaillée de nouvelles techniques d'estimation de pose à partir des images de couleur et de profondeur provenant de capteurs RGB-D. Etant donné que l'estimation de la pose nécessite d'établir une cartographie en simultanée, la reconstruction 3D de la scène sera aussi étudié dans cette thèse. La localisation et la cartographie ont été largement étudiés par la communauté de robotique et de vision par ordinateur, et ces techniques ont aussi été largement employés pour la robotique mobile et les systèmes autonomes afin d'exécuter des tâches telles que le suivi de caméra, la reconstruction 3D dense ou encore la localisation robuste. Le défi de l'estimation de pose réside dans la façon de relier les mesures des capteurs pour estimer l'état système en position et en orientation. Lorsqu'une multitude de capteurs fournisse différentes observations des mêmes variables, il devient alors complexe de fusionner au mieux ces informations acquises à des instants différents. De manière à développer un algorithme efficace pour traiter ces problèmes, une nouvelle méthode de recalage nommée Point-to-hyperplane sera introduite, analysée, comparée et appliquée à l'estimation de pose et à la cartographie basée sur des frames-clés. La méthode proposée permet de minimiser différentes métriques sous la forme d'un seul vecteur de mesure en n-dimensions, sans avoir besoin de définir un facteur d'échelle qui pondère l'influence de chaque terme durant la minimisation d'énergie. Au sein du concept Point-to-hyperplane, deux lignes principales ont été examinées. Premièrement, la méthode proposée sera employée dans des applications d'odométrie visuelle et de cartographie 3D. Compte-tenu des résultats expérimentaux, il a été montré que la méthode proposée permet d'estimer la pose localement avec précision en augmentant le domaine et la vitesse de convergence. L'invariance est mathématiquement prouvée et des résultats sont fournis à la fois pour environnements réels et synthétiques. Deuxièmement, une méthode pour la localisation globale a été proposée qui adresse les problèmes de reconnaissance et de détection de lieux. Cette méthode s'appuie sur l'utilisation du Point-to-hyperplane combinée à une optimisation Branch-and-bound pour estimer la pose globalement. Étant donné que les stratégies de Branch-and-Bound permettent d'obtenir des alignements grossiers sans la nécessité d'avoir la pose initiale entre les images, le Point-tohyperplane peut être utiliser pour raffiner l'estimation. Il sera démontré que cette stratégie est mieux contrainte quand davantage de dimensions sont utilisées. Cette stratégie s'avère être utile pour résoudre les problèmes de désalignement et pour obtenir des cartes 3D globalement consistantes. Pour finir cette thèse et pour démontrer la performance des méthodes proposées, des résultats sur des applications de SLAM visuel et de cartographie 3D sont présentés. / This thesis presents a detailed account of novel techniques for pose estimation by using both, color and depth information from RGB-D sensors. Since pose estimation simultaneously requires an environment map, 3D scene reconstruction will also be considered in this thesis. Localization and mapping has been extensively studied by the robotics and computer vision communities and it is widely employed in mobile robotics and autonomous systems for performing tasks such as tracking, dense 3D mapping and robust localization. The central challenge of pose estimation lies in how to relate sensor measurements to the state of position and orientation. When a variety of sensors, which provide different information about the same data points, are available, the challenge then becomes part of how to best fuse acquired information at different times. In order to develop an effective algorithm to deal with these problems, a novel registration method named Point-to-hyperplane Iterative Closest Point will be introduced, analysed, compared and applied to pose estimation and key-frame mapping. The proposed method allows to jointly minimize different metric errors as a single measurement vector with n-dimensions without requiring a scaling factor to tune their importance during the minimization process. Within the Point-to-hyperplane framework two main axes have been investigated. Firstly, the proposed method will be employed for performing visual odometry and 3D mapping. Based on actual experiments, it has been shown that the proposed method allows to accurately estimate the pose locally by increasing the domain of convergence and by speeding up the alignment. The invariance is mathematically proven and results in both, simulated and real environments, are provided. Secondly, a method is proposed for global localization for enabling place recognition and detection. This method involves using the point-to-hyperplane methods within a Branch-and-bound architecture to estimate the pose globally. Therefore, the proposed method has been combined with the Branch-and-bound algorithm to estimate the pose globally. Since Branch-and-bound strategies obtain rough alignments regardless of the initial position between frames, the Point-to-hyperplane can be used for refinement. It will be demonstrated that the bounds are better constrained when more dimensions are considered. This last approach is shown to be useful for solving mistracking problems and for obtaining globally consistent 3D maps. In a last part of the thesis and in order to demonstrate the proposed approaches and their performance, both visual SLAM and 3D mapping results are provided.

<div>In this thesis, we study invariants of graded modules over polynomial rings. In particular, we find bounds on the Hilbert functions and graded Betti numbers of certain modules. This area of research has been widely studied, and we discuss several well-known theorems and conjectures related to these problems. Our main results extend some known theorems from the case of homogeneous ideals of polynomial rings R to that of graded R-modules. In Chapters 2 & 3, we discuss preliminary material needed for the following chapters. This includes monomial orders for modules, Hilbert functions, graded Betti numbers, and generic initial modules.</div><div> </div><div> In Chapter 4, we discuss x_n-stability of submodules M of free R-modules F, and use this stability to examine properties of lexsegment modules. Using these tools, we prove our first main result: a general hypersurface restriction theorem for modules. This theorem states that, when restricting to a general hypersurface of degree j, the Hilbert series of M is bounded above by that of M^{lex}+x_n^jF. In Chapter 5, we discuss Hilbert series of local cohomology modules. As a consequence of our general hypersurface restriction theorem, we give a bound on the Hilbert series of H^i_m(F/M). In particular, we show that the Hilbert series of local cohomology modules of a quotient of a free module does not decrease when the module is replaced by a quotient by the lexicographic module M^{lex}.</div><div> </div><div> The content of Chapter 6 is based on joint work with Gabriel Sosa. The main theorem is an extension of a result of Caviglia and Sbarra to polynomial rings with base field of any characteristic. Given a homogeneous ideal containing both a piecewise lex ideal and an ideal generated by powers of the variables, we find a lex ideal with the following property: the ideal in the polynomial ring generated by the piecewise lex ideal, the ideal of powers, and the lex ideal has the same Hilbert function and Betti numbers at least as large as those of the original ideal. This bound on the Betti numbers is sharp, and is a closer bound than what was previously known in this setting.</div>

This dissertation has two topics. The rst one is about rotating a supporting hyperplane on the convex hull of a nite point set to arrive at one of its facets. We present three procedures for these rotations in multiple dimensions. The rst two procedures rotate a supporting hyperplane for the polytope starting at a lower dimensional face until the support set is a facet. These two procedures keep current points in the support set and accumulate new points after the rotations. The rst procedure uses only algebraic operations. The second procedure uses LP. In the third procedure we rotate a hyperplane on a facet of the polytope to a dierent adjacent facet. Similarly to the rst procedure, this procedure uses only algebraic operations. Some applications to these procedures include data envelopment analysis (DEA) and integer programming. The second topic is in the eld of containment problems for polyhedral sets. We present three procedures to nd a circumscribing simplex that contains a point set in any dimension. The rst two procedures are based on the supporting hyperplane rotation ideas from the rst topic. The third circumscribing simplex procedure uses polar cones and other geometrical properties to nd facets of a circumscribing simplex. One application of the second topic discussed in this dissertation is in hyperspectral unmixing.