Dans cette thèse nous avons étudié certaines questions mathématiques associées au calcul de l'action spectrale de Chamseddine--Connes sur des exemples fondamentaux de triplets spectraux non commutatifs, tels que le tore non commutatif et la 3-sphère quantique SUq(2). Nous avons montré en particulier qu'une condition diophantienne sur la matrice de déformation du tore est cruciale pour obtenir l'action spectrale en tenant compte de la structure réelle. <br />Nous avons aussi étudié la question de l'existence de tadpoles (termes linéaires par rapport au potentiel de jauge de la fluctuation de la métrique dans l'action spectrale) dans le cas de géométries riemanniennes commutatives, et la construction d'un calcul pseudodifférentiel global permettant une généralisation du produit de Weyl--Moyal sur un espace de Schwartz de sections rapidement décroissantes sur un fibré cotangent d'une variété avec linéarisation.

L'implicitisation d'une surface algébrique rationnelle, c'est-à-dire le passage de la paramétrisation à une représentation implicite, est un<br />problème géométrique classique. Dans ce travail de thèse, nous utilisons la théorie des syzygies pour représenter implicitement une surface par une matrice dont les mineurs de taille maximale ont l'équation implicite comme plus grand diviseur commun. Dans les deux premiers chapitres, nous traitons deux classes de surfaces spéciales pour lesquelles il est toujours possible de construire une matrice carrée qui correspond au résultant d'une μ-base : les surfaces réglées et les surfaces canales. Dans les chapitres suivants, le cas général de surfaces rationnelles paramétrées sur une variété torique de dimension 2 est étudié. Nous montrons qu'une telle matrice peut être construite en n'utilisant que des syzygies linéaires et nous décrivons un algorithme simple et efficace pour son calcul.

Dans cette thèse, nous montrons que le groupe de K-théorie $K_0$ de la $C^*$-algèbre associée aux pavages de type pinwheel est isomorphe à la somme de $\ZZ \oplus \ZZ^6$ et d'un groupe cohomologique $H$.\\ Cette $C^*$-algèbre est de plus munie d'une trace qui induit une application linéaire sur ce groupe de $K$-théorie.\\ Nous calculons explicitement l'image, sous cette application, du sommant $\ZZ \oplus \ZZ^6$, montrant que l'image de $\ZZ$ est nulle et que l'image de $\ZZ^6$ est contenue dans le module de fréquences des patchs du pavage de type pinwheel.\\ Nous montrons également que l'on peut appliquer le théorème de l'indice mesuré dû à A. Connes pour relier l'image de $H$ à une formule cohomologique plus calculable.\\ Pour l'étude de cette partie cohomologique, nous adaptons la cohomologie PV, introduite par J. Savinien et J. Bellissard, au cas des pavages de type pinwheel pour montrer que le groupe de cohomologie de \v{C}ech de dimension maximale de ces pavages est isomorphe au groupe des coinvariants entiers de la transversale canonique associée à ces pavages.\\ Ce résultat nous permet alors de prouver la conjecture du gap-labeling fait par J. Bellissard, dans le cas particulier des pavages de type pinwheel.\\ Nous terminons cette étude par un calcul explicite, montrant que le gap-labeling (ou module de fréquences des patchs) est donné par $\frac{1}{264}\ZZ \left [ \frac{1}{5} \right ]$.

The classical discriminant D_n of degree n polynomials detects whether a given univariate polynomial f has a repeated root. It is itself a polynomial in the coefficients of f which, according to several classical results by Bézout, Sylvester, Cayley, and others, may be expressed as the determinant of a matrix whose entries are much simpler polynomials in the coefficients of f. This thesis is concerned with the construction and classification of such determinantal formulae for D_n. In particular, all of the formulae for D_n appearing in the classical literature are equivalent in the sense that the cokernels of their associated matrices are isomorphic as modules over the associated polynomial ring. This begs the question of whether there exist formulae which are not equivalent to the classical formulae and not trivial in the sense of having the same cokernel as the 1 x 1 matrix (D_n). The results of this thesis lie in two directions. First, we construct an explicit non-classical formula: the presentation matrix of the open swallowtail first studied by Arnol'd and Givental. We study the properties of this formula, contrasting them with the properties of the classical formulae. Second, for the discriminant of the polynomial x^4+a_2x^2+a_3x+a_4 we embark on a classification of determinantal formulae which are homogeneous in the sense that the cokernels of their associated matrices are graded modules with respect to the grading deg a_i=i. To this end, we use deformation theory: a moduli space of such modules arises from the base spaces of versal deformations of certain modules over the E_6 singularity {x^4-y^3}. The method developed here can in principle be used to classify determinantal formulae for all discriminants, and, indeed, for all singularities.

The classical discriminant D_n of degree n polynomials detects whether a given univariate polynomial f has a repeated root. It is itself a polynomial in the coefficients of f which, according to several classical results by Bézout, Sylvester, Cayley, and others, may be expressed as the determinant of a matrix whose entries are much simpler polynomials in the coefficients of f. This thesis is concerned with the construction and classification of such determinantal formulae for D_n. In particular, all of the formulae for D_n appearing in the classical literature are equivalent in the sense that the cokernels of their associated matrices are isomorphic as modules over the associated polynomial ring. This begs the question of whether there exist formulae which are not equivalent to the classical formulae and not trivial in the sense of having the same cokernel as the 1 x 1 matrix (D_n). The results of this thesis lie in two directions. First, we construct an explicit non-classical formula: the presentation matrix of the open swallowtail first studied by Arnol'd and Givental. We study the properties of this formula, contrasting them with the properties of the classical formulae. Second, for the discriminant of the polynomial x^4+a_2x^2+a_3x+a_4 we embark on a classification of determinantal formulae which are homogeneous in the sense that the cokernels of their associated matrices are graded modules with respect to the grading deg a_i=i. To this end, we use deformation theory: a moduli space of such modules arises from the base spaces of versal deformations of certain modules over the E_6 singularity {x^4-y^3}. The method developed here can in principle be used to classify determinantal formulae for all discriminants, and, indeed, for all singularities.

Splines are piecewise polynomial functions of a given order of smoothness r on a triangulated region (or polyhedrally subdivided region) of Rd. The set of splines of degree at most k forms a vector space Crk() Moreover, a nice way to study Cr k()is to embed n Rd+1, and form the cone b of with the origin. It turns out that the set of splines on b is a graded module Cr b() over the polynomial ring R[x1; : : : ; xd+1], and the dimension of Cr k() is the dimension o This dissertation follows the works of Billera and Rose, as well as Schenck and Stillman, who each approached the study of splines from the viewpoint of homological and commutative algebra. They both defined chain complexes of modules such that Cr(b) appeared as the top homology module. First, we analyze the effects of gluing planar simplicial complexes. Suppose 1, 2, and = 1 [ 2 are all planar simplicial complexes which triangulate pseudomanifolds. When 1 \ 2 is also a planar simplicial complex, we use the Mayer-Vietoris sequence to obtain a natural relationship between the spline modules Cr(b), Cr (c1), Cr(c2), and Cr( \ 1 \ 2). Next, given a simplicial complex , we study splines which also vanish on the boundary of. The set of all such splines is denoted by Cr(b). In this case, we will discover a formula relating the Hilbert polynomials of Cr(cb) and Cr (b). Finally, we consider splines which are defined on a polygonally subdivided region of the plane. By adding only edges to to form a simplicial subdivision , we will be able to find bounds for the dimensions of the vector spaces Cr k() for k 0. In particular, these bounds will be given in terms of the dimensions of the vector spaces Cr k() and geometrical data of both and . This dissertation concludes with some thoughts on future research questions and an appendix describing the Macaulay2 package SplineCode, which allows the study of the Hilbert polynomials of the spline modules.

Chapter 0 In this introductory chapter, certain notational and terminological conventions are established and a summary given of background results that are needed in subsequent chapters. Chapter 1 In this chapter, the notion of a "weak conguence formula" [Tay72], [BB75] is introduced and used to characterize both subdirectly irreducible algebras and essential extensions. Special attention is paid to the role they play in varieties with definable principal congruences. The chapter focuses on residually small varieties; several of its results take their motivation from the so-called "Quackenbush Problem" and the "RS Conjecture". One of the main results presented gives nine equivalent characterizations of a residually small variety; it is largely due to W. Taylor. It is followed by several illustrative examples of residually small varieties. The connections between residual smallness and several other (mostly categorical) properties are also considered, e.g., absolute retracts, injectivity, congruence extensibility, transferability of injections and the existence of injective hulls. A result of Taylor that establishes a bound on the size of an injective hull is included. Chapter 2 Beginning with a proof of A. Day's Mal'cev-style characterization of congruence modular varieties [Day69] (incorporating H.-P. Gumm's "Shifting Lemma"), this chapter is a self-contained development of commutator theory in such varieties. We adopt the purely algebraic approach of R. Freese and R. McKenzie [FM87] but show that, in modular varieties, their notion of the commutator [α,β] of two congruences α and β of an algebra coincides with that introduced earlier by J. Hagemann and C. Herrmann [HH79] as well as with the geometric approach proposed by Gumm [Gum80a],[Gum83]. Basic properties of the commutator are established, such as that it behaves very well with respect to homomorphisms and sufficiently well in products and subalgebras. Various characterizations of the condition "(x, y) Є [α,β]” are proved. These results will be applied in the following chapters. We show how the theory manifests itself in groups (where it gives the familiar group theoretic commutator), rings, modules and congruence distributive varieties. Chapter 3 We define Abelian congruences, and Abelian and affine algebras. Abelian algebras are algebras A in which [A2, A2] = idA (where A2 and idA are the greatest and least congruences of A). We show that an affine algebra is polynomially equivalent to a module over a ring (and is Abelian). We give a proof that an Abelian algebra in a modular variety is affine; this is Herrmann's Funda- mental Theorem of Abelian Algebras [Her79]. Herrmann and Gumm [Gum78], [Gum80a] established that any modular variety has a so-called ternary "difference term" (a key ingredient of the Fundamental Theorem's proof). We derive some properties of such a term, the most significant being that its existence characterizes modular varieties. Chapter 4 An important result in this chapter (which is due to several authors) is the description of subdirectly irreducible algebras in a congruence modular variety. In the case of congruence distributive varieties, this theorem specializes to Jόnsson's Theorem. We consider some properties of a commutator identity (Cl) which is a necessary condition for a modular variety to be residually small. In the main result of the chapter we see that for a finite algebra A in a modular variety, the variety V(A) is residually small if and only if the subalgebras of A satisfy (Cl). This theorem of Freese and McKenzie also proves that a finitely generated congruence modular residually small variety has a finite residual bound, and it describes such a bound. Thus, within modular varieties, it proves the RS Conjecture. Conclusion The conclusion is a brief survey of further important results about residually small varieties, and includes mention of the recently disproved (general) RS Conjecture. / Thesis (M.Sc.)-University of Natal, Durban, 2000.

Le travail de cette thèse s'inscrit dans l'étude des fondements mathématiques de la théorie quantique de l'information et de la physique quantique, à travers l'étude de l'ensemble des points fixes (appelé aussi frontière de Poisson) d'opérateurs quantiques et l'étude des trajectoires quantiques en dimension infinie. Nous précisons en premier lieu la frontière de Poisson d'un opérateur quantique, puis nous répondons négativement aux conjectures soulevées par Arias et al. sur la frontière de Poisson d'un opérateur quantique. Dans un second temps, nous identifions la frontière de Poisson non-commutative d'un groupoïde s-discret mesuré permettant ainsi de retrouver un résultat de moyennabilité de l'extension de Poisson du groupoïde. Enfin nous obtenons des résultats de purification asymptotique des trajectoires quantiques à valeurs dans une algèbre fortement compacte.

In this thesis, we undertake a combinatorial study of certain aspects of triangulations of cartesian products of simplices, particularly in relation to their relevance in toric algebra and to their underlying product structure. The first chapter reports joint work with Samu Potka. The object of study is a class of homogeneous toric ideals called cut ideals of graphs, that were introduced by Sturmfels and Sullivant 2006. Apart from their inherent appeal to combinatorial commutative algebra, these ideals also generalize graph statistical models for binary data and are related to some statistical models for phylogenetic trees. Specifically, we consider minimal free resolutions for the cut ideals of trees. We propose a method to combinatorially estimate the Betti numbers of the ideals in this class. Using this method, we derive upper bounds for some of the Betti numbers, given by formulas exponential in the number of vertices of the tree. Our method is based on a common technique in commutative algebra whereby arbitrary homogeneous ideals are deformed to initial monomial ideals, which are easier to analyze while conserving some of the information of the original ideals. The cut ideal of a tree on n vertices turns out to be isomorphic to the Segre product of the cut ideals of its n-1 edges (in particular, its algebraic properties do not depend on its shape). We exploit this product structure to deform the cut ideal of a tree to an initial monomial ideal with a simple combinatorial description: it coincides with the edge ideal of the incomparability graph of the power set of the edges of the tree. The vertices of the incomparability graph are subsets of the edges of the tree, and two subsets form an edge whenever they are incomparable. In order to obtain algebraic information about these edge ideals, we apply an idea introduced by Dochtermann and Engström in 2009 that consists in regarding the edge ideal of a graph as the (monomial) Stanley-Reisner ideal of the independence complex of the graph. Using Hochster\'s formula for computting Betti numbers of Stanley-Reisner ideals by means of simplicial homology, the computation of the Betti numbers of these monomial ideals is turned to the enumeration of induced subgraphs of the incomparability graph. That the resulting values give upper bounds for the Betti numbers of the cut ideals of trees is an important well-known result in commutative algebra. In the second chapter, we focus on some combinatorial features of triangulations of the point configuration obtained as the cartesian product of two standard simplices. These were explored in collaboration with César Ceballos and Arnau Padrol, and had a two-fold motivation. On the one hand, we intended to understand the influence of the product structure on the set of triangulations of the cartesian product of two point configurations; on the other hand, the set of all triangulations of the product of two simplices is an intricate and interesting object that has attracted attention both in discrete geometry and in other fields of mathematics such as commutative algebra, algebraic geometry, enumerative geometry or tropical geometry. Our approach to both objectives is to examine the circumstances under which a triangulation of the polyhedral complex given by the the product of an (n-1)-simplex times the (k-1)-skeleton of a (d-1)-simplex extends to a triangulation of an (n-1)-simplex times a (d-1)-simplex. We refer to the former as a partial triangulation of the product of two simplices. Our main result says that if d >= k > n, a partial triangulation always extends to a uniquely determined triangulation of the product of two simplices. A somewhat unexpected interpretation of this result is as a finiteness statement: it asserts that if d is sufficiently larger than n, then all partial triangulations are uniquely determined by the (compatible) triangulations of its faces of the form “(n-1)-simplex times n-simplex”. Consequently, one can say that in this situation ‘\'triangulations of an (n-1)-simplex times a (d-1)-simplex are not much more complicated than triangulations of an (n-1)-simplex times an n-simplex\'\'. The uniqueness assertion of our main result holds already when d>=k>=n. However, the same is not true for the existence assertion; namely, there are non extendable triangulations of an (n-1)-simplex times the boundary of an n-simplex that we explicitly construct. A key ingredient towards this construction is a triangulation of the product of two (n-1)-simplices that can be seen as its ``second simplest triangulation\'\' (the simplest being its staircase triangulation). It seems to be knew, and we call it the Dyck path triangulation. This triangulation displays symmetry under the cyclic group of order n that acts by simultaneously cycling the indices of the points in both factors of the product. Next, we exhibit a natural extension of the Dyck path triangulation to a triangulation of an (n-1)-simplex times an n-simplex that, in a sense, enjoys some sort of ‘\'rigidity\'\' (it also seems new). Performing a ‘\'local modification\'\' on the restriction of this extended triangulation to the polyhedral complex given by (n-1)-simplex times the boundary of an n-simplex yields the non-extendable partial triangulation. The thesis includes two appendices on basic commutative algebra and triangulations of point configuration, included to make it slightly self-contained.

Le but de cette thèse est l'étude des invariants d'Iwasawa attachés aux p-groupes des classes généralisés dans les extensions de Lie p-adiques de corps de nombres.Ces invariants ont été introduits par Iwasawa pour les Zp-extensions. Les travaux de Venjakob sur la structure des modules sur l'algèbre d'Iwasawa d'un groupe de Lie p-adique ont permis d'en généraliser la définition à la théorie non-commutative. Par des techniques de descente et une étude algébrique fine de la structure des modules d'Iwasawa sur un groupe non-commutatif, on dégage des formules asymptotiques pour les p-groupes des classes généralisés le long d'une extension de corps de nombres de groupe de Galois p-valué. Ces formules ont pour paramètres les invariants d'Iwasawa de l'extension. Elles sont rendues plus précises dans le cas des Zp-extensions, où on remarque qu'un défaut de descente doit être pris en compte et est d'impact non négligeable sur le résultat final. Ces résultats asymptotiques sont ensuite exploités à l'aide de la théorie du miroir. Ceci conduit à des formules de dualité entre ramification et décomposition concernant les invariants d'Iwasawa.