Esta tese tem por base três artigos publicados pelo autor e colaboradores. O primeiro artigo trata do problema da quantização de modelos pseudoclássicos de partículas escalares em campos de fundo não-abelianos, cujo foco é a dedução desses modelos pseudo-clássicos usando métodos de integral de trajetória. O segundo artigo investiga a possibilidade de realizar modelos de gravitação dilatônica em variedades não-comutativas em duas dimensões. Para tanto, vale-se de um método de análise de vínculos e simetrias especialmente desenvolvido para gravitação não-comutativa em duas dimensões. O terceiro artigo discute modelos renormalizáveis em espaços-tempo não-comutativos com parâmetro de não-comutatividade bifermiônico em quatro dimensões. / This thesis is based on three published papers by the author and co-authors. The rst article treats the quantization problem of pseudoclassical models of scalar particles in non-Abelian backgrounds, which aims at deriving these models using path-integral methods. The second article examines the possibility of realizing dilaton gravity models in noncommutative two-dimensional manifolds. It relies upon a method of analysis of constraints and symmetries especially developed for non-commutative dilaton gravities in two dimensions. The third article discusses renormalizable models in noncommutative spacetime with bifermionic noncommutative parameter in four dimensions.

<div>Let A be a d by n integer matrix. Gel'fand et al.\ proved that most A-hypergeometric systems have an interpretation as a Fourier–Laplace transform of a direct image. The set of parameters for which this happens was later identified by Schulze and Walther as the set of not strongly resonant parameters of A. A similar statement relating A-hypergeometric systems to exceptional direct images was proved by Reichelt. In the first part of this thesis, we consider a hybrid approach involving neighborhoods U of the torus of A and consider compositions of direct and exceptional direct images. Our main results characterize for which parameters the associated A-hypergeometric system is the inverse Fourier–Laplace transform of such a "mixed Gauss–Manin system". </div><div><br></div><div>If the semigroup ring of A is normal, we show that every A-hypergeometric system is "mixed Gauss–Manin". </div><div><br></div><div>In the second part of this thesis, we use our notion of mixed Gauss–Manin systems to show that the projection and restriction of a normal A-hypergeometric system to the coordinate subspace corresponding to a face are isomorphic up to cohomological shift; moreover, they are essentially hypergeometric. We also show that, if A is in addition homogeneous, the holonomic dual of an A-hypergeometric system is itself A-hypergeometric. This extends a result of Uli Walther, proving a conjecture of Nobuki Takayama in the normal homogeneous case.</div>

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.

In this thesis we will try to explain and clarify the main concepts of R. Barthes's semiological method introduced in his book Système de la Mode. The key moments of his model will be confronted with critique of other authors such as J. Culler, T. Todorov or D. N. Rodowick. The principal aim of Barthes's book is to create a precise method which would lead the structural analysis of women's clothing described by fashion magazines. The author believes, that vestimentary features presented in fashion magazines are constituted into a system of signification. Barthes's main intention has been to reconstitute this system of meaning and to create a classification of written garment using linguistic approach and procedures. After considering the critique of the above-mentioned authors, we will try to apply Barthes's concepts to some examples of written garment, taken from current fashion magazines, in order to better understand author's semiological project.

The Signal protocol can be considered state-of-the-art when it comes to secure messaging, but advances in quantum computing stress the importance of finding post-quantum resistant alternatives to its asymmetric cryptographic primitives. The aim is to determine whether existing post-quantum cryptography can be used as a drop-in replacement for the public-key cryptography currently used in the Signal protocol and what the performance trade-offs may be. An implementation of the Signal protocol using commutative supersingular isogeny Diffie-Hellman (CSIDH) key exchange operations in place of elliptic-curve Diffie-Hellman (ECDH) is proposed. The benchmark results on a Samsung Galaxy Note 8 mobile device equipped with a 64-bit Samsung Exynos 9 (8895) octa-core CPU shows that it takes roughly 8 seconds to initialize a session using CSIDH-512 and over 40 seconds using CSIDH-1024, without platform specific optimization. To the best of our knowledge, the proposed implementation is the first post-quantum resistant Signal protocol implementation and the first evaluation of using CSIDH as a drop-in replacement for ECDH in a communication protocol.

Neste trabalho será provado que as álgebras comutativas de potências associativas de dimensão n e nilíndice n-3, assim como, álgebras de dimensão 9 sobre C, são solúveis, estendendo os resultados conhecidos ao famoso Problema de Albert. Logo depois de estudar o problema de Albert, será dada uma descrição das tabelas de multiplicação para as álgebras comutativas de potências associativas de dimensão n maior do que 12 e nilíndice n-1 sobre um corpo de característica diferente de 2,3 e 5. / We will prove that commutative power-associative nilalgebras both of dimension n and nilindex n-3, or of dimension 9 over C, are solvable. This solve an specific case of famous Albert\'s problem. After that, we will make a description about multiplications of commutative power-associative nilalgebras of dimension n (greater or igual that 12) and nilindex n-1 over a field of characteristic diferent from 2,3 and 5.

Es werden verschiedene Aspekte interner magnetischer Hintergrundfelder in Theorien offener Strings diskutiert. Phaenomenologisch und konzeptionell interessante Eigenschaften solcher Vakua, die Brechung von Supersymmetrie, Eichsymmetrie und chiraler Symmetrie, werden auf ganz generische Weise behandelt. Dann wird eine Spezialisierung auf Typ I Modelle, kompaktifiziert auf Tori und Bahnfaltigkeiten, durchgefuehrt. Daraus wird eine Methode gewonnen zur Konstruktion von Typ I Vakua mit attraktiven effektiven Feldtheorien als Niederenergienaeherungen, sowohl supersymmetrische wie nicht supersymmetrische Modelle mit chiralen Fermionspektren und Eichgruppen aehnlich dem Standardmodell oder einer vereinheitlichenden Verallgemeinerung desselben. Die am weitesten entwickelten Beispiele kombinieren magnetische Felder mit NSNS B-Feldern auf Bahnfaltigkeiten. Zuletzt wird noch eine verwandte Klasse von Modellen besprochen, die zwar eher weniger vielversprechende phaenomenologische Perspektiven bietet, aber einige konzeptionelle Spezialitaeten aufweist. In diesen Kompaktifizierungen werden asymmetrische Rotationen geeicht, so dass D-branen mit unterschiedlichen Werten fuer die magnetischen Felder auf ihrem Weltvolumen identifiziert werden, womit die Unterscheidung von kommutativen und nicht kommutativen Geometrien verlorengeht. / We discuss various aspects of internal magnetic background fields in open string theories. Phenomenologically and conceptually interesting properties of such string theory backgrounds, supersymmetry and gauge symmetry breaking, chiral fermion spectra and noncommutativity of the internal compactification manifolds, are treated in a rather generic framework. We then specialize to type I compactifications on tori and toroidal orbifolds with magnetic fields on the internal space. This allows to develop a strategy for constructing type I vacua with attractive low energy field theories which may either be supersymmetric or not and contain chiral spectra and gauge groups close to the Standard Model or some grand unified generalization thereof. The most sophisticated version uses magnetic fields and NSNS B-fields on orbifold spaces giving rise to a plethora of promising examples for semi-realistic string compactifications. We finally also present a related class of asymmetric orbifolds of type I which are of little phenomenological interest but still display certain interesting features. The asymmetric rotations which are gauged in these models identify D-branes with different values for the magnetic field on their world volume, such that the distinction of commutative and noncommutative internal geometries is lost.

Reposant sur l’analyse sociologique de quarante-neuf entretiens qualitatifs auprès de contribuables et spécialistes de l’évitement fiscal, ainsi que sur la lecture des philosophes majeurs de quatre grandes doctrines contemporaines de la justice sociale, cette thèse de doctorat vise à décrire et à comprendre les pratiques actuelles d’évitement fiscal des contribuables, personnes physiques et entreprises.La norme sociale s’avère plutôt favorable à l’évitement légal de l’impôt, et relativement permissive vis-à-vis de l’évitement illégal. Une analyse en sociologie de la déviance montrera les dangers que cette dynamique fait penser sur les finances publiques et sur la capacité de l’État à lever l’impôt dans le futur. Alors, l’étude des cadres axiologiques de ces phénomènes sociaux permettra d’établir une typologie idéal-typique des contribuables à partir des doctrines utilitariste, égalitariste libérale, libertarienne et communautarienne. Elle tentera de comprendre pour chaque groupe d’individus ainsi constitué les manières de penser et d’agir face à l’impôt, à partir notamment de la modélisation du « ras-le-bol fiscal ». Cette typologie pourra aussi être appliquée aux pratiques d’évitement des multinationales.Finalement, des recommandations originales et argumentées permettront de passer d’une vision comptable et juridique des finances publiques à une perspective de long terme basée sur les principes de justice. / This thesis is based on a qualitative survey among forty-nine taxpayers and tax specialists, as well as the study of the major philosophers of four different theories of social justice: utilitarianism, liberal egalitarianism, libertarianism and communitarianism. It aims at bringing to light tax avoidance and tax evasion strategies as well as the social and individual frames that allow such practices. This is conducted based on a social deviance analysis. Our work will confirm why taxpayers generally are in favour of (legal) tax avoidance. We have also observed a certain acceptance of (illegal) tax evasion. Our sociological and philosophical analysis will provide explanations for this situation. It will also offer new perspectives about the direct and side effects of this dynamic. We have established an ideal-type analysis grid that has allowed to better understand the social norm about tax, both from individual and firm perspectives. We have also created a theoretical model to explain the so-called tax “ras-le-bol”, as a breakeven threshold for taxes. We have eventually proposed original criteria for social justice based on our experience and research work that intends to open up new horizons for rebuilding a fair and sustainable tax and redistribution system.

Les signaux sujets à des contraintes non linéaires apparaissent dans un grand nombre d'applications physiques et techniques. Les travaux récents expriment une conscience croissante de l'importance des méthodes géométriques intrinsèques pour le traitement de tels signaux. La présente thèse s'inscrit dans cette orientation. Nous avons envisagé et résolu un certain nombre de problèmes en physique des ondes et en capture de mouvement. Les signaux sujets à des contraintes nonlinéaires sont modélisés comme des processus à valeurs dans les groupes de Lie matriciels. Nos problèmes correspondent alors à des problèmes d'estimation nonparamétrique et de filtrage. Afin de les résoudre nous avons mis en place des méthodes probabilistes et dynamiques, en particulier basées sur la notion de stabilité.

L'étude de certaines variables aléatoires, comme les paramètres additifs sur les arbres hyperquaternaires de points, ou encore le nombre de maxima au sein d'un ensemble de n points indépendants, et uniformément distribués dans [0,1]^d font apparaître des suites particulières, les sommes harmoniques multiples (SHM), extensions des nombres harmoniques classiques à des multi-indices.<br /><br />Nos travaux visant à appliquer des méthodes symboliques pour l'étude de ces variables aléatoires, nous remplaçons l'utilisation de multi-indices par des codages sur des alphabets distincts, et nous appuyons alors sur des résultats importants en combinatoire des mots pour les appliquer à nos suites de SHM, et aux fonctions polylogarithmes, qui sont des variantes des génératrices ordinaires des SHM. Dans les cas convergents, les deux objets convergent (respectivement lorsque z tend vers 1 et lorsque N tend vers l'infini) vers la même limite, appelée polyzêta. Pour les cas divergents, l'utilisation de séries génératrices non commutatives nous permet d'établir un théorème ``à l'Abel'', faisant apparaître une limite commune. Ce théorème permet de donner une forme explicite aux constantes d'Euler généralisées associées à des SHM divergentes et ainsi d'obtenir un algorithme très efficace pour calculer leur développement asymptotique.<br /><br />Finalement, nous proposons des applications des sommes harmoniques dans le domaine des structures de données multidimensionnelles, pour lesquelles notre approche donne naissance à des calculs exacts, qui peuvent par la suite être aisément évalués asymptotiquement.