Ce travail de thèse, composé de quatre parties, est consacré à l’étude de la géométrie algébrique en caractéristiques mixte et positive. Dans la première partie, motivés par une théorie conjecturale de la ramification pour les torseurs inséparables, nous étudions les modèles maximaux des torseurs sur un corps local, qui sont une généralisation des anneaux des entiers dans la théorie classique de la ramification. Nous prouvons la maximalité et la fonctorialité des modèles maximaux et nous les calculons explicitement pour les schémas en groupes finis plats d'ordre p. La deuxième partie est un travail en commun avec Giulio Orecchia et Matthieu Romagny. Nous étudions la perfection des algèbres et la coperfection des espaces et champs algébriques. Nous prouvons que l’espace des composantes connexes fournit la coperfection d’un espace algébrique et il représente la colimite du système de Frobenius relatifs. Dans le cas des champs algébriques, nous construisons le pro-groupoïde fondamental étale, nous prouvons qu'il fournit la coperfection, et il représente la colimite du système de Frobenius relatifs dans le cas de Deligne-Mumford. Dans la troisième partie, nous prouvons quelques résultats de platitude et de représentabilité des espaces de modules de torseurs sous certains schémas en groupes, qui découlent naturellement de l’espace de modules propre des p-revêtements galoisiens. Nous discutons également de la relation avec les jacobiennes généralisées des courbes ouvertes. Dans la dernière partie, nous nous intéressons à un nouveau type de géométrie analytique non-archimédienne, avec des valuations à valeurs dans des monoïdes commutatifs totalement ordonnés. Nous étudions quelques exemples de schémas et d’espaces adiques. / This thesis work, consisting of four parts, is devoted to the study of algebraic geometry in mixed and positive characteristics. In the first part, motivated by a conjectural ramification theory for inseparable torsors, we study the maximal model of a torsor over a local field, which is a generalization of integer rings in classical ramification theory. We prove the maximality and functoriality of maximal models, and calculate them explicitly for some finite flat group schemes of order p. The second part is a joint work with Giulio Orecchia and Matthieu Romagny. We study perfection of algebras and coperfection of algebraic spaces and stacks. We prove that the space of connected components provides the coperfection of an algebraic space, and it represents the colimit of relative Frobenii. In the case of algebraic stacks, we construct the étale fundamental pro-groupoid, and prove that it provides the coperfection, and it represents the colimit of relative Frobenii in Deligne-Mumford case. In the third part, we prove some results on flatness and representability of moduli spaces of torsors under certain group schemes, which naturally arise from the proper moduli space of Galois p-covers (stable p-torsors). We also discuss the relation with generalized Jacobians of open curves. In the last part, we are interested in a new kind of nonarchimedean analytic geometry, with valuations on totally ordered commutative monoids. We study some examples from schemes and adic spaces.

The irrational rotation algebra A_θ is known to be Poincaré self-dual in the KK-theoretic sense. The spectral triple representing the required K-homology fundamental class was constructed by Connes out of the Dolbeault operator on the 2-torus, but so far, there has not been an explicit description of the dual element. We geometrically construct, for certain elements g of the modular group, a finitely generated projective module L_g over A_θ ⊗ A_θ out of a pair of transverse Kronecker flows on the 2-torus. For upper triangular g, we find an unbounded cycle representing the dual of said module under Kasparov product with Connes' class, and prove that this cycle is invertible in KK(A_θ,A_θ), allowing us to 'untwist' L_g to an unbounded cycle representing the unit for the self-duality of A_θ. / Graduate

We initiate the study of a new class of semigroup C*-algebras arising from number-theoretic considerations; namely, we generalize the construction of Cuntz, Deninger, and Laca by considering the left regular C*-algebras of ax+b-semigroups from actions of congruence monoids on rings of algebraic integers in number fields. Our motivation for considering actions of congruence monoids comes from class field theory and work on Bost–Connes type systems. We give two presentations and a groupoid model for these algebras, and establish a faithfulness criterion for their representations. We then explicitly compute the primitive ideal space, give a semigroup crossed product description of the boundary quotient, and prove that the construction is functorial in the appropriate sense. These C*-algebras carry canonical time evolutions, so that our construction also produces a new class of C*-dynamical systems. We classify the KMS (equilibrium) states for this canonical time evolution, and show that there are several phase transitions whose complexity depends on properties of a generalized ideal class group. We compute the type of all high temperature KMS states, and consider several related C*-dynamical systems. / Graduate

No description available.

<p>Ova teza se bavi &curren;-kvazilinearnim varijetetima grupoida. Pokazano je da postoji ta&middot;cno dvadeset osam idempotentnih &curren;-kvazilinearnih varijeteta grupoida, od kojih dvadeset &middot;sest varijeteta imaju kona&middot;cnu bazu i te baze su i navedene, dok preostala dva varijeteta imaju inherentno beskona&middot;cnu bazu. U tezi je opisano ured enje svih idempotentnih &curren;-kvazilinearnih varijeteta grupoida i nalazimo male grupoide koji generi&middot;su svaki od navedenih varijeteta. Na kraju je pokazano da postoji kontinum mnogo &curren;-kvazilinearnih variejeteta grupoida.</p> / <p>The topic of this thesis are &curren;-quasilinear varieties of groupoids.<br />We show that there exist exactly twenty-eight idempotent &curren;-quasilinear varieties of groupoids, twenty-six of which are &macr;nitely based (and we explicitly<br />give &macr;nite bases for each of them), while two are inherently non&macr;nitely based.<br />We describe the ordering of these twenty-eight idempotent &curren;-quasilinear varieties of groupoids and &macr;nd small generating algebras for each of them. In<br />the end we show that there exist continuum many &curren;-quasilinear varieties of<br />groupoids, not all of which are even locally &macr;nite.</p>

A measure of the quality of Voronoi tessellations resulting from various mesh generators founded on feature-driven models is introduced in this work. A planar tessellation covers an image with polygons of various shapes and sizes. Tessellations have potential utility due to their geometry and the opportunity to derive useful information from them for object recognition, image processing and classification. Problem domains including images are generally feature-endowed, non-random domains. Generators modeled otherwise may easily guarantee quality of meshes but certainly bear no reference to features of the meshed problem domain. They are therefore unsuitable in point pattern identification, characterization and subsequently the study of meshed regions. We therefore found generators on features of the problem domain. This provides a basis for element quality studies and improvement based on quality criteria. The resulting polygonal meshes tessellating an n-dimensional digital image into convex regions are of varying element qualities. Given several types of mesh generating sets, a measure of overall solution quality is introduced to determine their effectiveness. Given a tessellation of general and mixed shapes, this presents a challenge in quality improvement. The Centroidal Voronoi Tessellation (CVT) technique is developed for quality improvement and guarantees of mixed, general-shaped elements and to preserve the validity of the tessellations. Mesh quality indicators and entropies introduced are useful for pattern studies, analysis, recognition and assessing information. Computed features of tessellated spaces are explored for image information content assessment and cell processing to expose detail using information theoretic methods. Tessellated spaces also furnish information on pattern structure and organization through their quality distributions. Mathematical and theoretical results obtained from these spaces help in understanding Voronoi diagrams as well as for their successful applications. Voronoi diagrams expose neighbourhood relations between pattern units. Given this realization, the foundation of near sets is developed for further applications. / February 2017