To the pair (E,c), where E is a countable Borel equivalence relation on a standard Borel space (X,A) and c a normalized Borel T-valued 2-cocycle on E, we associate a sequentially weakly closed Borel *-algebra Br*(E,c), contained in the bounded linear operators on L^2(E). Associated to Br*(E,c) is a natural (Borel) Cartan subalgebra (Definition 6.4.10) L(Bo(X)) isomorphic to the bounded Borel functions on X. Then L(Bo(X)) and its normalizer (the set of the unitaries u in Br*(E,c) such that u*fu in L(Bo(X)), f in L(Bo(X))) countably generates the Borel *-algebra Br*(E,c). In this thesis, we study Br*(E,c) and in particular prove that: i) If E is smooth, then Br*(E,c) is a type I Borel *-algebra (Definition 6.3.10). ii) If E is a hyperfinite, then Br*(E,c) is a Borel AF-algebra (Definition 7.5.1). iii) Generalizing Kumjian's definition, we define a Borel twist G over E and its associated sequentially closed Borel *-algebra Br*(G). iv) Let a Borel Cartan pair (B, Bo) denote a sequentially closed Borel *-algebra B with a Borel Cartan subalgebra Bo, where B is countably Bo-generated. Generalizing Feldman-Moore's result, we prove that any pair (B, Bo) can be realized uniquely as a pair (Br*(E,c), L(Bo(X))). Moreover, we show that the pair (Br*(E,c), L(Bo(X))) is a complete invariant of the countable Borel equivalence relation E. v) We prove a Krieger type theorem, by showing that two aperiodic hyperfinite countable equivalence relations are isomorphic if and only if their associated Borel *-algebras Br*(E1) and Br*(E2) are isomorphic.

To the pair (E,c), where E is a countable Borel equivalence relation on a standard Borel space (X,A) and c a normalized Borel T-valued 2-cocycle on E, we associate a sequentially weakly closed Borel *-algebra Br*(E,c), contained in the bounded linear operators on L^2(E). Associated to Br*(E,c) is a natural (Borel) Cartan subalgebra (Definition 6.4.10) L(Bo(X)) isomorphic to the bounded Borel functions on X. Then L(Bo(X)) and its normalizer (the set of the unitaries u in Br*(E,c) such that u*fu in L(Bo(X)), f in L(Bo(X))) countably generates the Borel *-algebra Br*(E,c). In this thesis, we study Br*(E,c) and in particular prove that: i) If E is smooth, then Br*(E,c) is a type I Borel *-algebra (Definition 6.3.10). ii) If E is a hyperfinite, then Br*(E,c) is a Borel AF-algebra (Definition 7.5.1). iii) Generalizing Kumjian's definition, we define a Borel twist G over E and its associated sequentially closed Borel *-algebra Br*(G). iv) Let a Borel Cartan pair (B, Bo) denote a sequentially closed Borel *-algebra B with a Borel Cartan subalgebra Bo, where B is countably Bo-generated. Generalizing Feldman-Moore's result, we prove that any pair (B, Bo) can be realized uniquely as a pair (Br*(E,c), L(Bo(X))). Moreover, we show that the pair (Br*(E,c), L(Bo(X))) is a complete invariant of the countable Borel equivalence relation E. v) We prove a Krieger type theorem, by showing that two aperiodic hyperfinite countable equivalence relations are isomorphic if and only if their associated Borel *-algebras Br*(E1) and Br*(E2) are isomorphic.

To the pair (E,c), where E is a countable Borel equivalence relation on a standard Borel space (X,A) and c a normalized Borel T-valued 2-cocycle on E, we associate a sequentially weakly closed Borel *-algebra Br*(E,c), contained in the bounded linear operators on L^2(E). Associated to Br*(E,c) is a natural (Borel) Cartan subalgebra (Definition 6.4.10) L(Bo(X)) isomorphic to the bounded Borel functions on X. Then L(Bo(X)) and its normalizer (the set of the unitaries u in Br*(E,c) such that u*fu in L(Bo(X)), f in L(Bo(X))) countably generates the Borel *-algebra Br*(E,c). In this thesis, we study Br*(E,c) and in particular prove that: i) If E is smooth, then Br*(E,c) is a type I Borel *-algebra (Definition 6.3.10). ii) If E is a hyperfinite, then Br*(E,c) is a Borel AF-algebra (Definition 7.5.1). iii) Generalizing Kumjian's definition, we define a Borel twist G over E and its associated sequentially closed Borel *-algebra Br*(G). iv) Let a Borel Cartan pair (B, Bo) denote a sequentially closed Borel *-algebra B with a Borel Cartan subalgebra Bo, where B is countably Bo-generated. Generalizing Feldman-Moore's result, we prove that any pair (B, Bo) can be realized uniquely as a pair (Br*(E,c), L(Bo(X))). Moreover, we show that the pair (Br*(E,c), L(Bo(X))) is a complete invariant of the countable Borel equivalence relation E. v) We prove a Krieger type theorem, by showing that two aperiodic hyperfinite countable equivalence relations are isomorphic if and only if their associated Borel *-algebras Br*(E1) and Br*(E2) are isomorphic.

Pour une représentation automorphe cuspidale de GL(2,F) avec F un corps de nombres totalement réel, tel que est de type (k, r) et satisfait une condition de pente non critique, l’on construit une distribution p-adique sur le groupe de Galois de l’extension abélienne maximale de F non ramifiée en dehors de p et 1. On démontre que la distribution obtenue est admissible et interpole les valeurs critiques de la fonction L complexe de la représentation automorphe. Cette construction est basée sur l’étude de la cohomologie de la variété modulaire de Hilbert à coefficients surconvergents. / For each cohomological cuspidal automorphic representation for GL(2,F) where F is a totally real number field, such that is of type (k, r) tand satisfies the condition of non critical slope we construct a p-adic distribution on the Galois group of the maximal abelian extension of F unramified outside p and 1. We prove that the distribution is admissible and interpolates the critical values of L-function of the automorphic representation. This construction is based on the study of the overconvergent cohomology of Hilbert modular varieties.

Dans cette Thèse, nous considérons dans le plan complexe l’équation de la chaleur avec la condition initiale singulière u(0,z)=1/(1-exp(z)). Ce problème de Cauchy possède une unique solution formelle série entière, laquelle peut être sommée par des procédés de sommation différents. Le but est d’établir des relations existant entre les différentes sommes ainsi étudiées: d’une part la somme de Borel de celle-ci et, de l’autre, deux versions q-analogues de la somme de Borel qui sont obtenuesrespectivement avec le noyau de la chaleur et la fonction thêta de Jacobi. Notre analyse sur le phénomène de Stokes correspondant nous conduit à une généralisation d’un résultat de Mordell sur le nombre de classes des formes quadratiques binaires définies et positives. / In this thesis, we consider the heat equation with the singular initial condition u(0,z)=1/(1-exp(z)), where z is a complex variable. The aim is to establish relations among three sums of a divergent formal solution to this Cauchy problem: its Borel-sum and two q-Borel-sums obtained by means of heat kernel and theta function respectively. This Stokes analysis allows us to give a generalization to a classical result of Mordell related to the class numbers of the binary positive-definite quadratic forms.

O presente trabalho tem como objetivo apresentar os conjuntos fortemente nulos e fortemente magros. Mais especicamente, iremos apresentar algumas aplicações e avaliar a independência de ZFC de armações envolvendo tais conjuntos. Com relação às aplicações, daremos alguns exemplos de conjuntos fortemente nulos e fortemente magros, estudaremos a aditividade do ideal formado pelos subconjuntos fortemente nulos da reta real, apresentaremos uma análise da relação entre a propriedade fortemente nulo e translações de subconjuntos da reta, mostraremos equivalências da Conjectura de Borel em espaços métricos, com a armação R-BC e com uma armação envolvendo jogos. Com relação a análise de independência de armações de ZFC, mostraremos que a Conjectura Dual de Borel é independente de ZFC e que a negação da Conjectura de Borel é consistente com ZFC. / The present work aims to present the strongly null and strongly meager sets. More specically, we will present some applications and evaluate the independence of ZFC from statements involving such sets. With respect to the applications, we will give some examples of strongly null and strongly meager sets, we will study the additivity of the ideal formed by the strongly null subsets of the real line, we will present an analysis of the relation between the strongly null property and the subsets of the line, of the Borel Conjecture in metric spaces, with the statement R-BC and with a statement involving games. Regarding the analysis of the independence of ZFC statements, we will show that the Borel Dual Conjecture is independent of ZFC and that the negation of the Borel Conjecture is consistent with ZFC.

No description available.

Thesis (Ph. D.)--Auburn University, 2008. / Abstract. Vita. Includes bibliographical references (p. 76-78).

University, Diss., 2002--Kiel.

The study of Borel equivalence relations on Polish spaces has become a major area of focus within descriptive set theory. Primarily, work in this area has been carried out using the standard methods of descriptive set theory. In this work, however, we develop a model-theoretic framework suitable for the study of Borel equivalence relations, introducing a class of objects we call Borel structurings. We then use these structurings to examine conditions under which marker sets for Borel equivalence relations can be concluded to exist or not exist, as well as investigating to what extent the Compactness Theorem from first-order logic continues to hold for Borel structurings.