Esta tese aborda, como a despeito da rigidez da álgebra de Weyl An(k), suas subálgebras de invariantes possuem uma rica teoria de invariantes: do ponto de vista de estrutura, se fizermos um estudo de equivalência birracional dentro da filosofia de Gelfand-Kirillov, temos o Problema de Noether Não-Comutativo, sobre o qual obtemos vários novos resultados (Capítulo 4). Do ponto de vista de representações, obtemos que suas subálgebras de invariantes, em vários casos, herdam de maneira natural a estrutura de módulos de Gelfand-Tsetlin da álgebra de Weyl (Capítulo 5), assim como uma noção natural de módulos holonômicos (Capítulo 6). Analisaremos resultados similares para outras álgebras semelhantes a Álgebra de Weyl, como anéis de operadores diferenciais no toro e álgebras de Weyl generalizadas (Capítulos 2, 4 e 5). Como aplicações, temos uma Conjectura de Gelfand-Kirillov para subálgebras esféricas de Cherednik (Capítulo 4); para a Conjectura de Gelfand-Kirillov para várias álgebras de Galois (Capítulos 5 e 7); e o problema de realizar U(L), em que L é uma algebra de Lie simples de tipo B,C,D, como uma ordem de Galois generalizando o caso de gln (Capítulo 5). Um Capítulo sobre o Problema de Noether Quântico e um resumo do artigo de Futorny e Schwarz, \"Quantum Linear Galois Algebras\", encerram a tese. / This thesis discussess how, given the rigidity results on the Weyl Algebra An(k), its invariant subrings can nonetheless have an interesting invariant theory: from the structural point of view, a birrational equivalence study under the Gelfand-Kirillov philosophy gives us the Noncommutative Noether Problem, of which we obtain many new results (Chapter 4). From the point of view of representations, we obtain that their invariant rings, in many cases, have a natural theory of Gelfand-Tsetlin modules just like the Weyl Algebra (Chapter 5), and a natural notion of holonomic modules (Chapter 6). We discuss analogues results for algebras which are similar to the Weyl Algebra, such as the ring of differential operators on the torus and the generalized Weyl algebras (Chapters 2,4,5). As applications, we have a Gelfand-Kirillov Conjecture for spherical subalgebras of Cherednik (Chapter 4); for the Gelfand-Kirillov Conjecture of many Galois algebras (Chapter 5 and 7); and the problem to give a Galois structure to the algebra U(L), where L is a simple Lie algebra of type B,C,D -generalizing the case A (Chapter 5). A chapter about the Quantum Noether Problem and a resume of the article Quantum Linear Galois Algebras\" ends the thesis.

Esta tese aborda, como a despeito da rigidez da álgebra de Weyl An(k), suas subálgebras de invariantes possuem uma rica teoria de invariantes: do ponto de vista de estrutura, se fizermos um estudo de equivalência birracional dentro da filosofia de Gelfand-Kirillov, temos o Problema de Noether Não-Comutativo, sobre o qual obtemos vários novos resultados (Capítulo 4). Do ponto de vista de representações, obtemos que suas subálgebras de invariantes, em vários casos, herdam de maneira natural a estrutura de módulos de Gelfand-Tsetlin da álgebra de Weyl (Capítulo 5), assim como uma noção natural de módulos holonômicos (Capítulo 6). Analisaremos resultados similares para outras álgebras semelhantes a Álgebra de Weyl, como anéis de operadores diferenciais no toro e álgebras de Weyl generalizadas (Capítulos 2, 4 e 5). Como aplicações, temos uma Conjectura de Gelfand-Kirillov para subálgebras esféricas de Cherednik (Capítulo 4); para a Conjectura de Gelfand-Kirillov para várias álgebras de Galois (Capítulos 5 e 7); e o problema de realizar U(L), em que L é uma algebra de Lie simples de tipo B,C,D, como uma ordem de Galois generalizando o caso de gln (Capítulo 5). Um Capítulo sobre o Problema de Noether Quântico e um resumo do artigo de Futorny e Schwarz, \"Quantum Linear Galois Algebras\", encerram a tese. / This thesis discussess how, given the rigidity results on the Weyl Algebra An(k), its invariant subrings can nonetheless have an interesting invariant theory: from the structural point of view, a birrational equivalence study under the Gelfand-Kirillov philosophy gives us the Noncommutative Noether Problem, of which we obtain many new results (Chapter 4). From the point of view of representations, we obtain that their invariant rings, in many cases, have a natural theory of Gelfand-Tsetlin modules just like the Weyl Algebra (Chapter 5), and a natural notion of holonomic modules (Chapter 6). We discuss analogues results for algebras which are similar to the Weyl Algebra, such as the ring of differential operators on the torus and the generalized Weyl algebras (Chapters 2,4,5). As applications, we have a Gelfand-Kirillov Conjecture for spherical subalgebras of Cherednik (Chapter 4); for the Gelfand-Kirillov Conjecture of many Galois algebras (Chapter 5 and 7); and the problem to give a Galois structure to the algebra U(L), where L is a simple Lie algebra of type B,C,D -generalizing the case A (Chapter 5). A chapter about the Quantum Noether Problem and a resume of the article Quantum Linear Galois Algebras\" ends the thesis.

L’objet de cette thèse est l’étude de la propagation de la 2-birationalité pour les 2-extensions de corps de nombres. Le problème étudié se présente comme suit : étant donnés un corps 2-rationnel totalement réel K, une extension quadratique totalement imaginaire L de K, et une 2-extension totalement réelle de K de K, à quelles conditions la 2-birationalité du compositum L = KL se lit-elle sur L ? La thèse se structure en trois parties : l’étude du cas absolument quadratique d’abord, le cas relativement quadratique ensuite ; le cas général enfin. Le résultat principal de la thèse résout complètement le problème posé en toute généralité. En fin de thèse, diverses illustrations numériques sont proposées à l’aide du PARI, ainsi qu’une étude des tours d’extensions 2-birationnelles. / This thesis deals with the propagation of 2-birationality for 2-extensions of numbers fields. More precisely, le t K be a 2-rational totally real number field, L a CM quadratic extension of K, and let K be a totally real 2-extension of K. Under which conditions can one read the 2-birationaltiy of the compositum L = LK from L ? This work is divided into three parts : we first study the absolute quadratic case, then the relatively quadratic case, then finally the general case. The thesis’s main result solves the whole problem. We also illustrate the result with various numeric examples, obtained with PARI and a focus at the end on 2-birational extensions’ towers.

Cette thèse est consacrée à l'étude de la géométrie des variétés de Fano complexes en utilisant les propriétés des sous-faisceaux du fibré tangent et la géométrie du diviseur fondamental. Les résultats principaux compris dans ce texte sont : (i) Une généralisation de la conjecture de Hartshorne: une variété lisse projective est isomorphe à un espace projectif si et seulement si son fibré tangent contient un sous-faisceau ample.(ii) Stabilité du fibré tangent des variétés de Fano lisses de nombre de Picard un : à l'aide de théorèmes d'annulation sur les espaces hermitiens symétriques irréductibles de type compact M, nous montrons que pour presque toute intersection complète générale dans M, le fibré tangent est stable. La même méthode nous permet de donner une réponse sur la stabilité de la restriction du fibré tangent de l'intersection complète à une hypersurface générale.(iii) Non-annulation effective pour des variétés de Fano et ses applications : nous étudions la positivité de la seconde classe de Chern des variétés de Fano lisses de nombre de Picard un. Ceci nous permet de montrer un théorème de non-annulation pour les variétés de Fano lisses de dimension n et d'indice n-3. Comme application, nous étudions la géométrie anticanonique des variétés de Fano et nous calculons les constantes de Seshadri des diviseurs anticanoniques des variétés de Fano d'indice grand.(iv) Diviseurs fondamentaux des variétés de Moishezon lisses de dimension trois et de nombre de Picard un : nous montrons l'existence d'un diviseur lisse dans le système fondamental dans certain cas particulier. / This thesis is devoted to the study of complex Fano varieties via the properties of subsheaves of the tangent bundle and the geometry of the fundamental divisor. The main results contained in this text are:(i) A generalization of Hartshorne's conjecture: a projective manifold is isomorphic to a projective space if and only if its tangent bundle contains an ample subsheaf.(ii) Stability of tangent bundles of Fano manifolds with Picard number one: by proving vanishing theorems on the irreducible Hermitian symmetric spaces of compact type M, we establish that the tangent bundles of almost all general complete intersections in M are stable. Moreover, the same method also gives an answer to the problem of stability of the restriction of the tangent bundle of a complete intersection on a general hypersurface.(iii) Effective non-vanishing for Fano varieties and its applications: we study the positivity of the second Chern class of Fano manifolds with Picard number one, this permits us to prove a non-vanishing result for n-dimensional Fano manifolds with index n-3. As an application, we study the anticanonical geometry of Fano varieties and calculate the Seshadri constants of anticanonical divisors of Fano manifolds with large index.(iv) Fundamental divisors of smooth Moishezon threefolds with Picard number one: we prove the existence of a smooth divisor in the fundamental linear system in some special cases.