La conjecture de Kadison-Singer traitant de l’existence et de l’unicité d’extension d’état pur de la C*-algèbre des opérateurs diagonaux dans B(H) sur B(H) fut émise en 1959 par Kadison et Singer. Il faut attendre jusqu’en 2013 pour que l’une de ses équivalences soit finalement résolue. La première partie de ce mémoire étudie le lien entre la conjecture et le résultat prouvé via deux autres équivalences. La seconde partie traite en profondeur de la preuve du résultat en passant par plusieurs concepts, tels que les familles entrelacées, la notion de stabilité et le polynôme caractéristique mixte. Enfin, la dernière partie porte sur une équivalence particulière, soit la conjecture de Feichtinger.

Edmonds and Giles conjectured that the maximum number of directed joins in a packing is equal to the minimum weight of a directed cut, for any weighted directed graph. This is a generalization of Woodall's Conjecture (which is still open). Schrijver found the first known counterexample to the Edmonds-Giles Conjecture, while Cornuejols and Guenin found the next two. In this thesis we introduce new counterexamples, and prove that all minimal counterexamples of a certain type have now been found.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Dans leur article de 2015 intitulé "On quantitative operator K-theory", H. Oyono-Oyono et G. Yu introduisent un raffinement de la K-théorie opératorielle adapté au cadre desC*-algèbres filtrées, appelé K-théorie quantitative ou contrôlée. Dans cette thèse, nous généralisons la notion de filtration de C_-algèbres. Nous montrons ensuite que ce cadre contient celui déjà traité par G. Yu et H. Oyono-Oyono, tout en se révélant assez souple pour traiter les produits croisés de groupoïdes étalés et de groupes quantiques discrets. Nous construisons ensuite des applications d'assemblage _a valeurs dans les groupes de K-théorie contrôlée associés, pour les C*-algèbres de Roe à coefficients et les produits croisés de groupoïdes étalés. Nous montrons que ces applications factorisent les applications d'assemblage usuelles de Baum-Connes. Nous prouvons ensuite ce que nous appelons des énoncés quantitatifs, et nous montrons qu'une version contrôlée de la conjecture de Baum-Connes est vérifiée pour une large classe de groupoïdes étalés. La fin de la thèse est consacrée à plusieurs applications de ces résultats. Nous montrons que l'application d'assemblage contrôlée coarse est équivalente à son analogue à coefficients pour le groupoïde coarse introduit par G. Skandalis, J-L. Tu et G. Yu. Nous donnons ensuite une preuve que les espaces coarses qui admettent un plongement hilbertien fibré vérifient la version maximale de la conjecture de Baum-Connes coarse contrôlée. Enfin nous étudions les groupoïdes étalés dont toutes les actions propres sont localement induites par des sous-groupoïdes compacts ouverts, dont un exemple est donné par les groupoïdes amples introduits par J. Renault. Nous développons un principe de restriction pour cette classe de groupoïdes, et prouvons que, sous des hypothèses raisonnables, leurs produits croisés vérifient la formule de Künneth en K-théorie contrôlée / In their paper entitled "On quantitative operator K-theory", H. Oyono-Oyono and G. Yu introduced a refinement of operator K-theory, called quantitative or controlled K-theory, adapted to the setting of filtered C_-algebras. In this thesis, we generalize filtration of C*-algebras. We show that this setting contains the theory developed by H. Oyono-Oyono and G. Yu, and is general enough to be applied to the setting of crossed products by étale groupoids and discrete quantum groups. We construct controlled assembly maps with values into this controlled K-groups, for Roe C*-algebras and crossed products by étale groupoids. We show that these controlled assembly maps factorize the usual Baum-Connes and coarse Baum-Connes assembly maps. We prove statements called quantitative statements, and we show that a controlled version of the Baum-Connes conjecture is satisfied for a large class of étale groupoids. The end of the thesis is devoted to several applications of these results. We show that the controlled coarse assembly map is equivalent to its analog with coefficients for the coarse groupoid introduced by G. Skandalis, J-L. Tu and G. Yu. We give a proof that coarse spaces which admit a _bred coarse embedding into Hilbert space satisfy the maximal controlled coarse Baum-Connes conjecture. Finally, we study étale groupoids whose proper actions are locally induced by compact open subgroupoids, e.g. ample groupoids introduced by J. Renault. We develop a restriction principle for these groupoids, and prove that under suitable assumptions, their crossed products satisfy the controlled Künneth formula

The birational variant of Grothendieck's section conjecture proposes a characterisation of the rational points of a curve over a finitely generated field over Q in terms of the sections of the absolute Galois group of its function field. While the p-adic version of the birational section conjecture has been proven by Jochen Koenigsmann, and improved upon by Florian Pop, the conjecture in its original form remains very much open. One hopes to deduce the birational section conjecture over number fields from the p-adic version by invoking a local-global principle, but if this is achieved the problem remains to deduce from this that the conjecture holds over all finitely generated fields over Q. This is the problem that we address in this thesis, using an approach which is inspired by a similar result by Mohamed Saïdi concerning the section conjecture for étale fundamental groups. We prove a conditional result which says that, under the condition of finiteness of certain Shafarevich-Tate groups, the birational section conjecture holds over finitely generated fields over Q if it holds over number fields.

First, we consider the group integrals where integrands are the monomials of matrix elements of irreducible representations of classical groups. These group integrals are invariants under the group action. Based on analysis on Young tableaux, we investigate some related duality theorems and compute the asymptotics of the group integrals for fixed signatures, as the rank of the classical groups go to infinity. We also obtain the Viraosoro constraints for some partition functions, which are power series of the group integrals. Second, we show that the proof of Witten's conjecture can be simplified by using the fermion-boson correspondence, i.e., the KdV hierarchy and Virasoro constraints of the partition function in Witten's conjecture can be achieved naturally. Third, we consider the partition function involving the invariants that are intersection numbers of the moduli spaces of holomorphic maps in nonlinear sigma model. We compute the commutator of the representation of Virasoro algebra and give a fat graph(ribbon graph) interpretation for each term in the diferential operators.

Edmonds and Giles conjectured that the maximum number of directed joins in a packing is equal to the minimum weight of a directed cut, for any weighted directed graph. This is a generalization of Woodall's Conjecture (which is still open). Schrijver found the first known counterexample to the Edmonds-Giles Conjecture, while Cornuejols and Guenin found the next two. In this thesis we introduce new counterexamples, and prove that all minimal counterexamples of a certain type have now been found.

In 1986, Gupta and Murty proved the Lang-Trotter conjecture in the case of elliptic curves having complex multiplication, conditional on the generalized Riemann hypothesis. That is, given a non-torsion point P∈E(ℚ), they showed that P (mod p) generates E(𝔽p) for infinitely many primes p, conditional on the generalized Riemann hypothesis. We demonstrate that Gupta's and Murty's result can be translated into an unconditional result in the language of Drinfeld modules. We follow the example of Hsu and Yu, who proved Artin's conjecture unconditionally in the case of sign normalized rank one Drinfeld modules. Further, we will cover all necessary background information.

In 1986, Gupta and Murty proved the Lang-Trotter conjecture in the case of elliptic curves having complex multiplication, conditional on the generalized Riemann hypothesis. That is, given a non-torsion point P∈E(ℚ), they showed that P (mod p) generates E(𝔽p) for infinitely many primes p, conditional on the generalized Riemann hypothesis. We demonstrate that Gupta's and Murty's result can be translated into an unconditional result in the language of Drinfeld modules. We follow the example of Hsu and Yu, who proved Artin's conjecture unconditionally in the case of sign normalized rank one Drinfeld modules. Further, we will cover all necessary background information.

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