The influence of Chow groups on singular cohomology is motivated by classical results by Mumford and Roitman and has been investigated extensively. On the other hand, the converse influence is rather conjectural and it takes place in the framework of the ``philosophy of mixed motives'', which is mainly due to Grothendieck, Bloch and Beilinson. In the spirit of exploring this influence, Voisin formulated in 1996 a conjecture on 0--cycles on the self--product of surfaces of geometric genus one. There are few examples in which Voisin's conjecture has been verified, but it is still open for a general $K3$ surface. Our aim is to present a new example in which Voisin's conjecture is true, a family of Todorov surfaces. We give an explicit description of the family as quotient of complete intersection of four quadrics in $mathbb{P}^{6}$. We verify Voisin's conjecture for the family of Todorov surfaces of type $(2,12)$. Our main tool is Voisin's ``spreading of cycles'', we use it to establish a relation between 0--cycles on the Todorov surface and on the associated K3 surface. We give a motivic version of this result and some interesting motivic applications.

No description available.

<p dir="ltr">In this thesis, we prove the rationality results for the ratio of the critical values of certain <i>L</i>-functions, which appear in the constant term of Eisenstein series associated with the exceptional group <i>G</i>_<em>2</em> over a totally imaginary field. Our methodology builds upon the works of Harder and Raghuram, who established rationality results for special values of Rankin-Selberg <i>L</i>-functions for<i> </i><i>GL</i>_<em>n</em><i>× GL</i>_<em>n'</em> by studying the rank-one Eisenstein cohomology of the ambient group <i>GL</i>_n+n' over a totally real field, as well as its generalization by Raghuram [35] for the case over a totally imaginary field.</p><p dir="ltr">The <i>L</i>-functions in this thesis were constructed using the Langlands-Shahidi method for <i>G</i>_<em>2</em> over a totally imaginary field, attached to maximal parabolic subgroups. This is the first instance of applying the Harder-Raghuram method to an exceptional group, and the first case involving more than one function appearing in the constant term. Our results demonstrate the relationship between the rationality of different <i>L</i>-functions appearing in the constant term, allowing one to prove the rationality of one <i>L</i>-function based on the known rationality result of another <i>L</i>-functions.</p>

In this thesis we discuss linear dependence of translations which is intimately related to the zero divisor conjecture. We also discuss the square integrable representations of the generalized Wyle-Heisenberg group in 𝑛² dimensions and its relations with Gabor's question from Gabor Analysis in the light of the time-frequency equation. We study the zero divisor conjecture in relation to the reduced 𝐶*-algebras and operator norm 𝐶*-algebras. For certain classes of groups we address the zero divisor conjecture by providing an isomorphism between the the reduced 𝐶*-algebra and the operator norm 𝐶*-algebra. We also provide an isomorphism between the algebra of weak closure and the von Neumann algebra under mild conditions. Finally, we prove some theorems about the injectivity of some spaces as ℂ𝐺 modules for some groups 𝐺. / Master of Science

Cette thèse s'articule principalement autour de la théorie des codes et des fonctions booléennes liés à la cryptographie. Deux axes sont suivis : la première partie est dédiée à la non-linéarité des fonctions booléennes, alors que la deuxième partie présente des applications en cryptographie d'objets provenant de ces théories. Motivé par la conjecture de Patterson et Wiedemann, nous proposons une généralisation de la construction par réunions d'orbites suivant l'action d'un groupe, où la minimisât!on de l'amplitude spectrale se ramène à deux sous-problèmes que nous étudions : l'estimation de sommes de Gauss et l'estimation de sommes d'exponentielles incomplètes. Plusieurs conditions et pistes de résolution de la conjecture sont alors détaillées. Ce travail nous permet de construire a sympto tique ment des fonctions de non-linéarité plus élevée que la moyenne et nous obtenons de plus, suivant ce principe, un exemple de recollement quadratique hautement non-linéaire proche de la borne de Patterson et Wiedemann en dimension 15. Dans la deuxième partie, nous portons tout d'abord notre attention sur des protocoles cryptographiques dits à faibles ressources. Des fonctions booléennes résistantes à la cryptanalyse différentielle sont utilisées afin de protéger le protocole HB+ d'une attaque par le milieu. À partir d'un deuxième protocole basé sur un principe de bruitage, nous effectuons un parallèle avec la théorie du canal à jarretière de Wyner, ce qui permet d'accroître la sécurité. D'autre part, dans le cadre de l'authentification de données variables dans le temps, une adaptation du cryptosystème de McEliece est détaillée afin de contrôler l'accès aux fonctions de vérification.

We consider the finite two-dimensional Ising model on a lattice with periodic boundaryconditions. Kaufman determined the spectrum of the transfer matrix on the finite,periodic lattice, and her derivation was a simplification of Onsager's famous result onsolving the two-dimensional Ising model. We derive and rework Kaufman's resultsby applying representation theory, which give us a more direct approach to computethe spectrum of the transfer matrix. We determine formulas for the spin correlationfunction that depend on the matrix elements of the induced rotation associated withthe spin operator. The representation of the spin matrix elements is obtained byconsidering the spin operator as an intertwining map. We wrap the lattice aroundthe cylinder taking the semi-infinite volume limit. We control the scaling limit of themulti-spin Ising correlations on the cylinder as the temperature approaches the criticaltemperature from below in terms of a Bugrij-Lisovyy conjecture for the spin matrixelements on the finite, periodic lattice. Finally, we compute the matrix representationof the spin operator for temperatures below the critical temperature in the infinite-volume limit in the pure state defined by plus boundary conditions.

Cette thèse porte sur la famille des nombres dits multizêtas, et sur les relations qu'ils vérifient.Le premier chapitre est une introduction générale au domaine et se donne pour objectif de présenter brièvement les différents cadres dans lesquels s'inscrivent les résultats des trois autres chapitres, et d'énoncer ces résultats.Dans le chapitre 2, on étudie les relations linéaires entre zêtas simples et zêtas doubles, en établissant un lien rigoureux entre ces relations, les relations linéaires entre crochets de Poisson d'éléments de profondeur 1 de l'algèbre de Lie libre à deux générateurs, et l'espace des formes modulaires. Il s'agit en grande partie d'algèbre linéaire élémentaire sur des matrices définies explicitement.Le résultat principal du chapitre 3 a trait à une algèbre de Lie de dérivations déduite de l'étude de la catégorie des motifs elliptiques mixtes introduite par Hain et Matsumoto. Il démontre l'existence de relations linéaires observées par Pollack dans cette algèbre et provenant elles aussi des formes modulaires. Les démonstrations consistent majoritairement à adapter des techniques introduites par Ecalle à l'étude des propriétés de certains polynômes non commutatifs.Le quatrième et dernier chapitre propose une construction d'une algèbre de multizêtas elliptiques formels, en analogie avec les travaux de Hain et Matsumoto sur les motifs elliptiques mixtes et d'Enriquez sur les associateurs elliptiques. Celle-ci se place dans le formalisme écallien des moules ; on prouve deux résultats partiels qui corroborent la validité de cette dernière construction.

Neste trabalho procuramos realizar uma apresentação autocontida sobre os conceitos da teoria algébrica de formas quadráticas e sobre os anéis graduados que surgiram no desenvolvimento desta teoria. Iniciamos procurando esclarecer o sentido da equivalência entre as várias acepções do conceito de forma quadrática. Após a apresentação de ingredientes e resultados geométricos, fazemos um extrato da teoria dos anéis de Witt, conceito que originou a moderna teoria algébrica de formas quadráticas. Disponibilizamos os elementos fundamentais para a formulação das teorias de cohomologia, nos concentrado no desenvolvimento da teoria de cohomologia profinita e, sobretudo, galoisiana. Descrevemos os funtores K0, K1 e K2 da K-teoria clássica e também a K-teoria de Milnor, que é mais adequada para formular questões sobre formas quadráticas. Finalizamos o trabalho com a apresentação de alguns conceitos da Teoria dos Grupos Especiais, uma codificação em primeira-ordem da teoria algébrica das formas quadráticas e exemplificamos sua importância, fornecendo um extrato da prova realizada por Dickmann-Miraglia da conjectura de Marshall sobre assinaturas, que se baseia fortemente nesta teoria. / In this work I try to provide a self-contained presentation on the concepts of algebraic theory of quadratic forms and on the graded rings that have emerged in the development of this theory. I started trying to clarify the meaning of \"equivalence\"between the various meanings of the concept of quadratic form. After the presentation of geometrical ingredients and results, we make an extract of the theory of Witt rings, a concept that originated the modern algebraic theory of quadratic forms. It is provided the key elements for the formulation of cohomology theories, focusing on the development of profinite cohomology theory and, especially, on galoisian cohomology. Are described the functors K0, K1 and K2 of classical K-theory and also the Milnor K-theory, which is more appropriate to formulate questions about quadratic forms. The dissertation is finished with the presentation of some concepts of the Theory of Special Groups, a first-order encoding of algebraic theory of quadratic forms, and with an example its importance by providing an extract of proof by Dickmann-Miraglia of the Marshalls conjecture on signatures, which relies heavily on this theory.

Made available in DSpace on 2016-04-27T17:27:08Z (GMT). No. of bitstreams: 1 Rogerio Bettin.pdf: 901827 bytes, checksum: a7d6c66d907176975e5b667d88dcdc29 (MD5) Previous issue date: 2014-09-01 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This paper aims at analyzing the plurality of realities of worlds of knowledge in Karl Popper. In the first section, we have chosen to analyze the object studied respecting the chronological order of the Popperian publications, bearing in mind the verification of the development of the three worlds thesis in Popper. The author rejects both monistic and dualistic positions and hence proposes a notion of a tripartite reality, claiming that reality is made up by the interaction among three worlds: World 1, of physical objects and material states; World 2, of states of consciousness or mental states or, maybe, of behavioral willingness to act, the world of subjective knowledge; and World 3, of objective autonomous knowledge, which doesn't depend on the subject who knows. World 3 is inhabited by problems, critical arguments and theories, as a result of the evolution of human language. It contains the history of our ideas, of how we invent and react to such products of our own elaboration of objective contents of thinking. In the second section, aiming at better understanding the three worlds theory, even though it is metaphysical, we present a connection between this thesis and the Popperian epistemology, known as critical rationalism. For the author, scientific knowledge is fallible, correctable and provisional, thus making criticism assume a crucial role in the development of knowledge. Therefore, as we analyse the thesis of the three worlds inserted in Popperian epistemology, we can better understand some aspects of the theory of the three worlds, as well as how knowledge grows, according to the presuppositions defended by Karl Popper / Esta pesquisa tem o objetivo de analisar a pluralidade de realidades de mundos do conhecimento em Karl Popper. Na primeira seção, optamos por analisar o objeto aqui estudado respeitando a ordem cronológica das publicações popperianas, tendo em vista a verificação do desenvolvimento da tese dos três mundos em Popper. O autor não aceita as posições monistas e dualistas, por isso que ele propõe uma noção de realidade tripartite, ao afirmar que a realidade é composta pela interação de três mundos: mundo um, dos objetos físicos ou de estados materiais; mundo dois, de estados de consciência ou de estados mentais, ou, talvez, de disposições comportamentais para agir é o mundo do conhecimento subjetivo; e, mundo três, do conhecimento objetivo e autônomo que independe do sujeito que conhece. Este é habitado pelos problemas, argumentos críticos e teorias, como resultado da evolução da linguagem humana. O mundo três é a história de nossas ideias, de como a inventamos e reagimos diante desses produtos de nossas próprias elaborações de conteúdos objetivos de pensamento. Na segunda seção, com o intuito de melhor compreendermos a tese dos três mundos, mesmo sendo uma teoria metafísica, apresentaremos uma conexão desta tese em relação a epistemologia popperiana - o racionalismo crítico. Para o autor, todo conhecimento científico é falível, corrigível e provisório, tendo a crítica papel fundamental para o desenvolvimento do conhecimento. Portanto, ao analisarmos a o estatuto da tese dos três mundos inserida na epistemologia popperiana, nos será permitido compreender melhor alguns aspectos da teoria dos três mundos, assim como de que forma ocorre o crescimento do conhecimento, segundo os pressupostos defendidos por Karl Popper

Nous présentons des résultats obtenus dans cinq directions autour des représentations des algèbres affines quantiques $\U_q(\hat{\Glie})$. En premier lieu nous prouvons la conjecture de Kirillov-Reshetikhin, c'est-à-dire des formules de caractères pour certaines représentations de dimension finie de $\U_q(\hat{\Glie})$, et nous étendons le résultat à des affinisations minimales; nous étendons le modèle monomial des cristaux aux représentations extrémales et nous y interprétons des automorphismes de Kashiwara. Ensuite, à l'interface avec la géométrie algébrique, nous définissons une notion de groupes de lacets analytiques avec une factorisation de Riemann-Hilbert qui permet de réaliser géométriquement le centre de $\U_q(\hat{\Glie})$ aux racines de $1$. Comme application, nous paramétrisons des classes d'équivalences de représentations de $\U_q(\hat{\Glie})$ par des $G$-fibrés sur une courbe elliptique. On résoud le problème de petitesse géométrique posé par Nakajima pour des résolutions de variétés carquois. Troisièmement, nous établissons une nouvelle dualité de Langlands pour des représentations de $\Glie$ et de $\U_q(\hat{\Glie})$ et nous définissons des groupes quantiques d'interpolation pour l'interpréter. Quatrièmement, nous construisons une catégorie tensorielle pour les algèbres affinisées quantiques et des représentations de dimension finie d'algèbres toroïdales quantiques (et de Cherednik); nous proposons un analogue en théorie de Lie des algèbres de réflexion symplectiques. Enfin, nous obtenons des catégorifications monoïdales d'algèbres cluster en terme d'une catégorie $\mathcal{C}_1$ de représentations de $\U_q(\hat{\Glie})$. Pour ce faire, nous établissons notamment la factorisation en modules premiers de modules simples de $\mathcal{C}_1$.