String diagrams turn algebraic equations into topological moves that have recurring shapes, involving the sliding of one diagram past another. We individuate, at the root of this fact, the dual nature of polygraphs as presentations of higher algebraic theories, and as combinatorial descriptions of "directed spaces". Operations of polygraphs modelled on operations of topological spaces are used as the foundation of a compositional universal algebra, where sliding moves arise from tensor products of polygraphs. We reconstruct several higher algebraic theories in this framework. In this regard, the standard formalism of polygraphs has some technical problems. We propose a notion of regular polygraph, barring cell boundaries that are not homeomorphic to a disk of the appropriate dimension. We define a category of non-degenerate shapes, and show how to calculate their tensor products. Then, we introduce a notion of weak unit to recover weakly degenerate boundaries in low dimensions, and prove that the existence of weak units is equivalent to a representability property. We then turn to applications of diagrammatic algebra to quantum theory. We re-evaluate the category of Hilbert spaces from the perspective of categorical universal algebra, which leads to a bicategorical refinement. Then, we focus on the axiomatics of fragments of quantum theory, and present the ZW calculus, the first complete diagrammatic axiomatisation of the theory of qubits. The ZW calculus has several advantages over ZX calculi, including a computationally meaningful normal form, and a fragment whose diagrams can be read as setups of fermionic oscillators. Moreover, its generators reflect an operational classification of entangled states of 3 qubits. We conclude with generalisations of the ZW calculus to higher-dimensional systems, including the definition of a universal set of generators in each dimension.

The aim of this work is to study the ring-theoretic properties of the diagonals of a Rees algebra, which from a geometric point of view are the homogenous coordinate rings of embeddings of blow-ups of projective varieties along a subvariety. First we are going to introduce the subject and the main problems. After that we shall review the known results about these problems, and finally we will give a summary of the contents and results obtained in this work. / L’objectiu d’aquesta memòria és l’estudi de les propietats aritmètiques de les diagonals d’una àlgebra de Rees o, des d’un punt de vista geomètric, dels anells de coordenades homogenis d’immersions d’explosions de varietats projectives al llarg d’una subvarietat. En primer lloc, anem a introduir el tema i els principals problemes que tractarem. A continuació, exposarem els resultats coneguts sobre aquests problemes i finalment farem un resum dels resultats obtinguts en aquesta memòria.

In the paper [HSS] Herrlich, Salicrup and Strecker were able to show that Kuratowski / Mrowka's Theorem concerning compactness for topological spaces could be applied to a wider setting. In this dissertation, which is based on the paper [F subscript 1], we interpret Kuratowski / Mrowka's result in the category R-Mod. Chapter One deals mainly with the preliminary definitions and results and we also show that there is a 1-1 correspondence between torsion theories and standard factorisation systems. In Chapter Two we, obtain for every torsion theory T, a theory of T-compactness which is an extension of the definition of compactness found in [HSS]. We then obtain a characterisation of T-compactness under certain conditions on the ring R and torsion theory T. In Chapter Three we examine the class of T-compact R-modules more closely when the ring R is T-hereditary and T-noetherian. We also obtain further characterisation of T-compactness under these additional conditions. In Chapter Four we show that many topological results have analogues in R-Mod. / Mathematical Sciences / M. Sc. (Mathematics)

Orientador: Marcos Benevenuto Jardim / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T16:48:31Z (GMT). No. of bitstreams: 1 Cardoso_NunoFilipedeAndrade_M.pdf: 1858794 bytes, checksum: bbe47182338feb3de60b480df87b52a7 (MD5) Previous issue date: 2014 / Resumo: Os teoremas de degenerescência de Hodge e de anulamento de Kodaira, Akizuki e Nakano são de suma importância na teoria de variedades complexas. Usando o teorema de comparação de Serre, ambos podem ser traduzidos para o contexto de esquemas projetivos e suaves sobre um corpo de característica zero. Para corpos de característica positiva, no entanto, os dois deixam de valer sem hipóteses adicionais, sendo que os primeiros contra-exemplos foram encontrados por Mumford e Raynaud. O objetivo desta dissertação é apresentar um teorema devido a Deligne e Illusie que assegura a degenerescência da seqüência espectral de Hodge-de Rham e uma versão do teorema de Kodaira, Akizuki e Nakano para certos esquemas projetivos e suaves sobre um corpo perfeito de característica positiva. Nos propusemos a dar um tratamento, na medida do possível, auto-suficiente / Abstract: The Hodge degeneration theorem and the Kodaira, Akizuki and Nakano's vanishing theorem are of paramount importance in the theory of complex manifolds. Using Serre's comparison theorem, both can be translated to the context of smooth projective schemes over a field of characteristic zero. For fields of positive characteristic, however, both fail to hold without additional hypothesis, and the first counterexamples were found by Mumford and Raynaud. Our goal in this dissertation is to present a theorem due to Deligne and Illusie that ensures the degeneration of the Hodge-de Rham spectral sequence and a version of the theorem of Kodaira, Akizuki and Nakano for certain smooth projective schemes over a perfect field of positive characteristic. We tried to keep the treatment as self-contained as possible / Mestrado / Matematica / Mestre em Matemática