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We discuss the quantum Poincar\'e symmetries of the $\varrho$-Minkowski spacetime, a space characterised by an angular form of noncommutativity. We show that it is possible to give them both a bicrossproduct and a Drinfel'd twist structure. We also obtain a new noncommutative $\star$-product, which is cyclic with respect to the standard integral measure., Comment: 38 pages, including appendices

We study general symmetric powers $S^k L^{[n]}$ of a tautological bundle $L^{[n]}$ on the Hilbert scheme $X^{[n]}$ of $n$ points over a smooth quasi-projective surface $X$, associated to a line bundle $L$ on $X$. Let $V_L$ be the $\mathfrak{S}_n$-vector bundle on $X^n$ defined as the exterior direct sum $L \boxplus \cdots \boxplus L$. We prove that the Bridgeland-King-Reid transform $\mathbf{\Phi}(S^k L^{[n]})$ of symmetric powers $S^k L^{[n]}$ is quasi isomorphic to the last term of a finite decreasing filtration on the natural vector bundle $S^k V_L$, defined by kernels of operators $D^l_L$, which operate locally as higher order restrictions to pairwise diagonals. We use this description and the natural filtration on $(S^k V_L)^{\mathfrak{S}_n}$ induced by the decomposition in direct sum, to obtain, for $n =2$ or $k \leq 4$, a finite decreasing filtration $\mathcal{W}^\bullet$ on the direct image $\mu_*(S^k L^{[n]})$ for the Hilbert-Chow morphism whose graded sheaves we control completely. As a consequence of this structural result, we obtain a chain of cohomological consequences, like a spectral sequence abutting to the cohomology of symmetric powers $S^k L^{[n]}$, an effective vanishing theorem for the cohomology of symmetric powers $S^k L^{[n]} \otimes \mathcal{D}_A$ twisted by the determinant, in presence of adequate positivity hypothesis on $L$ and $A$, as well as universal formulas for their Euler-Poincar\'e characteristic., Comment: pdflatex, 70 pages

This paper describes, from an intra-dialect point of view, the phonological traits of pronoun clitization in Algherese. The approach is based on the restrictions upon the syllabic structure of this variety, which tends to avoid the coda between clitics and verbs and between one clitic and another, It almost does not allow any type of coda in cases of clitization, which are solved through rules of assimilation and simplification, through the insertion of an epenthetic [a] between the verb and the clitic, or by modifying the ending, which leads to a wide range of homophonic sequences. The study also presents an update on the syntactic restrictions upon pronoun combinations and the phonetic peculiarities of some clitic combinations, more specifically those including ne and se, with a wealth of examples.

This paper describes, from an intra-dialect point of view, the phonological traits of pronoun clitization in Algherese. The approach is based on the restrictions upon the syllabic structure of this variety, which tends to avoid the coda between clitics and verbs and between one clitic and another, It almost does not allow any type of coda in cases of clitization, which are solved through rules of assimilation and simplification, through the insertion of an epenthetic [a] between the verb and the clitic, or by modifying the ending, which leads to a wide range of homophonic sequences. The study also presents an update on the syntactic restrictions upon pronoun combinations and the phonetic peculiarities of some clitic combinations, more specifically those including ne and se, with a wealth of examples.

We consider the $\varrho$-Minkowski spacetime, a model with linear noncommutativity involving the time and the azimuthal angle. We study its quantum symmetries, the $\varrho$-Poincar\'e quantum group, and analyse the concepts of localizability and quantum observers., Comment: Minor typos corrected in v2

Orientador: Marcos Benevenuto Jardim Tese (doutorado) Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica Resumo: O espaço de moduli das representações do quiver ADHM na categoria de espaços vetoriais complexos é isomorfo ao espaço de moduli dos feixes livres de torção no plano projetivo. Em particular, quando são considerados os feixes livres de torção de posto 1, os dados ADHM dão uma descrição do esquema de Hilbert de n pontos no plano complexo, para algum inteiro n. De modo semelhante, afirmamos e provamos uma relação entre o espaço de moduli das representações estáveis do quiver ADHM aumentado e o esquema de Hilbert aninhado Abstract: The moduli space of representations of the ADHM quiver in the category of complex vector spaces is isomorphic to moduli space of framed torsion-free sheaves on the projective plane. In particular, when we consider rank 1 torsion-free sheaves, the ADHM data describe the Hilbert scheme of n points in the complex plane, for some given integer n. Similarly, we state and prove a relation between the moduli space of stable framed representations of the enhanced ADHM quiver and the nested Hilbert scheme Doutorado Matemática Doutora em Matemática FAPESP 2009/16646-1

Orientador: Marcos Benevenuto Jardim Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica Resumo: Nesta dissertação faremos uma apresentação das equações de Seiberg-Witten. Mostraremos a não existência de soluções não triviais em variedades com curvatura escalar não negativa. Este último resultado será apresentado como uma consequência da fórmula de Weizenböck para o operador de Dirac Abstract: In this work we do an presentation of the equation of Seiberg-Witten. We show the non-existence of non-trivial solutions in manifolds with scalar curvature non-negative. This latter result is displayed as a result of Weizenböck formula for the Dirac operator Mestrado Matemática Mestre em Matemática FAPESP 2014/18855-5 CNPQ