Made available in DSpace on 2016-04-27T16:58:17Z (GMT). No. of bitstreams: 1 Rute da Cunha Pires.pdf: 2876100 bytes, checksum: 0b3c757d2f3a86d214340a76204d7046 (MD5) Previous issue date: 2006-08-11 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This work aimed to establish the importance of the French group called Nicolas Bourbaki in the development of the mathematical research and teaching instruction at the Faculdade de Filosofia, Ciências e Letras da Universidade de São Paulo. The reasons for the presence, for intermittent periods, from 1945 to 1966, of some of the most important mathematicians of the Bourbaki group at the college math department is discussed and how the perception and assimilation of the bourbakist structure of the mathematics has been transmitted and re-read by the academic community of the University of São Paulo at the time. Documents and bibliographical material were acquired in order to constitute, characterized and built the research goal. Bourbaki presence at São Paulo University was mainly due to two factors, the II world war and the bounds between professors from USP and those from other parts of the world that have been here around the time of its foundation. A great number of courses and conferences was given by the group, while here in Brazil, and through those we can raise parameters that point out to the influence of the Bourbaki perspective on the mathematics production at the university / Este trabalho teve como objetivo retratar o grupo francês Nicolas Bourbaki, e o Departamento de Matemática da Faculdade de Filosofia, Ciências e Letras da Universidade de São Paulo, bem como investigar a que se deve a presença, por períodos intermitentes, entre os anos de 1945 e 1966, de alguns dos mais importantes membros do grupo Bourbaki, neste Departamento e de que modos a perspectiva estruturalista bourbakista da matemática, teria sido, por um lado transmitida por eles e, por outro lado, recebida, apropriada e re-significada pela comunidade acadêmico-institucional de professores do Departamento de Matemática da USP, no que diz respeito à produção da pesquisa em Matemática e à formação do bacharel em matemática e do professor de matemática. Para o desenvolvimento do trabalho, a base documental e bibliográfica foi escolhida com o intuito de levantar, caracterizar e constituir o objeto da pesquisa. A presença de Bourbaki na Universidade de São Paulo se deve a dois fatores: a Segunda Guerra Mundial e as relações entre os professores da USP e os professores estrangeiros que nela estiveram quando da criação da mesma. Inúmeros cursos e conferências foram realizados durante a permanência no departamento destes membros do grupo Bourbaki, onde puderam transmitir seu ponto de vista estrutural da Matemática. Através das concepções de Bourbaki e dos tópicos de matemática contemplados nos cursos e conferências ministrados pelos membros do grupo junto ao Departamento, pode-se levantar parâmetros que pudessem indicar a influência da perspectiva bourbakista da matemática, nas teses para professor catedrático, nas teses de doutoramento e nos programas para o curso de Matemática. Concluiu-se que esta influência é incontestável

<div>In the first part of this thesis, we study Rees algebras of modules. We investigate their Cohen-Macaulay property and their defining ideal, using <i>generic Bourbaki ideals</i>. These were introduced by Simis, Ulrich and Vasconcelos in [65], in order to characterize the Cohen-Macaulayness of Rees algebras of modules. Thanks to this technique, the problem is reduced to the case of Rees algebras of ideals. Our main results are the following.</div><div><br></div><div><div>In Chapters 3 and 4 we consider a finite module <i>E</i> over a Gorenstein local ring <i>R</i>. In Theorem 3.2.4 and Theorem 4.3.2, we give sufficient conditions for <i>E</i> to be of linear type, while Theorem 4.2.4 provides a sufficient condition for the Rees algebra <i>R(E)</i> of <i>E</i> to be Cohen-Macaulay. These results rely on properties of the residual intersections of a generic Bourbaki ideal <i>I</i> of<i> E</i>, and generalize previous work of Lin (see [46, 3.1 and 3.4]). In the case when <i>E</i> is an ideal, Theorem 4.2.4 had been previously proved independently by Johnson and Ulrich (see [39, 3.1]) and Goto, Nakamura and Nishida (see [20, 1.1 and 6.3]).</div></div><div><br></div><div><div>In Chapter 5, we consider a finite module <i>E</i> of projective dimension one over <i>k</i>[X₁, . . . , X_n]. Our main result, Theorem 5.2.6, describes the defining ideal of <i>R(E)</i>, under the assumption that the presentation matrix φ of <i>E</i> is <i>almost linear</i>, i.e. the entries of all but one column of φ are linear. This theorem extends to modules a known result of Boswell and Mukundan on the Rees algebra of almost linearly presented perfect ideals of height 2 (see [5, 5.3 and 5.7]).</div></div><div><br></div><div><div>The second part of this thesis studies the Cohen-Macaulay property of the special fiber ring<i> F(E)</i> of a module <i>E</i>. In Theorem 6.2.14, we prove that the generic Bourbaki ideals of Simis, Ulrich and Vasconcelos allow to reduce the problem to the case of fiber cones of ideals, similarly as for Rees algebras. We then provide sufficient conditions for <i>F(E)</i> to be Cohen-Macaulay. Our Theorems 6.2.15, 6.1.3 and 6.2.18 are module versions of results proved for the fiber cone of an ideal by Corso, Ghezzi, Polini and Ulrich (see [10, 3.1] and [10, 3.4]) and by Monta˜no (see [47, 4.8]), respectively.</div></div><div><br></div>

What exactly can we understand by "structural mathematics"? Mathematical structures was a concept which dominated images of mathematics of the twentieth century. It was followed by an effort to transpose the work of Nicolas Bourbaki and his contemporaries into high school education. Progress and results of so-called "modernization of school mathematics" were often received with criticism and hesitation. Using didactical-mathematical concepts of constructivism, languages of mathematics and their various representations and instrumental realism we analyze the principal reasons of that failure. Concurrently we show how it is possible to revive its teaching by broader conception of "structuring" in mathematics. So from mathematical structures we gradually pass to structural mathematics.

<p>The defining equations of Rees algebras provide a natural pathway to study these rings. However, information regarding these equations is often elusive and enigmatic. In this dissertation we study Rees algebras of particular classes of ideals and modules over hypersurface rings. We extend known results regarding Rees algebras of ideals and modules to this setting and explore the properties of these rings.</p> <p><br></p> <p>The majority of this thesis is spent studying Rees algebras of ideals in hypersurface rings, beginning with perfect ideals of grade two. After introducing certain constructions, we arrive in a setting similar to the one encountered by Boswell and Mukundan in [3]. We establish a similarity between Rees algebras of ideals with linear presentation in hypersurface rings and Rees algebras of ideals with <em>almost</em> linear presentation in polynomial rings. Hence we adapt the methods developed by Boswell and Mukundan in [3] to our setting and follow a path parallel to theirs. We introduce a recursive algorithm of <em>modified Jacobian dual iterations</em> which produces a minimal generating set for the defining ideal of the Rees algebra.</p> <p><br></p> <p>Once success has been achieved for perfect ideals of grade two, we consider perfect Gorenstein ideals of grade three in hypersurface rings and their Rees algebras. We follow a path similar to the one taken for the previous class of ideals. A recursive algorithm of <em>gcd-iterations</em> is introduced and it is shown that this method produces a minimal generating set of the defining ideal of the Rees algebra. </p> <p><br></p> <p>Lastly, we extend our techniques regarding Rees algebras of ideals to Rees algebras of modules. Using <em>generic Bourbaki ideals</em> we study Rees algebras of modules with projective dimension one over hypersurface rings. For such a module $E$, we show that there exists a generic Bourbaki ideal $I$, with respect to $E$, which is perfect of grade two in a hypersurface ring. We then adapt the techniques used by Costantini in [9] to our setting in order to relate the defining ideal of $\mathcal{R}(E)$ to the defining ideal of $\mathcal{R}(I)$, which is known from the earlier work mentioned above.</p> <p><br></p> <p>In all three situations above, once the defining equations have been determined, we investigate certain properties of the Rees algebra. The depth, Cohen-Macaulayness, relation type, and Castelnuovo-Mumford regularity of these rings are explored.</p>

Les marginaux ont tendance à être facilement jugés pour leur anticonformisme face à la société. Le recueil de nouvelles intitulé Funambules présente des personnages marginaux sans les identifier comme tels, de façon à ce que le lecteur ne les perçoivent pas comme étant différents. L'essai critique se penche quant à lui sur la problématique de l'écriture de la marginalité et décortique, en mettant à profit la narratologie, la façon dont les textes de Funambules et les nouvelles Son dernier amant de Hans-Jùrgen Greif et Quelle est la longueur de la côte gaspésienne? d'Alexandre Bourbaki se construisent pour en arriver à démarginaliser les personnages. Le but est donc de déterminer comment la narration a pu rendre normaux des personnages marginaux.