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## Abordagem histórico-epistemológica do ensino da geometria fazendo uso da geometria dinâmica; Historical-epistemological approach geometry teaching making use of dynamic geometry.

Waldomiro, Tatiana de Camargo
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Dissertação de Mestrado Formato: application/pdf
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A presente pesquisa, de cunho quantitativo, tem como propósito responder a seguinte questão: De que modo e em que alcance o trabalho pedagógico articulado com a história, geometria e meio computacional tem refletido sobre posturas e caminhos que levassem os alunos a se envolver com o conhecimento matemático? Desse modo, fizemos uma investigação e análise sobre os efeitos de uma articulação entre o ensino da história da matemática e o uso de ferramentas computacionais como solução para as dificuldades apresentadas no Ensino de Geometria, principalmente no Ensino Médio. Utilizamos a obra de Lakatos e a primeira proposição (do livro 1) de Euclides para realizar a verificação de sua demonstração através de um software de Geometria dinâmica. Os resultados serão utilizados para a construção de um novo software que envolva o ensino e aprendizagem de história da matemática e geometria. Outros objetivos podem ser assim colocados: Refletir sobre as condições e viabilidade da integração de recursos computacionais para o ensino da Matemática no âmbito Ensino Médio em especial a partir do produtos/softwares propostos para a educação matemática; Compreender o potencial de softwares de geometria dinâmica para a educação matemática escolar; Analisar as necessidades matemáticas de uma instrumentação eficaz...

## O componente espacial da habilidade matematica de alunos do ensino medio e as relações com o desempenho escolar e as atitudes em relação a matematica e a geometria

Odaléa Aparecida Viana
Fonte: Biblioteca Digital da Unicamp Publicador: Biblioteca Digital da Unicamp
Tipo: Tese de Doutorado Formato: application/pdf
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Considerando a influência de fatores cognitivos e afetivos no desempenho escolar em geometria, este trabalho teve como objetivos analisar o componente espacial da habilidade matemática e verificar a existência de relações entre este componente, o raciocínio espacial, as atitudes em relação à matemática e à geometria e o desempenho escolar. Foram sujeitos 177 alunos de ensino médio de uma escola particular, tendo sido aplicadas duas provas tipo lápis e papel, um teste psicológico de raciocínio espacial e duas escalas de atitudes em relação à matemática e geometria. A análise fatorial das operações do componente espacial da habilidade matemática (contagem de cubos, formação e identificação de polígonos no espaço, secção, planificação, projeção e revolução) indicou a existência de um único fator, o que comprova que a prova avaliou a habilidade geral dos sujeitos em lidar com conceitos geométricos espaciais trabalhados no ensino médio, com base nas tarefas propostas. As atitudes em relação à matemática estavam relacionadas com as atitudes em relação à geometria. O desempenho em geometria estava relacionado com o raciocínio espacial, com o componente espacial da habilidade matemática e com as atitudes em relação à geometria. O trabalho faz referência aos processos de formação...

## Infinite-Dimensional Geometry of the Universal Deformation of the Complex Disk

Juriev, D.
Tipo: Artigo de Revista Científica
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The universal deformation of the complex disk is studied from the viewpoint of infinite-dimensional geometry. The structure of a subsymmetric space on the universal deformation is described. The foliation of the universal deformation by subsymmetry mirrors is shown to determine a real polarization. The subject of the paper maybe of interest to specialists in algebraic geometry and representation theory as well as to researchers dealing with mathematical problems of modern quantum field theory. Contents. I. The infinite-dimensional geometry of the flag manifold of the Virasoro-Bott group (the base of the universal deformation of the complex disk). II. The infinite-dimensional geometry of the skeleton of the flag manifold of the Virasoro-Bott group. III. The infinite-dimensional geometry of the universal deformation of the complex disk.; Comment: 9 pages AMSTEX, to appear in RUSSIAN J. MATH. PHYS. V.2. N.1 (1994)

## Finiteness Problems in Diophantine Geometry

Zarhin, Yuri G.; Parshin, Alexey N.
Tipo: Artigo de Revista Científica
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This survey contains an exposition of ideas and results related to Faltings' proof of the conjectures of Shafarevich, Tate and Mordell. This paper originally appeared in 1986 as an Appendix to the Russian translation of Serge Lang, "Fundamentals of Diophantine Geometry" (Springer Verlag, 1983) published by "Mir", Moscow (MR0854670, 88a:11054). A history of the publication of the Appendix is briefly described by Lang in Section 4 of his paper "Mordell's review, Siegel's letter to Mordell, Diophantine geometry, and 20th century mathematics" that was published (in 1995) simultaneously in Notices of the AMS and Gazette des Math\'ematiciens (SMF) (MR1316025, 96g:11002a; MR1316133, 96g:11002b) http://smf.emath.fr/Publications/Gazette/1995/63/smf_gazette_63_17-36.pdf . Later an expanded version of the Appendix was translated into English by Neal Koblitz and published in 1989 by the American Mathematical Society as part of the collection "Eight papers translated from the Russian", AMS Translations, Series 2, Vol. 143 http://www.ams.org/bookstore-getitem/item=TRANS2-143 (MR1008476, 90b:00011). We put this paper on the arXiv with the kind permission of the American Mathematical Society. For this version we slightly updated the bibliography and added a few short notes (marked as "Added in December 2009"). We also corrected inaccuracies that were kindly pointed out to us by J.-P. Serre - one of the few people to read this paper (in English) twenty years ago.; Comment: 68 pages...

## Noncommutative geometry and motives (a quoi servent les endomotifs?)

Consani, Caterina
Tipo: Artigo de Revista Científica
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This paper gives a short and historical survey on the theory of pure motives in algebraic geometry and reviews some of the recent developments of this theory in noncommutative geometry. The second part of the paper outlines the new theory of endomotives and some of its relevant applications in number-theory.; Comment: 36 pages

## Six mathematical gems from the history of Distance Geometry

Liberti, Leo; Lavor, Carlile
Tipo: Artigo de Revista Científica
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This is a partial account of the fascinating history of Distance Geometry. We make no claim to completeness, but we do promise a dazzling display of beautiful, elementary mathematics. We prove Heron's formula, Cauchy's theorem on the rigidity of polyhedra, Cayley's generalization of Heron's formula to higher dimensions, Menger's characterization of abstract semi-metric spaces, a result of Goedel on metric spaces on the sphere, and Schoenberg's equivalence of distance and positive semidefinite matrices, which is at the basis of Multidimensional Scaling.; Comment: 22 pages, 8 figures, submitted to ITOR special issue on distance geometry

## Axiomatization of geometry employing group actions

Dydak, Jerzy
Tipo: Artigo de Revista Científica
Português
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The aim of this paper is to develop a new axiomatization of planar geometry by reinterpreting the original axioms of Euclid. The basic concept is still that of a line segment but its equivalent notion of betweenness is viewed as a topological, not a metric concept. That leads quickly to the notion of connectedness without any need to dwell on the definition of topology. In our approach line segments must be connected. Lines and planes are unified via the concept of separation: lines are separated into two components by each point, planes contain lines that separate them into two components as well. We add a subgroup of bijections preserving line segments and establishing unique isomorphism of basic geometrical sets, and the axiomatic structure is complete. Of fundamental importance is the Fixed Point Theorem that allows for creation of the concepts of length and congruency of line segments. The resulting structure is much more in sync with modern science than other axiomatic approaches to planar geometry. For instance, it leads naturally to the Erlangen Program in geometry. Our Conditions of Homogeneity and Rigidity have two interpretations. In physics, they correspond to the basic tenet that independent observers should arrive at the same measurement and are related to boosts in special relativity. In geometry...

## Counting Algebraic Curves with Tropical Geometry

Block, Florian
Tipo: Artigo de Revista Científica
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Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to significant progress. In this survey, we give an introduction to tropical geometry techniques for algebraic curve counting problems. We also survey some recent developments, with a particular emphasis on the computation of the degree of the Severi varieties of the complex projective plane and other toric surfaces as well as Hurwitz numbers and applications to real enumerative geometry. This paper is based on the author's lecture at the Workshop on Tropical Geometry and Integrable Systems in Glasgow, July 2011.; Comment: 14 pages, 6 figures. To appear in Contemporary Mathematics (Proceedings), "Tropical Geometry and Integrable Systems", Glasgow, July 2011

## The symplectic and algebraic geometry of Horn's problem

Knutson, Allen
Tipo: Artigo de Revista Científica
Português
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Horn's problem was the following: given two Hermitian matrices with known spectra, what might be the eigenvalue spectrum of the sum? This linear algebra problem is exactly of the sort to be approached with the methods of modern Hamiltonian geometry (which were unavailable to Horn). The theorem linking symplectic quotients and geometric invariant theory lets one also bring algebraic geometry and representation theory into play. This expository note is intended to elucidate these connections for linear algebraists, in the hope of making it possible to recognize what sort of problems are likely to fall to the same techniques that were used in proving Horn's conjecture.; Comment: 16 pages, 1 figure; expository conference paper (second version has inessential cosmetic changes)

## A blueprinted view on $\mathbb F_1$-geometry

Lorscheid, Oliver
Tipo: Artigo de Revista Científica
Português
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This overview paper has two parts. In the first part, we review the development of $\mathbb F_1$-geometry from the first mentioning by Jacques Tits in 1956 until the present day. We explain the main ideas around $\mathbb F_1$, embedded into the historical context, and give an impression of the multiple connections of $\mathbb F_1$-geometry to other areas of mathematics. In the second part, we review (and preview) the geometry of blueprints. Beyond the basic definitions of blueprints, blue schemes and projective geometry, this includes a theory of Chevalley groups over $\mathbb F_1$ together with their action on buildings over $\mathbb F_1$; computations of the Euler characteristic in terms of $\mathbb F_1$-rational points, which involve quiver Grassmannians; $K$-theory of blue schemes that reproduces the formula $K_i(\mathbb F_1)=\pi^{st}_i(S^0)$; models of the compactifications of $\Spec \mathbb Z$ and other arithmetic curves; and explanations about the connections to other approaches towards $\mathbb F_1$ like monoidal schemes after Deitmar, $B_1$-algebras after Lescot, $\Lambda$-schemes after Borger, relative schemes after To\"en and Vaqui\'e, log schemes after Kato and congruence schemes after Berkovich and Deitmar.; Comment: 58 pages; correction of section 7.2 and other minor modifications

## Lectures on Arithmetic Noncommutative Geometry

Marcolli, Matilde
Tipo: Artigo de Revista Científica
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This is the text of a series of five lectures given by the author at the "Second Annual Spring Institute on Noncommutative Geometry and Operator Algebras" held at Vanderbilt University in May 2004. It is meant as an overview of recent results illustrating the interplay between noncommutative geometry and arithmetic geometry/number theory.; Comment: 129 pages LaTeX, 28 figures

## Introduction to tropical algebraic geometry

Maclagan, Diane
Tipo: Artigo de Revista Científica
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This is an expository introduction to tropical algebraic geometry based on my lectures at the Workshop on Tropical Geometry and Integrable Systems in Glasgow, July 4-8, 2011, and at the ELGA 2011 school on Algebraic Geometry and Applications in Buenos Aires, August 1-5, 2011.; Comment: To appear in AMS Contemporary Mathematics Volume, "Tropical Geometry and Integrable Systems"

## On the works of Euler and his followers on spherical geometry

Tipo: Artigo de Revista Científica
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We review and comment on some works of Euler and his followers on spherical geometry. We start by presenting some memoirs of Euler on spherical trigonometry. We comment on Euler's use of the methods of the calculus of variations in spherical trigonometry. We then survey a series of geometrical resuls, where the stress is on the analogy between the results in spherical geometry and the corresponding results in Euclidean geometry. We elaborate on two such results. The first one, known as Lexell's Theorem (Lexell was a student of Euler), concerns the locus of the vertices of a spherical triangle with a fixed area and a given base. This is the spherical counterpart of a result in Euclid's Elements, but it is much more difficult to prove than its Euclidean analogue. The second result, due to Euler, is the spherical analogue of a generalization of a theorem of Pappus (Proposition 117 of Book VII of the Collection) on the construction of a triangle inscribed in a circle whose sides are contained in three lines that pass through three given points. Both results have many ramifications, involving several mathematicians, and we mention some of these developments. We also comment on three papers of Euler on projections of the sphere on the Euclidean plane that are related with the art of drawing geographical maps.; Comment: To appear in Ganita Bharati (Indian Mathematics)...

## Algebraic Geometry and Physics

Roan, Shi-shyr
Tipo: Artigo de Revista Científica
Português
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This article is an interdisciplinary review and an on-going progress report over the last few years made by myself and collaborators in certain fundamental subjects on two major theoretic branches in mathematics and theoretical physics: algebraic geometry and quantum physics. I shall take a practical approach, concentrating more on explicit examples rather than formal developments. Topics covered are divided in three sections: (I) Algebraic geometry on two-dimensional exactly solvable statistical lattice models and its related Hamiltonians: I will report results on the algebraic geometry of rapidity curves appeared in the chiral Potts model, and the algebraic Bethe Ansatz equation in connection with quantum inverse scattering method for the related one-dimensional Hamiltonion chain, e.g., XXZ, Hofstadter type Hamiltonian. (II) Infinite symmetry algebras arising from quantum spin chain and conformal field theory: I will explain certain progress made on Onsager algebra, the relation with the superintegrable chiral Potts quantum chain and problems on its spectrum. In conformal field theory, mathematical aspects of characters of N=2 superconformal algebra are discussed, especially on the modular invariant property connected to the theory. (III). Algebraic geometry problems on orbifolds stemming from string theory: I will report recent progress on crepant resolutions of quotient singularity of dimension greater than or equal to three. The direction of present-day research of engaging finite group representations in the geometry of orbifolds is briefly reviewed...

## Non-Archimedean analytic geometry as relative algebraic geometry

Ben-Bassat, Oren; Kremnizer, Kobi
Tipo: Artigo de Revista Científica
Português
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We show that Berkovich analytic geometry can be viewed as relative algebraic geometry in the sense of To\"{e}n--Vaqui\'{e}--Vezzosi over the category of non-Archimedean Banach spaces. For any closed symmetric monoidal quasi-abelian category we can define a topology on certain subcategories of the of the category of affine schemes with respect to this category. By examining this topology for the category of Banach spaces we recover the G-topology or the topology of admissible subsets on affinoids which is used in analytic geometry. This gives a functor of points approach to non-Archimedean analytic geometry and in this way we also get definitions of (higher) non-Archimedean analytic stacks. We demonstrate that the category of Berkovich analytic spaces embeds fully faithfully into the category of varieties in our version of relative algebraic geometry. We also include a treatment of quasi-coherent sheaf theory in analytic geometry. Along the way, we use heavily the homological algebra in quasi-abelian categories developed by Schneiders.; Comment: added material on quasi-coherent modules, connection to derived analytic geometry, corrected mistakes

## Foundations of Rigid Geometry I

Fujiwara, Kazuhiro; Kato, Fumiharu
Tipo: Artigo de Revista Científica
Português
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In this research oriented manuscript, foundational aspects of rigid geometry are discussed, putting emphasis on birational side of formal schemes and topological feature of rigid spaces. Besides the rigid geometry itself, topics include the general theory of formal schemes and formal algebraic spaces, based on a theory of complete rings which are not necessarily Noetherian (cf. introduction). The manuscript is encyclopedic and almost self-contained, and contains plenty of new results. A discussion on relationship with J. Tate's rigid analytic geometry, V. Berkovich's analytic geometry and R. Huber's adic spaces is also included. As a model example of applications, a proof of Nagata's compactification theorem for schemes is given in the appendix.; Comment: 706 pages containing TOC, Index, and Symbol List - 2nd version (Feb. 4, 2014): Changes are made for correcting inaccuracies and replacing some arguments in more detail - 3rd version (Feb. 7, 2014): We have corrected a few minor errors

## Amoebas of algebraic varieties and tropical geometry

Mikhalkin, Grigory
Tipo: Artigo de Revista Científica
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This survey consists of two parts. Part 1 is devoted to amoebas. These are images of algebraic subvarieties in the complex torus under the logarithmic moment map. The amoebas have essentially piecewise-linear shape if viewed at large. Furthermore, they degenerate to certain piecewise-linear objects called tropical varieties whose behavior is governed by algebraic geometry over the so-called tropical semifield. Geometric aspects of tropical algebraic geometry are the content of Part 2. We pay special attention to tropical curves. Both parts also include relevant applications of the theories. Part 1 of this survey is a revised and updated version of an earlier prepreint of 2001.; Comment: 40 pages, 15 figures, a survey for the volume "Different faces in Geometry"

## Enumerative tropical algebraic geometry in R2

Mikhalkin, Grigory
Tipo: Artigo de Revista Científica
Português
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The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in http://arxiv.org/abs/math.AG/0209253. The result is established with the help of the so-called tropical algebraic geometry. This geometry allows one to replace complex toric varieties with the Euclidean n-space and holomorphic curves with certain piecewise-linear graphs there.; Comment: 83 pages, 20 figures, Version 4, to appear in the Journal of the AMS

## Notes on noncommutative geometry

Nikolaev, Igor
Tipo: Artigo de Revista Científica