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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.

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

Publicado em 09/06/2011
Português

Relevância na Pesquisa
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#ensino de geometria#epistemological obstacle and dynamic geometry#história da geometria#history of geometry#obstáculo epistemológico e geometria dinâmica#teaching geometry

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...

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## ‣ 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

Fonte: Biblioteca Digital da Unicamp
Publicador: Biblioteca Digital da Unicamp

Tipo: Tese de Doutorado
Formato: application/pdf

Publicado em 08/08/2005
Português

Relevância na Pesquisa
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#Psicologia da educação#Matematica#Geometria - Estudo e ensino#Capacidade matematica#Raciocinio (Psicologia)#Educação matematica#Psychology or mathematics education#Geometry teaching#Mathematical abilities#Spatial sense

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...

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/01/1994
Português

Relevância na Pesquisa
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#Mathematics - Functional Analysis#High Energy Physics - Theory#Mathematics - Algebraic Geometry#Mathematics - Operator Algebras

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)

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## ‣ Finiteness Problems in Diophantine Geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/12/2009
Português

Relevância na Pesquisa
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#Mathematics - Number Theory#Mathematics - Algebraic Geometry#Mathematics - History and Overview#11G35, 11G10, 14H25, 14K15, 11D41, 14D10 (Primary) 11G25, 14K10,
14L15, 14G25 (Secondary)

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...

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/11/2007
Português

Relevância na Pesquisa
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#Mathematics - Quantum Algebra#Mathematics - Algebraic Geometry#Mathematics - Number Theory#Mathematics - Operator Algebras#58B34, 11G35, 14G40

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

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## ‣ Six mathematical gems from the history of Distance Geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/02/2015
Português

Relevância na Pesquisa
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#Mathematics - History and Overview#Mathematics - Combinatorics#Mathematics - Metric Geometry#Mathematics - Optimization and Control#01-02, 51K05

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

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## ‣ Axiomatization of geometry employing group actions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa
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#Mathematics - Metric Geometry#Mathematics - Algebraic Topology#Mathematics - General Topology#Mathematics - Geometric Topology#51K05, Secondary 51H99

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...

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## ‣ Counting Algebraic Curves with Tropical Geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/06/2012
Português

Relevância na Pesquisa
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#Mathematics - Algebraic Geometry#Mathematics - Combinatorics#Primary: 14N35. Secondary: 14T05, 14N10

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa
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#Mathematics - Rings and Algebras#Mathematics - Algebraic Geometry#Mathematics - Representation Theory#Mathematics - Symplectic Geometry

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)

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa
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#Mathematics - Algebraic Geometry#Mathematics - Combinatorics#Mathematics - K-Theory and Homology#Mathematics - Number Theory#Mathematics - Representation Theory

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

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## ‣ Lectures on Arithmetic Noncommutative Geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/09/2004
Português

Relevância na Pesquisa
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#Mathematics - Quantum Algebra#Mathematical Physics#Mathematics - Algebraic Geometry#Mathematics - Number Theory#Mathematics - Operator Algebras#58B34, 46L87, 14G40, 11G35, 37B10, 82A15, 81E30

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

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## ‣ Introduction to tropical algebraic geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/07/2012
Português

Relevância na Pesquisa
46.9103%

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"

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/09/2014
Português

Relevância na Pesquisa
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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)...

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## ‣ Algebraic Geometry and Physics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa
46.93799%

#Mathematical Physics#Condensed Matter - Statistical Mechanics#High Energy Physics - Theory#Mathematics - Algebraic Geometry#Mathematics - Quantum Algebra#Nonlinear Sciences - Exactly Solvable and Integrable Systems

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...

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## ‣ Non-Archimedean analytic geometry as relative algebraic geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa
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#Mathematics - Algebraic Geometry#Mathematics - Category Theory#Mathematics - Functional Analysis#Mathematics - Number Theory#Mathematics - Representation Theory#14A20, 13J07, 14G22, 14E25, 46M99, 18D10, 19D23, 14F20

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

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## ‣ Foundations of Rigid Geometry I

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa
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#Mathematics - Algebraic Geometry#Mathematics - Commutative Algebra#Mathematics - Number Theory#11-02, 11G99, 13-02, 13F30, 13J07, 13J15, 13J20, 14-02, 14A15, 14A20

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

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## ‣ Amoebas of algebraic varieties and tropical geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 29/02/2004
Português

Relevância na Pesquisa
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#Mathematics - Algebraic Geometry#Mathematics - Geometric Topology#Mathematics - Symplectic Geometry#14-02#14N35#14P25

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"

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## ‣ Enumerative tropical algebraic geometry in R2

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa
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#Mathematics - Algebraic Geometry#Mathematical Physics#Mathematics - Combinatorics#Mathematics - Geometric Topology#Mathematics - Symplectic Geometry

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

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## ‣ Notes on noncommutative geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/03/2015
Português

Relevância na Pesquisa
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#Mathematics - Operator Algebras#Mathematics - Algebraic Geometry#Mathematics - Number Theory#11F72, 11G15, 11G45, 11J81, 14A22, 14F42, 14G15, 14H10, 14H52,
14H55, 18Dxx, 16R10, 46L85, 55S35, 57M27, 58F10

The book covers basics of noncommutative geometry and its applications in
topology, algebraic geometry and number theory. A brief survey of main parts of
noncommutative geometry with historical remarks, bibliography and a list of
exercises is attached. Our notes are intended for the graduate students and
faculty with interests in noncommutative geometry; they can be read by
non-experts in the field.; Comment: 309 pages, 43 figures

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## ‣ Algebraic combinatorial geometry: the polynomial method in arithmetic combinatorics, incidence combinatorics, and number theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa
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#Mathematics - Combinatorics#Mathematics - Algebraic Geometry#Mathematics - Number Theory#05B25, 11T06, 12D10, 51H10

Arithmetic combinatorics is often concerned with the problem of bounding the
behaviour of arbitrary finite sets in a group or ring with respect to
arithmetic operations such as addition or multiplication. Similarly,
combinatorial geometry is often concerned with the problem of bounding the
behaviour of arbitrary finite collections of geometric objects such as points,
lines, or circles with respect to geometric operations such as incidence or
distance. Given the presence of arbitrary finite sets in these problems, the
methods used to attack these problems have primarily been combinatorial in
nature. In recent years, however, many outstanding problems in these questions
have been solved by algebraic means (and more specifically, using tools from
algebraic geometry and/or algebraic topology), giving rise to an emerging set
of techniques which is now known as the polynomial method.
While various instances of the polynomial method have been known for decades
(e.g. Stepanov's method, the combinatorial nullstellensatz, or Baker's
theorem), the general theory of this method is still in the process of
maturing; in particular, the limitations of the polynomial method are not well
understood, and there is still considerable scope to apply deeper results from
algebraic geometry or algebraic topology to strengthen the method further. In
this survey we present several of the known applications of these methods...

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