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## Special Kaehler geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/02/2000

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The geometry that is defined by the scalars in couplings of Einstein-Maxwell
theories in N=2 supergravity in 4 dimensions is denoted as special Kaehler
geometry. There are several equivalent definitions, the most elegant ones
involve the symplectic duality group. The original construction used conformal
symmetry, which immediately clarifies the symplectic structure and provides a
way to make connections to quaternionic geometry and Sasakian manifolds.; Comment: Contribution to the Proceedings of the Second Meeting on Quaternionic
Structures in Mathematics and Physics, Rome 6-10 September 1999; 17 pages, 1
figure, using color (colordvi.sty)

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## Geometry of Distributions and $F-$Gordon equation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/08/2009

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In this paper we describe the geometry of distributions by their symmetries,
and present a simplified proof of the Frobenius theorem and some related
corollaries. Then, we study the geometry of solutions of $F-$Gordon equation; A
PDE which appears in differential geometry and relativistic field theory.; Comment: 28 pages

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## Geometry of Maslov cycles

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 02/01/2013

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#Mathematics - Symplectic Geometry#Mathematics - Algebraic Geometry#Mathematics - Optimization and Control

We introduce the notion of induced Maslov cycle, which describes and unifies
geometrical and topological invariants of many apparently unrelated problems,
from Real Algebraic Geometry to sub-Riemannian Geometry.

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## Exceptional Calabi--Yau spaces: the geometry of $\mathcal{N}=2$ backgrounds with flux

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/09/2015

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In this paper we define the analogue of Calabi--Yau geometry for generic
$D=4$, $\mathcal{N}=2$ flux backgrounds in type II supergravity and M-theory.
We show that solutions of the Killing spinor equations are in one-to-one
correspondence with integrable, globally defined structures in
$E_{7(7)}\times\mathbb{R}^+$ generalised geometry. Such "exceptional
Calabi--Yau" geometries are determined by two generalised objects that
parametrise hyper- and vector-multiplet degrees of freedom and generalise
conventional complex, symplectic and hyper-Kahler geometries. The integrability
conditions for both hyper- and vector-multiplet structures are given by the
vanishing of moment maps for the "generalised diffeomorphism group" of
diffeomorphisms combined with gauge transformations. We give a number of
explicit examples and discuss the structure of the moduli spaces of solutions.
We then extend our construction to $D=5$ and $D=6$ flux backgrounds preserving
eight supercharges, where similar structures appear, and finally discuss the
analogous structures in $O(d,d)\times\mathbb{R}^+$ generalised geometry.; Comment: 68 pages plus appendices

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## Algorithms in Real Algebraic Geometry: A Survey

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/09/2014

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#Mathematics - Algebraic Geometry#Computer Science - Computational Complexity#Computer Science - Computational Geometry#Computer Science - Symbolic Computation#Primary 14P10, 14P25, Secondary 68W30

We survey both old and new developments in the theory of algorithms in real
algebraic geometry -- starting from effective quantifier elimination in the
first order theory of reals due to Tarski and Seidenberg, to more recent
algorithms for computing topological invariants of semi-algebraic sets. We
emphasize throughout the complexity aspects of these algorithms and also
discuss the computational hardness of the underlying problems. We also describe
some recent results linking the computational hardness of decision problems in
the first order theory of the reals, with that of computing certain topological
invariants of semi-algebraic sets. Even though we mostly concentrate on exact
algorithms, we also discuss some numerical approaches involving semi-definite
programming that have gained popularity in recent times.; Comment: 41 pages, 4 figures. Based on survey talk given at the Real Algebraic
Geometry Conference, Rennes, June 20-24, 2011. Some references updated and
some newer material added

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## Key developments in geometry in the 19th Century

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/01/2013

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This paper describes several key discoveries in the 19th century that led to
the modern theory of manifolds in the twentieth century: intrinsic differential
geometry, projective geometry and higher dimensional manifolds and Riemannian
geometry.

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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

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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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## An XML-Format for Conjectures in Geometry (Work-in-Progress)

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/07/2012

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With a large number of software tools dedicated to the visualisation and/or
demonstration of properties of geometric constructions and also with the
emerging of repositories of geometric constructions, there is a strong need of
linking them, and making them and their corpora, widely usable. A common
setting for interoperable interactive geometry was already proposed, the i2g
format, but, in this format, the conjectures and proofs counterparts are
missing. A common format capable of linking all the tools in the field of
geometry is missing. In this paper an extension of the i2g format is proposed,
this extension is capable of describing not only the geometric constructions
but also the geometric conjectures. The integration of this format into the
Web-based GeoThms, TGTP and Web Geometry Laboratory systems is also discussed.; Comment: Conferences on Intelligent Computer Mathematics, CICM 2012, 8.-13.
July 2012, Jacobs University, Bremen, Germany. 12 pages 2 figures

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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

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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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## String geometry vs. spin geometry on loop spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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#Mathematics - Differential Geometry#Mathematical Physics#Mathematics - Algebraic Topology#Primary 53C27, Secondary 81T30, 58B05

We introduce various versions of spin structures on free loop spaces of
smooth manifolds, based on a classical notion due to Killingback, and
additionally coupled to two relations between loops: thin homotopies and loop
fusion. The central result of this article is an equivalence between these
enhanced versions of spin structures on the loop space and string structures on
the manifold itself. The equivalence exists in two settings: in a purely
topological one and a in geometrical one that includes spin connections and
string connections. Our results provide a consistent, functorial, one-to-one
dictionary between string geometry and spin geometry on loop spaces.; Comment: 54 pages. In v2 two errorneous lemmata (2.3.3 and 3.1.3) have been
removed, with corresponding changes in Prop. 2.3.4 and Def. 3.1.4 (now Prop.
2.3.3 and Def. 3.1.3, respectively); otherwise minor changes. v3 comes with
few minor changes and is the published version

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## Radar orthogonality and radar length in Finsler and metric spacetime geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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The radar experiment connects the geometry of spacetime with an observers
measurement of spatial length. We investigate the radar experiment on Finsler
spacetimes which leads to a general definition of radar orthogonality and radar
length. The directions radar orthogonal to an observer form the spatial equal
time surface an observer experiences and the radar length is the physical
length the observer associates to spatial objects. We demonstrate these
concepts on a forth order polynomial Finsler spacetime geometry which may
emerge from area metric or pre-metric linear electrodynamics or in quantum
gravity phenomenology. In an explicit generalisation of Minkowski spacetime
geometry we derive the deviation from the euclidean spatial length measure in
an observers rest frame explicitly.; Comment: 18 pages, 7 figures, axes label in figures corrected, journal
references added

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## Geometry of contact transformations and domains: orderability versus squeezing

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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Gromov's famous non-squeezing theorem (1985) states that the standard
symplectic ball cannot be symplectically squeezed into any cylinder of smaller
radius. Does there exist an analogue of this result in contact geometry? Our
main finding is that the answer depends on the sizes of the domains in
question: We establish contact non-squeezing on large scales, and show that it
disappears on small scales. The algebraic counterpart of the (non)-squeezing
problem for contact domains is the question of existence of a natural partial
order on the universal cover of the contactomorphisms group of a contact
manifold. In contrast to our earlier beliefs, we show that the answer to this
question is very sensitive to the topology of the manifold. For instance, we
prove that the standard contact sphere is non-orderable while the real
projective space is known to be orderable. Our methods include a new embedding
technique in contact geometry as well as a generalized Floer homology theory
which contains both cylindrical contact homology and Hamiltonian Floer
homology. We discuss links to a number of miscellaneous topics such as topology
of free loops spaces, quantum mechanics and semigroups.
An erratum is attached whose purpose is to is to correct a number of
inconsistencies in the main paper. These are related to the grading of
generalized Floer homology and do not affect formulations and proofs of the
main results of the paper.; Comment: This is the version published by Geometry & Topology on 28 October
2006 and includes the erratum published 1 February 2009

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## When symplectic topology meets Banach space geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/04/2014

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In this paper we survey some recent works that take the first steps toward
establishing bilateral connections between symplectic geometry and several
other fields, namely, asymptotic geometric analysis, classical convex geometry,
and the theory of normed spaces.; Comment: Submitted to Proceedings of the ICM 2014, 23 pages

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## On geometry of curves of flags of constant type

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 02/10/2011

Relevância na Pesquisa

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We develop an algebraic version of Cartan method of equivalence or an analog
of Tanaka prolongation for the (extrinsic) geometry of curves of flags of a
vector space $W$ with respect to the action of a subgroup $G$ of the $GL(W)$.
Under some natural assumptions on the subgroup $G$ and on the flags, one can
pass from the filtered objects to the corresponding graded objects and describe
the construction of canonical bundles of moving frames for these curves in the
language of pure Linear Algebra. The scope of applicability of the theory
includes geometry of natural classes of curves of flags with respect to
reductive linear groups or their parabolic subgroups. As simplest examples,
this includes the projective and affine geometry of curves. The case of
classical groups is considered in more detail.; Comment: 39 pages

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## On different curvatures of spheres in Funk geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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We compute the series expansions for the normal curvatures of hyperspheres,
the Finsler and Rund curvatures of circles in Funk geometry as the radii tend
to infinity. These three curvatures are different at infinity in Funk geometry.; Comment: 15 pages, revised version - references have been corrected. arXiv
admin note: substantial text overlap with arXiv:1105.1865

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## Geometry of manifolds with area metric: multi-metric backgrounds

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#High Energy Physics - Theory#General Relativity and Quantum Cosmology#Mathematics - Differential Geometry

We construct the differential geometry of smooth manifolds equipped with an
algebraic curvature map acting as an area measure. Area metric geometry
provides a spacetime structure suitable for the discussion of gauge theories
and strings, and is considerably more general than Lorentzian geometry. Our
construction of geometrically relevant objects, such as an area metric
compatible connection and derived tensors, makes essential use of a
decomposition theorem due to Gilkey, whereby we generate the area metric from a
finite collection of metrics. Employing curvature invariants for multi-metric
backgrounds we devise a class of gravity theories with inherently stringy
character, and discuss gauge matter actions.; Comment: 34 pages, REVTeX4, journal version

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## Fractional Almost Kahler - Lagrange Geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematical Physics#High Energy Physics - Theory#Mathematics - Differential Geometry#26A33, 32Q60, 53C60, 53C99, 70S05

The goal of this paper is to encode equivalently the fractional Lagrange
dynamics as a nonholonomic almost Kahler geometry. We use the fractional Caputo
derivative generalized for nontrivial nonlinear connections (N-connections)
originally introduced in Finsler geometry, with further developments in
Lagrange and Hamilton geometry and, in our approach, with fractional
derivatives. For fundamental geometric objects induced canonically by regular
Lagrange functions, we construct compatible almost symplectic forms and linear
connections completely determined by a "prime" Lagrange (in particular,
Finsler) generating function. We emphasize the importance of such constructions
for deformation quantization of fractional Lagrange geometries and applications
in modern physics.; Comment: latex2e, 17 pages, v3 performed following requests of referee with
additional references and explanations; accepted to "Nonlinear Dynamics"

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## Birational geometry for number theorists

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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Awfully idiosyncratic lecture notes from CMI summer school in arithmetic
geometry July 31-August 4, 2006.
Does not include: rationality problems, techniques of the minimal model
problem and much of the rest.
Includes: Lecture 0: geometry and arithmetic of curves
Lecture 1: Kodaira dimension and properties, rational connectendess, Lang's
and Campana's conjectures.
Lecture 2: Campana's program; Campana constellations framed in terms of
b-divisors, to allow for a definition of Kodaira dimension directly on the
base. A speculative notion of firmaments which may deserve further
investigation, especially the arithmetic side.
Lecture 3: the minimal model program: very short discussion of
bend-and-break; even shorter discussion of finite generation and the existence
of flip.
Lecture 4: Vojta's conjectures, Campana's conjectures, and ABC.; Comment: CMI summer school in arithmetic geometry, Gottingen, 2006. Numerous
corrections following referee's report

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## One dimensional metrical geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 11/01/2007

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One dimensional metrical geometry may be developed in either an affine or
projective setting over a general field using only algebraic ideas and
quadratic forms. Some basic results of universal geometry are already present
in this situation, such as the Triple quad formula, the Triple spread formula
and the Spread polynomials, which are universal analogs of the Chebyshev
polynomials of the first kind. Chromogeometry appears here, and the related
metrical and algebraic properties of the projective line are brought to the
fore.; Comment: 19 pages

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## Group-like objects in Poisson geometry and algebra

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/01/2007

Relevância na Pesquisa

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#Mathematics - Symplectic Geometry#High Energy Physics - Theory#Mathematics - Differential Geometry#Mathematics - Rings and Algebras#20L05#16W30, 53D17

A group, defined as set with associative multiplication and inverse, is a
natural structure describing the symmetry of a space. The concept of group
generalizes to group objects internal to other categories than sets. But there
are yet more general objects that can still be thought of as groups in many
ways, such as quantum groups. We explain some of the generalizations of groups
which arise in Poisson geometry and quantization: the germ of a topological
group, Poisson Lie groups, rigid monoidal structures on symplectic
realizations, groupoids, 2-groups, stacky Lie groups, and hopfish algebras.; Comment: 21 pages, based on lectures at School on Poisson Geometry and Related
Topics, Keio University, 2006

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