Hodge theory applied to tropical curves.
problem Developing Hodge theory for tropical curves.
method Analytical approach using tropical differential forms and cohomologies.
result Construction of Hodge theory analog on tropical curves.
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48 results for “curve theory”
This paper extends Jacobi field theory to Jacobi curves and their curvatures.
problem Characterizing and understanding Jacobi curves and their curvatures.
method Developed a new theory of Jacobi curves and associated curvatures, derived Ricci curvature, and presented a Cartan-like theory.
result Jacobi curves are fully characterized by a family of conformal symplectic invariant curvatures.
Rectangular peg problem solved for many curves.
problem Rectangular peg problem for continuous Jordan curves.
method Microlocal sheaf theory and recent work of Greene and Lobb.
result Affirmative answer for a large class of rectifiable curves.
The paper compactifies stability conditions on curves, akin to Teichmüller theory.
problem Compactifying spaces of stability conditions on curves.
method Constructing a Thurston-like compactification.
result Obtained a compactification of stability conditions space, analogous to Teichmüller theory.
Extends curve theory to non-smooth data with finite curvature and torsion.
problem Applying classical curve theory to non-smooth data.
method Using distributional derivative measures of functions of bounded variation.
result Essentially unique non-smooth curve solution with finite total curvature and torsion.
Study of curves in dual space with constant curvature and torsion.
problem Classifying curves in dual space with specific geometric properties.
method Defined curvature and torsion for curves in dual space, classified curves with constant properties, and proved existence theorems.
result Established fundamental theorem of existence for dual curves with prescribed curvature and torsion.
New algebraic theory classifies symplectic curves in complex projective space.
problem Classifying symplectic curves with specific singularities.
method Developed a novel algebraic theory of positive braids and conjugacy classes in the braid group.
result Established a complete classification of isotopy classes of degree three symplectic curves with -singularities.
The curve graph's model theory reveals its central role in surface study.
problem Why is the curve graph central in surface and mapping class group studies?
method Developed a bridge between model theory, topology, and group theory; bi-interpreted curve graph with mapping class group.
result Proved the curve graph's first-order theory is -stable and has quantifier elimination.
Lie algebroids and curved Lie algebras are equivalent categories.
problem Understanding the relationship between Lie algebroids and curved Lie algebras.
method Developed a method to study the -category of curved Lie algebras using homotopy theory of algebras over a complete operad.
result Equivalence of -categories between Lie algebroids and certain kinds of curved Lie algebras.
Curves with constant torsion can be deformed arbitrarily.
problem Deforming curves of constant torsion in Euclidean space.
method Convex integration and degree theory.
result Existence of knots with constant torsion in each isotopy class.
Derives path-integrals for superstrings on curved backgrounds using string geometry theory.
problem Calculating path-integrals for superstrings on curved backgrounds.
method Derives path-integrals from string geometry theory by considering fluctuations around string backgrounds.
result Derives path-integrals for perturbative superstrings on all string backgrounds.
Study S-duality and supersymmetry on curved manifolds using localization.
problem Understanding S-duality and supersymmetry on curved manifolds.
method Localization and Fourier transform interpretation of S-duality.
result Evidence for interpreting S-duality as Fourier transform.
Bershadsky-Cecotti-Ooguri-Vafa (BCOV) proposed that the B-model of mirror symmetry should be described by a quantum field theory on a Calabi-Yau variety, which they called the Kodaira-Spenser theory (we call it the BCOV theory). This is the first of three papers in which we construct and analyze the quantum BCOV theory…
Study of holomorphic curves and surfaces using singularity theory.
problem Understanding the geometry of holomorphic curves and complex surfaces.
method Application of singularity theory to holomorphic curves and surfaces.
result Definition of geometric invariants for curves and surfaces.
Characterizes covers using simple closed curves on surfaces.
problem Tackles the equivalence of covers via simple closed curves.
method Uses Teichmüller theory and the complex of curves.
result Two covers are equivalent if and only if the same curves lift to simple curves.
We propose a new approach to the value distribution theory of entire holomorphic curves. We define a ``packing density'' of an entire holomorphic curve, and show that it has various non-trivial properties. We prove a ``gap theorem'' for holomorphic maps from elliptic curves to the complex projective space, and study th…
Novel multisymplectic framework for pseudo-Fueter curves in Hamiltonian field theory.
problem Generalizing Floer theory to multisymplectic geometry.
method Introducing pseudo-Fueter curves in a compatible almost hyperkähler structure.
result Gradient lines of multisymplectic action functional are pseudo-Fueter curves.
The paper defines evolutes and involutes for framed curves and their properties.
problem Defining evolutes and involutes for framed curves with singular points.
method Using the theory of framed curves and Bertrand type curves.
result Conditions for evolutes and involutes being inverse operations of framed curves.
Generalizes Higgs bundles theory using a vector bundle twist.
problem Extending Higgs bundles theory to incorporate vector bundle twists.
method Defined a Hitchin map and spectral correspondence, stated Hitchin-Kobayashi correspondence.
result Established a theory halfway between curve and higher-dimensional variety Higgs bundles.
Proves minimum number of normals to curves in 3D space.
problem Finding the minimum number of normals to closed curves in 3D.
method Morse theory for squared distance function and self intersections of the focal surface.
result For generic curves, points have at least 6, 8, or 10 normals depending on knotting.
We develop the structure theory of full isometry groups of locally compact non-positively curved metric spaces. Amongst the discussed themes are de Rham decompositions, normal subgroup structure and characterising properties of symmetric spaces and Bruhat--Tits buildings. Applications to discrete groups and further dev…
This study defines finite-type invariants for curves on surfaces and reveals the construction of these finite-type invariants for stable homeomorphism classes of curves on compact oriented surfaces without boundaries. These invariants are a higher-order generalisation of a part of Arnold's invariants that are first-ord…
By carrying out a rational transformation on the base curve of the Seiberg-Witten curve for supersymmetric pure -gauge theory, we obtain a family of Jacobian elliptic K3 surfaces of Picard rank 17. The isogeny relating the Seiberg-Witten curve for pure -ga…
The main purpose of this paper is to summarize the basic ingredients, illustrated with examples, of a pseudoholomorphic curve theory for symplectic 4-orbifolds. These are extensions of relevant work of Gromov, McDuff and Taubes on symplectic 4-manifolds concerning pseudoholomorphic curves and Seiberg-Witten theory. The…
Grupos y geometría cubularmath.GR
Survey on non-positively curved cube complexes and geometric group theory.
problem None explicitly stated, focuses on introduction.
method Lecture notes and mini-course teaching.
result Introduction to non-positively curved cube complexes and geometric group theory.
We discuss Ghys' theorem on 4 zeroes of the Schwarzian derivative and its relation with flattening points of Legendrian curves and Sturm theory.
The field of multiple view geometry has seen tremendous progress in reconstruction and calibration due to methods for extracting reliable point features and key developments in projective geometry. Point features, however, are not available in certain applications and result in unstructured point cloud reconstructions.…
In classical curve theory, the geometry of a curve in three dimensions is essentially characterized by their invariants, curvature and torsion. When they are given, the problem of finding a corresponding curve is known as 'solving natural equations'. Explicit solutions are known only for a handful of curve classes, inc…
Develops a new theory of width for embedded circles in Riemannian manifolds.
problem Defining and understanding the width of embedded circles in Riemannian manifolds.
method Morse-Lusternik-Schnirelmann theory applied to geodesics and minimising configurations.
result Classifies configurations of minimising geodesics intersecting embedded circles.
The paper extends Heintze-Kobayashi-Wolf theory to negatively curved homogeneous Finsler manifolds.
problem Understanding negatively curved homogeneous Finsler manifolds.
method Generalizing Heintze-Kobayashi-Wolf theory to homogeneous Finsler geometry, proving two main theorems.
result Negatively curved homogeneous Finsler manifolds are isometric to Lie groups with specific properties.
The main purpose of this paper is to show that ideas of deformation theory can be applied to "infinite dimensional geometry". We develop the deformation theory of Brody curves. Brody curve is a kind of holomorphic map from the complex plane to the projective space. Since the complex plane is not compact, the parameter …
New discrete curves defined in space forms with geometric properties.
problem Defining discrete elastic and constrained elastic curves in space forms.
method Extending discrete Euclidean curvature to space forms and using Bäcklund transformations.
result Discrete elastic and constrained elastic curves are elements of a curve hierarchy.
New knot models analyze local entanglement for robust curve analysis.
problem Lack of local structural information in classical knot theory.
method Proposed multiscale and persistent Jones polynomials.
result Models are stable to small perturbations, robust for real-world applications.
We present the general theory of curves in conformal geometry using tractor calculus. This primarily involves a tractorial determination of distinguished parametrizations and relative and absolute conformal invariants of generic curves. The absolute conformal invariants are defined via a tractor analogue of the classic…
Equal-volume polygons are obtained from adequate discretizations of curves in 3-space, contained or not in surfaces. In this paper we explore the similarities of these polygons with the affine arc-length parameterized smooth curves to develop a theory of discrete affine invariants. Besides obtaining discrete affine inv…
This is a revision of some expository lecture notes written originally for a 5-hour minicourse on the intersection theory of punctured holomorphic curves and its applications in 3-dimensional contact topology. The main lectures are aimed primarily at students and require only a minimal background in holomorphic curve t…
Study algebraic curves in C^2 using Floer theory.
problem Configurations of singular points on algebraic curves.
method Floer theory applied to knot Floer complexes.
result Formula for -action on knot Floer complex.
New Morse functions on curve moduli space via geodesics.
problem Understanding the moduli space of curves via geometric and combinatorial methods.
method Introducing Morse functions based on geodesic lengths and analyzing their critical points and indices.
result Found new explicit Morse functions on , leading to a combinatorial cell decomposition.
The paper proves a generalized inverse function theorem for curved spaces.
problem Proving a generalized inverse function theorem for curved spaces.
method Obstruction theory for homomorphisms and homotopy transfer theorem for curved algebras.
result A morphism of curved spaces which is a quasi-isomorphism at a point has a local homotopy inverse.
In classical curve theory, the geometry of a curve in three dimensions is essentially characterized by their invariants, curvature and torsion. When they are given, the problem of finding a corresponding curve is known as 'solving natural equations'. Explicit solutions are known only for a handful of curve classes, inc…
We study a smooth analogue of jumping curves of a holomorphic vector bundle, and use Yang-Mills theory over to show that any non-trivial, smooth Hermitian vector bundle over a smooth simply connected manifold, must have such curves. This is used to give new examples complex manifolds for which a non-tri…
Computes colored HOMFLYPT invariants using holomorphic curves.
problem Counting holomorphic curves in Calabi-Yau 3-folds.
method Computes contributions of multiple covers of holomorphic annuli.
result Agrees with topological string theory predictions and proves Ooguri-Vafa formula.
Study null surfaces of pseudo-spherical curves in anti-de Sitter space.
problem Characterize null surfaces of pseudo-spherical spacelike framed curves in anti-de Sitter 3-space.
method Introduced nullcone fronts, classified singularities, defined Anti-de Sitter distance-squared functions.
result Relate singularities of nullcone fronts to those of framed curves.
Study of height jumps in Ceresa cycle using asymptotic Hodge theory.
problem Understanding height jumps in the Ceresa cycle.
method Analysis of asymptotic behavior of Hain-Reed beta-invariant in degenerating families of curves.
result Height jump of Ceresa cycle is equal to the slope of the dual graph of the curve.
Study presents a twistor correspondence for specific geometric structures.
problem Twistor theory for almost-Grassmannian manifolds.
method Utilizes moduli of curves-with-boundary for global correspondence.
result Foundational results in complex setting, global correspondence for real Grassmannian.
New invariants measure entanglement of open curves in 3D space.
problem Characterizing the complexity of open curves in 3-space.
method Introducing linkoids and virtual knots, and new invariants.
result Strong invariants of linkoids independent of virtual closure.
The Hessian Topology is a subject having interesting relations with several areas, for instance, differential geometry, implicit differential equations, analysis and singularity theory. In this article we study the problem of realization of a real plane curve as the Hessian curve of a smooth function. The plane curves …
The paper proves existence of minimal homotopies for immersed planar curves.
problem Existence of area-minimizing homotopies between homotopic curves in the plane.
method Geometric and variational approach, lifting curves into higher co-dimension, applying Douglas's solution of the Plateau problem.
result Uniform convergence of Douglas minimizers and minimal homotopy area minimization.