Paper classifies bifurcations of Minkowski symmetry sets for plane curves.
problem Classifying bifurcations of Minkowski symmetry sets.
method Classification of bifurcations based on geometric criteria.
result Different bifurcation types for Minkowski symmetry sets compared to Euclidean symmetry sets.
Curved 10-manifolds with torus symmetry are spheres or complex projective spaces.
problem Characterizing positively curved manifolds with torus symmetry.
method Analyzing actions of 3-dimensional tori on closed, simply connected 10-manifolds. result Closed, simply connected, positively curved 10-manifolds with T3-symmetry are homotopy spheres or complex projective spaces. The paper examines the geometry of a curve's centre symmetry set.
problem Global geometrical properties of a curve's centre symmetry set.
method Study of the envelope of affine chords.
result Number of singularities and asymptotes of the centre symmetry set.
Classifies 6-manifolds with maximal symmetry up to diffeomorphism.
problem Classifying 6-manifolds with maximal symmetry.
method Classification based on curvature and symmetry properties.
result Classification of 6-manifolds with almost maximal symmetry rank.
Generalizes symmetries of curved manifolds.
problem Maximizing symmetries in curved manifolds.
method Replaces torus with abelian group, generalizes results.
result Generalizes symmetry results for positively curved manifolds.
The symmetry-rank of a riemannian manifold is by definition the rank of its isometry group. We determine precisely which smooth closed manifolds admit a positively curved metric with maximal symmetry-rank.
Researchers compute c-projective symmetry algebras for Kähler surfaces.
problem Understanding symmetries in Kähler surfaces.
method Defined and analyzed c-projective vector fields and computed their symmetries.
result Computed c-projective symmetry algebras for Kähler surfaces with essential c-projective vector fields.
Study constructs closed curves with constant curvature on cylinders and tori.
problem Creating closed curves with constant curvature.
method ODEs and symmetry arguments, starting with cylinders, then tori, and finally Frenet-Serret equations.
result Closed constant curvature space curves constructed on cylinders and tori.
Study symmetry groups and curves from sums of exponentials.
problem Understanding the geometry and symmetry of curves from sums of exponentials.
method Analysis of symmetry groups, winding numbers, and parametrization of the unit circle.
result Unified method for constructing curves with specific properties.
ccc-Autoevolutes are closed curves congruent to their evolutes, constructed via symmetry.
problem Constructing closed curves congruent to their evolutes.
method Modified Frenet equation, numerical solutions, symmetry.
result Found the smallest autoevolute as a trefoil knot.
Classifies charge-3 monopoles with symmetry, identifying new spectral curves.
problem Classifying charge-3 monopoles with symmetry.
method Constructing Nahm data and identifying spectral curves with elliptic quotients.
result New monopole spectral curves with D6 and V4 symmetry identified. Analytic curves are classified w.r.t. their symmetry under a regular and separately analytic Lie group action on an analytic manifold. We show that an analytic curve is either exponential or splits into countably many analytic immersive curves, each of them discretely generated by the symmetry group (i.e., each such cu…
Researchers derive symmetry operators from twistor spinors in curved spacetime.
problem Deriving symmetry operators for gauged twistor spinors in curved backgrounds.
method Using gauged twistor spinors and conformal Killing-Yano forms, symmetry operators are constructed.
result Symmetry operators can be obtained from ordinary twistor spinors in constant curvature backgrounds.
New insights into (2,3,5)-distributions via Legendrian curves.
problem Understanding symmetries of (2,3,5)-distributions. method Exploiting a correspondence between distributions and lines on contact manifolds.
result One-to-one correspondence between equivalence classes of (2,3,5)-distributions and Legendrian curves. Study shows non-negative curvature on 4-manifolds with torus symmetry.
problem Classifying 4-manifolds with torus symmetry under non-negative curvature.
method Investigated invariant metrics on 4-manifolds with circle and torus actions.
result Found that almost non-negative curvature implies non-negative curvature for 4-manifolds with torus symmetry.
Study shows Calabi-Yau manifolds are non-hyperbolic via mirror symmetry.
problem Proving non-hyperbolicity of Calabi-Yau manifolds.
method Using mirror symmetry, entire curves are found on Calabi-Yau manifolds.
result Calabi-Yau manifolds are Kobayashi non-hyperbolic.
New rank 3 distributions with exponentially growing symmetries.
problem Constructing distributions with very large symmetries.
method Using rational normal curves and symplectic flags.
result Symmetry groups grow exponentially in dimension.
We find a general solution to the unique 7th order ODE admitting ten dimensional group of contact symmetries. The integral curves of this ODE are rational contact curves in $\PP^3$ which give rise to rational plane curves of degree six. The moduli space of these curves is a real form of the homogeneous space $Sp(4)/SL(…
Classifies 4D spaces with circle symmetry and positive curvature.
problem Classifying spaces with specific geometric properties.
method Equivariant homeomorphism classification, infinitesimal geometry analysis.
result Classification of 4D Alexandrov spaces and orbifolds with circle symmetry.
Symmetry groups help define solitons in curved spaces.
problem Understanding solitons in curved spaces.
method Defined generalized solitons using symmetry groups.
result Affine solutions are self-similar.
Geometric approach to mixed symmetry tensors using Z2n-manifolds.
problem Understanding mixed symmetry tensors on manifolds.
method Using Z2n-manifolds to study mixed symmetry tensors. result Geometric aspects of mixed symmetry tensors on both flat and curved spaces.
Study infinitesimal symmetries in planar differential equations.
problem Qualitative study of differential equations and their integral curves.
method Use meromorphic connections methods to analyze infinitesimal symmetries.
result Infinitesimal symmetries of planar d-webs are understood. New method detects projective equivalences and symmetries in rational 3D curves.
problem Detecting projective equivalences and symmetries in rational 3D curves.
method Using differential invariants and Möbius transformations to avoid solving large polynomial systems.
result Efficient algorithm for detecting projective equivalences and symmetries without solving large polynomial systems.
The study explores the symmetries of exotic negatively curved manifolds.
problem Exploring the extent of symmetry in exotic manifolds.
method Combining techniques from smoothing theory and hyperbolic manifold construction.
result Examples of manifolds with maximal and minimal symmetry are found.
Study Picard groups of curves with symmetry, focusing on abelian groups and hyperelliptic curves.
problem Understanding the Picard groups of moduli spaces of curves with symmetry.
method Theory of symmetric mapping class groups, finitely generated Picard groups computation.
result Finitely generated Picard groups for moduli spaces of curves with abelian automorphisms.
Given a pair of planar curves, one can define its generalized area distance, a concept that generalizes the area distance of a single curve. In this paper, we show that the generalized area distance of a pair of planar curves is an improper indefinite affine spheres with singularities, and, reciprocally, every indefini…
Study projective structures and rational curves to understand Painlevé equations.
problem Analyzing projective structures and rational curves on surfaces.
method Analytic classification, normal forms, pencil/fibration decomposition, infinitesimal symmetries.
result Deduced transcendental results about Painlevé equations.
We determine the spectral curve of charge 3 BPS su(2) monopoles with C_3 cyclic symmetry. The symmetry means that the genus 4 spectral curve covers a (Toda) spectral curve of genus 2. A well adapted homology basis is presented enabling the theta functions and monopole data of the genus 4 curve to be given in terms of g…
Study the symmetry and winding numbers of curves defined by sums of exponentials.
problem Understanding the geometry and topology of curves defined by sums of exponentials.
method Analyzing the continuous transition of the graph of the curve as a parameter changes.
result Determined winding numbers and cusp points for curves defined by sums of exponentials.
We prove an obstruction at the level of rational cohomology in small degrees to the existence of positively curved metrics with large symmetry rank. The symmetry rank bound is logarithmic in the dimension of the manifold. As an application, we provide evidence for a generalized conjecture of Hopf that says that no symm…
We propose a spectral curve describing torus knots and links in the B-model. In particular, the application of the topological recursion to this curve generates all their colored HOMFLY invariants. The curve is obtained by exploiting the full Sl(2, Z) symmetry of the spectral curve of the resolved conifold, and should …
Manifolds admitting positive sectional curvature are conjectured to have rigid homotopical structure and, in particular, comparatively small Euler charateristics. In this article, we obtain upper bounds for the Euler characteristic of a positively curved Riemannian manifold that admits a large isometric torus action. W…
Method learns symmetries in curves without augmentation.
problem Symmetries in datasets like rotations and scalings.
method Geometric learning using principal fiber bundles.
result 2-parameter family of canonical curve parameterizations.
New hexagonal circular 3-webs with reducible curves classified.
problem Classifying hexagonal circular 3-webs with reducible polar curves of degree 3.
method New examples and classifications presented.
result Classification of hexagonal circular 3-webs with reducible polar curves of degree 3.
Study examines properties of curved spaces with special symmetry.
problem Understanding fundamental groups of curved spaces with specific symmetry.
method Analyzes universal covers with cohomology similar to Bazaikin spaces, proving structural results for fundamental groups with torus symmetry.
result Structural results for fundamental groups in spaces with torus symmetry.
We classify nonnegatively curved simply connected 4-manifolds with circle symmetry up to equivariant diffeomorphisms. The main problem is rule out knotted curves in the singular set of the orbit space. As an extension of this work we classify all knots in S^3 which can be realized as an extremal set with respect to an …
New aesthetic curves in equiaffine geometry include the quadratic and logarithmic spiral.
problem Designing aesthetic shapes in equiaffine geometry.
method Introducing a new symmetry (ESA) to characterize planar curves.
result The new class of curves includes the quadratic curve and logarithmic spiral.
New steady solitons found with SO(3) symmetry.
problem Finding steady solitons with specific symmetries.
method Proved existence of a family of solitons with SO(3) symmetry.
result Existence of a one-parameter family of solitons.
Extends Euler's problem to Lorentz-Minkowski plane.
problem Finding critical points of moment of inertia in Lorentz-Minkowski space.
method Explicit solutions for stationary curves, symmetries, inversions, and energy maximization.
result Explicit solutions for stationary spacelike and timelike curves, and methods to transform between them.
Investigates the vertex curve of smooth surfaces in 3D space, connecting geometry and image analysis.
problem Understanding the geometry of smooth surfaces in 3D space.
method Analyzes the vertex curve, related to differential geometry and symmetry sets of isophote curves.
result Establishes connections between the vertex curve and other geometric curves like parabolic and flecnodal curves.
Motivated by Legendrian curve shortening flows in R3, we study the curve shortening flow of figure-eight curves in the plane. We show that, under some symmetry and curvature conditions, a figure-eight curve will shrink to a point at the first singular time.
We classify closed, simply-connected non-negatively curved 5-manifolds admitting an (almost) effective, isometric T3 or T2 action. As a direct consequence, we show that for any manifold, of dimensions up to and including 9 under the same hypotheses, the maximal symmetry rank is equal to [2n/3] and the free rank…
New symmetries discovered in Kepler's orbit family.
problem Symmetry properties of Kepler orbits and related subfamilies.
method Projective geometry and Lie's infinitesimal point symmetries.
result Kepler orbits form a flat family with a 7-dimensional local symmetry group.
Criteria for extending degree-2 Azumaya algebras with C2-actions over curves.
problem Determining when degree-2 Azumaya algebras with C2-actions extend to entire curves.
method Criteria for extension of algebra and new condition for extension with action, testable by computer algebra systems.
result New conditions for extending degree-2 Azumaya algebras with C2-actions over curves.
We discuss SU(2) Bogomolny monopoles of arbitrary charge k invariant under various symmetry groups. The analysis is largely in terms of the spectral curves, the rational maps, and the Nahm equations associated with monopoles. We consider monopoles invariant under inversion in a plane, monopoles with cyclic symmetry…
The paper shows that almost every path structure is not variational.
problem Determining if a path structure is variational.
method Generalized Douglas's result to higher dimensions and analyzed path geometries with infinitesimal symmetries.
result Almost every path structure is not variational.
New findings on non-congruent curves with identical signatures.
problem Identifying congruence of non-degenerate curves with non-simple signatures.
method Associated directed graphs to signatures and used paths to reflect global and local symmetries.
result Non-congruent, non-degenerate curves can have identical signatures.
In this paper, we consider the intensity surface of a 2D image, we study the evolution of the symmetry sets (and medial axes) of 1-parameter families of iso-intensity curves. This extends the investigation done on 1-parameter families of smooth plane curves (Bruce and Giblin, Giblin and Kimia, etc.) to the general case…