New methods to construct curve pairs and their applications.
problem Constructing curve pairs and their properties.
method Using integral curves to study direction and donor curves.
result New methods to construct partner curves of unit speed curves.
We study pairs of curves with Poncelet's porism properties and compute their vertex curves.
problem Understanding pairs of curves with Poncelet's porism properties.
method Developed formulas to compute vertex curves for given envelope curves and vice versa, for all sufficiently regular pairs of Poncelet curves.
result Formulas produce all possible sufficiently regular pairs of Poncelet curves, including sets of curves analogous to pencils of conic sections.
Study examines equivalence relations on the pair of pants, proving k-equivalence implies 1-equivalence and 2-equivalence.
problem Understanding equivalence classes of closed curves on the pair of pants.
method Examined k-equivalence, proving it implies 1-equivalence and 2-equivalence, and deepened understanding of 1-equivalence.
result k-equivalence implies 1-equivalence and 2-equivalence on the pair of pants.
Classifies pro-p PD2 pairs and builds a pro-p curve complex.
problem Classifying pro-p Poincaré duality pairs in dimension two. method Using classification of pro-p PD2 pairs to build a pro-p curve complex. result Established basic properties of the pro-p curve complex. The paper presents new representations and spherical indicatrices of Bertrand curves in Lie groups.
problem Understanding geometric properties of Bertrand curves in Lie groups.
method New representations and spherical indicatrices of Bertrand curves in three Lie groups with bi-invariant metrics are derived.
result Relations between spherical indicatrices and new representations of Bertrand curves are established.
Paper finds infinite pairs of fiber-type curves with same topology but different embeddings.
problem Conditions for curves in projective surfaces to have specific fundamental groups.
method Examine fiber-type curves in P2 and use twisted Alexander polynomials. result Infinite Zariski pairs of fiber-type curves with non-isomorphic fundamental groups.
Origami edge-paths connect coherent curves on surfaces.
problem Understanding coherent curves on surfaces.
method Origami structure and edge-paths.
result Existence of origami edge-paths connecting coherent curves.
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…
In this work, we studied the properties of the spherical indicatrices of a Bertrand curve and its mate curve and presented some characteristic properties in the cases that Bertrand curve and its mate curve are slant helices, spherical indicatrices are slant helices and we also researched that whether the spherical indi…
The study finds pairs of curves at distance 5 in surface curve graphs.
problem Finding pairs of curves at distance 5 in the curve graph of closed surfaces.
method Applying Dehn twists to fixed curves and characterizing conditions for distance 5.
result Characterization of pairs of curves at distance 5 in surface curve graphs.
Study on the minimum length of curves on once-punctured hyperbolic surfaces.
problem Finding the minimum length of filling pairs on once-punctured hyperbolic surfaces.
method Analyzing the topology and geometry of the surface to derive a lower bound for the length of filling pairs.
result A lower bound for the length of filling pairs on once-punctured hyperbolic surfaces is derived, depending only on the surface's topology.
New method to construct partner curves of non-lightlike curves.
problem Constructing partner curves for non-lightlike curves.
method Using integral curves in Minkowski 3-space, direction curve, and donor curve.
result New methods to construct partner curves of a unit speed non-lightlike curve.
Curved flats linked to pairs of Lie applicable surfaces.
problem Understanding curved flats in Lie sphere geometry.
method One-to-one correspondence with pairs of Demoulin families of Lie applicable surfaces via Darboux transformation.
result Curved flats correspond to specific Lie applicable surface pairs.
The paper examines singular points in Wigner caustics and affine equidistants of planar curves.
problem Analyzing singular points in Wigner caustics and affine equidistants of planar curves.
method Generalizing the Blaschke-Süss theorem to study convex curves and their antipodal pairs.
result Existence of antipodal pairs in convex curves is generalized.
Abstract reviews applications of parabolic structures to nodal curve sheaves.
problem Understanding torsion-free sheaves and Hitchin pairs on nodal curves.
method Examines the relation between nodal curve fundamental groups and moduli spaces of parabolic bundles.
result Establishes connections between representations and moduli spaces.
In this paper, complement-equivalent arithmetic Zariski pairs will be exhibited answering in the negative a question by Eyral-Oka on these curves and their groups. A complement-equivalent arithmetic Zariski pair is a pair of complex projective plane curves having Galois-conjugate equations in some number field whose co…
A compact Riemannian homogeneous space G/H, with a bi--invariant orthogonal decomposition g=h+m is called positively curved for commuting pairs, if the sectional curvature vanishes for any tangent plane in TeH(G/H) spanned by a linearly independent commuting pair in $\mathfrak{…
Study fills a surface with odd, non-3 curves.
problem Determining filling pairs for surfaces with odd, non-3 punctures.
method Constructed minimally intersecting pairs of curves.
result Completed the classification of filling pairs for all surfaces.
A pair of distinct free homotopy classes of closed curves in an orientable surface F with negative Euler characteristic is said to be length equivalent if for any hyperbolic structure on F, the length of the geodesic representative of one class is equal to the length of the geodesic representative of the other clas…
Study on ordering geodesics on hyperbolic surfaces.
problem Determining the order of lengths of closed geodesics on hyperbolic surfaces.
method Using Teichmüller space and properties of curves on pairs of pants.
result Order of lengths of curves determine a point in Teichmüller space and identify classes of curves with constant order.
The study finds an upper limit for the number of minimal origami pairs on a surface.
problem Counting the minimal origami pairs on a surface of genus g.
method Algorithm to count minimal origami pairs and using Ménage Problem to establish an upper bound.
result Established a new upper bound for the count of minimal origami pairs.
New examples of translation surfaces on hyperelliptic curves with many automorphisms.
problem Determining when translation surfaces are supported on the same algebraic curve.
method Analyzing eigenforms of automorphisms on hyperelliptic curves with many automorphisms.
result Presentation of infinitely many examples of translation surfaces on hyperelliptic curves.
The article finds conditions for separating filling pairs on surfaces and constructs a Morse function.
problem Conditions for separating filling pairs on surfaces.
method Combinatorial study of mapping class group action and construction of a Morse function.
result Cardinality of global minima of the Morse function equals the number of orbits of filling pairs.
Study isotopy of rational cuspidal curves in 4-manifolds.
problem Isotopy of rational cuspidal curves in 4-manifolds.
method Tame symplectic curves, pseudo-holomorphic curves, log pairs, 4-dimensional topology.
result Every rational cuspidal curve is isotopic to a complex curve in degrees up to 5.
Continuous curves inscribe isosceles trapezoids in complex plane.
problem Proving periodic curves inscribe isosceles trapezoids.
method Lagrangian intersection problem and convergence argument.
result Continuous curves inscribe trapezoids with any similarity type.
The paper constructs minimal coherent filling pairs on surfaces.
problem Finding minimal intersecting coherent filling pairs on surfaces.
method Geometric procedure starting from a torus filling pair.
result Construction of minimal intersecting coherent filling pairs on Sg for g≥3. Study trisections on rational elliptic surfaces to find new Zariski pairs.
problem Constructing trisections and related plane curves on rational elliptic surfaces.
method Utilized Mumford representations of semi-reduced divisors to construct trisections and plane curves.
result Existence of a family of Zariski pairs degenerating to the same conic-line arrangement.
Study Cremona transformations in weighted projective planes to find rational cuspidal curves and Zariski pairs.
problem Finding rational cuspidal curves and Zariski pairs in weighted projective planes.
method Construct families of curves using Cremona transformations, compute fundamental groups, and use blow-up-down decompositions.
result Discover new examples of rational cuspidal curves and Zariski pairs in weighted projective planes.
A quaternionic calculus for surface pairs in the conformal 4-sphere is elaborated. This calculus is then used to discuss the relation between curved flats in the symmetric space of point pairs and Darboux and Christoffel pairs of isothermic surfaces. A new viewpoint on relations between surfaces of constant mean curvat…
Study shows open manifolds can have non-homeomorphic souls with positive curvature.
problem Existence of non-homeomorphic open manifolds with positive curvature souls.
method Extended known existence results for simply connected manifolds with positive sectional curvature.
result Open manifolds can have non-homeomorphic simply connected and positively-curved souls.
Improved trading strategy using macroeconomic forecasts.
problem Optimizing trading strategies based on yield curve mean-reversion.
method Factored in machine learning forecasts of macroeconomic variables to optimize trading signals.
result Clear improvement in APR over evaluation period.
This paper explores how pairs of multicurves can be realized as cylinders on translation surfaces.
problem Understanding when pairs of multicurves can be realized as cylinders on translation surfaces.
method Surface topology and flat grafting deformation.
result Pairs of multicurves can be realized as cylinders on some translation surface.
In this study, we investigate Bertrand curves in three dimensional dual space D3 and we obtain the characterizations of these curves in dual space D3. Also we show that involutes of a curve constitute Bertrand pair curves.
The paper studies Möbius energy gradient of helix pairs and finds limiting behavior as coiling ratio increases.
problem Characterizing the limiting behavior of Möbius energy gradient for symmetric helix pairs.
method Complex asymptotics
result The gradient diverges in opposing directions based on radius, approaching 1/2 as coiling ratio increases.
Study conic line arrangements of degree 7, finding their topology and connected components.
problem Understanding the topology of conic line arrangements of degree 7.
method Identifying a π1-equivalent Zariski pair to prove the existence of a conic line arrangement with specific combinatorics. result Determine the number of connected components of conic line arrangements of degree 7.
Constructs universal local deformations for curves and differential forms.
problem Local deformations of curves and differential forms under preservation of periods.
method Develops Kuranishi families for pairs of curves and meromorphic 1-forms, focusing on hyperelliptic cases.
result First paper in a series developing a deformation theory for spectral curve data of integrable systems.
The distortion of a curve is the supremum, taken over distinct pairs of points of the curve, of the ratio of arclength to spatial distance between the points. Gromov asked in 1981 whether a curve in every knot type can be constructed with distortion less than a universal constant C. Answering Gromov's question seems to…
Study shows K-moduli spaces of curves on quadrics and K3 surfaces match with VGIT quotients.
problem Understanding K-moduli spaces of curves on quadrics and K3 surfaces.
method Using log Fano pairs and VGIT quotients, the study compares K-moduli spaces of curves on P1imesP1 and quartic hyperelliptic K3 surfaces. result K-moduli spaces of curves on quadrics and K3 surfaces form a natural interpolation.
Identifies spectral curves for SU(3) coadjoint orbits.
problem Understanding the geometry of coadjoint orbits in SU(3).
method Using Hitchin pairs and spectral curves, identifies a Hamiltonian circle action and finds Darboux coordinates.
result Identifies a differential equation for the Hamiltonian.
This paper generalizes the envelope of mid-lines to intermediate lines for a plane curve.
problem Understanding the envelope of intermediate lines for a plane curve.
method Using singularity theory techniques to analyze the local behavior of the envelope of intermediate lines.
result The envelope of intermediate lines (EIL) is formed by three disconnected sets: AEIL, the curve itself, and IPTL. For smooth families of projective algebraic curves, we extend the notion of intersection pairing of metrized line bundles to a pairing on line bundles with flat relative connections. In this setting, we prove the existence of a canonical and functorial "intersection" connection on the Deligne pairing. A relationship is…
Paper connects surfaces in 4D and 3D spacetime.
problem Finding relations between Lorentz surfaces in different spacetime dimensions.
method Weierstrass-type representations for null curves and surfaces.
result Relation between minimal Lorentz surfaces in R24 and R13. Study on Mannheim curves in 3D space with modified frame.
problem Characterizing Mannheim curves in modified orthogonal frames.
method Investigation of Mannheim pairs, Frenet-Mannheim, and Weakened Mannheim curves.
result Characterizations of Mannheim curves in modified frames.
We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. This invariant is shown to be a generalization of the I-invariant of line arrangements developed by the first author with Artal and Florens. We give two practical tools for computin…
Study conditions for curvature functions of closed planar curves.
problem Conditions for curvature functions of closed planar curves.
method Equivalent conditions and periodic behaviors shown; explicit construction of pairs.
result Characterization of curvature functions and limitations of 4-vertex theorem.
The paper computes a pairing for knot concordance and finds non-slice knots.
problem Computing and understanding the twisted Blanchfield pairing for knot concordance.
method Combinatorial algorithm for computing the twisted Blanchfield pairing; using zero-framed surgery and Casson-Gordon representations.
result Some satellites of genus two ribbon knots are non-slice.
In this study, we define a family of null curves in Minkowski 3-space and called null similar curves. We obtain some properties of these special curves. We show that two null curves are null similar curves if and only if these curves form a null Bertrand pair. Moreover, we obtain that the family of null geodesics and n…
Study shows how lengths of geodesic arcs determine linking number of Legendrian knots.
problem Linking number of Legendrian knots on negatively curved surfaces.
method Analyzes Poincaré series on negatively curved surfaces.
result Explicit rational value of Poincaré series at 0 interprets linking number of Legendrian knots.