The paper constructs examples of complex manifolds with singularities and jumps.
problem Understanding the structure and behavior of complex manifolds with singularities.
method Explicit construction of twistor spaces with jumping rational curves and singularities.
result The existence of normalisable solutions through folds in certain cases.
We study deformations of irreducible Hermitian symmetric spaces S of the compact type, known to be locally rigid, as projective-algberaic manifolds and prove that no jump of complex structures can occur. For each S of rank ≥2 there is an associated reductive linear group G such that S admits a holomorphic …
The paper defines normal forms for rational 3-tangles and shows a sequence of moves to transform one form to another.
problem Understanding and manipulating rational 3-tangles.
method Definition of normal forms and sequence of normal jump moves.
result There is a sequence of normal jump moves leading to equivalent normal forms of rational 3-tangles.
RationalNet improves graph convolutional networks by approximating jump discontinuities more efficiently.
problem Graph convolutional networks struggle with approximating jump discontinuities, leading to oscillations and high computational costs.
method RationalNet uses rational functions to approximate graph signals, avoiding oscillations and reducing computational complexity.
result RationalNet effectively characterizes jump discontinuities, outperforming other methods on both synthetic and real-world graphs.
We study the following question: given a set P of 3d-2 points and an immersed curve G in the real plane R^2, all in general position, how many real rational plane curves of degree d pass through these points and are tangent to this curve. We count each such curve with a certain sign, and present an explicit formula for…
Study reveals strong co-jumping behavior in U.S. yield curves compared to Europe.
problem Understanding co-jumps in interest rate futures markets.
method Localized co-jumps through wavelet coefficients, identified statistically significant ones, and analyzed using high frequency data.
result Stronger co-jumping behavior in U.S. yield curves compared to European ones.
Study Alexander invariants and cohomology jump loci in group extensions with trivial monodromy.
problem Understanding Alexander invariants and cohomology jump loci in group extensions with specific conditions.
method Analyzing integral, rational, and modular Alexander invariants and cohomology jump loci of groups as extensions with trivial algebraic monodromy.
result Established a tight relationship between Alexander invariants, characteristic varieties, and resonance varieties, leading to an inequality between Chen ranks.
We describe a graph parametrization of rational quadratic differentials with presence of a simple pole, whose critical trajectories form a network depending on parameters focusing on the network topological jumps. Obtained bifurcation diagrams are associated with the Stasheff polytopes.
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.
The paper explores Newton-Cartan structures with torsion on Kodaira moduli spaces.
problem Understanding Newton-Cartan spacetimes and their deformations.
method Construction of connections and frames on Kodaira moduli spaces, generalizing canonical connections to include torsion.
result Novel twistor theories of Newton-Cartan spacetimes in three and five dimensions, including torsion.
We give here some extensions of Gromov's and Polterovich's theorems on $\karea$ of CPn, particularly in the symplectic and Hamiltonian context. Our main methods involve Gromov-Witten theory, and some connections with Bott periodicity, and loop groups. The argument is closely connected with study of jump…
New varieties found without smooth curves.
problem Finding varieties without smooth rational curves.
method Constructing normal rationally connected varieties.
result Found varieties of arbitrary large dimensions without smooth rational curves.
Classifies real rational knots and curves in a specific quadric space.
problem Classifying real rational knots and curves in a quadric space of signature (3,2). method Classification through a study of real rational curves of low degree in the quadric.
result Provides representatives of all real rational knots of degree ≤5 in the quadric. A theory of signatures for odd-dimensional links in rational homology spheres is studied via their generalized Seifert surfaces. The jump functions of signatures are shown invariant under appropriately generalized concordance and a special care is given to accommodate 1-dimensional links with mutual linking. Furthermor…
We study a smooth analogue of jumping curves of a holomorphic vector bundle, and use Yang-Mills theory over S2 to show that any non-trivial, smooth Hermitian vector bundle E over a smooth simply connected manifold, must have such curves. This is used to give new examples complex manifolds for which a non-tri…
Study constraints on singular points of rational cuspidal curves using Heegaard Floer theory.
problem Constraints on singular points of rational cuspidal curves.
method Involutive Heegaard Floer homology theory.
result Results do not apply to rational cuspidal curves of even degree.
Three methods solve spatial rational curves with rational arc length.
problem Construct all spatial rational curves with rational arc length.
method Three different methods: PH curve adaptation, zero-residue conditions, and dual approach.
result Three methods share quaternion-based representation.
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.
Survey on minimal rational curves and their geometric structures.
problem Germ-equivalence problem of minimal rational curves on uniruled projective manifolds.
method Analysis of isotrivial families of projective varieties and G-structures.
result Natural G-structure on Zariski-open subset of uniruled projective manifolds.
New findings on prime theta-curves with simple tangles.
problem Understanding prime theta-curves with specific unknotting numbers.
method Analyzing composite theta-curves and their components.
result Composite theta-curves with unknotting number one are prime.
Survey on rational curves on complex surfaces, highlighting different approaches.
problem Existence of rational curves on complex surfaces.
method Classification of complex surfaces and systematic study of rational curves in each class.
result Highlighting the different approaches to study rational curves on complex surfaces.
A conjecture about rational curves' formal principle and convergence proved for Goursat type families.
problem Formal principle and convergence for rational curves of Goursat type.
method Natural ODEs and Cartan connections constructed by Doubrov-Komrakov-Morimoto.
result The conjecture is proved for rational curves of Goursat type.
Study pseudo-hyperkähler geometry of curves in hyperkähler twistor spaces.
problem Understanding the geometry of rational curves in twistor spaces.
method Investigate pseudo-hyperkähler geometry of higher degree rational curves.
result Characterize the pseudo-hyperkähler structure of rational curves.
The study of symplectic fillings for rational cuspidal curves.
problem Understanding symplectic fillings of contact manifolds associated with rational cuspidal curves.
method Exploration through Stein handlebodies and rational blow-downs.
result Examples of contact manifolds that are links of normal surface singularities, and those that do not admit symplectic fillings.
Study of rational curves in complex manifolds with specific normal bundles.
problem Characterizing rational curves in complex manifolds with given normal bundles.
method Analyzing differential and projective geometric properties of rational curves and their tangents.
result Classification of rational curves into Goursat and Cartan types based on their geometric properties.
New proof for curved 3-cohom manifold rational ellipticity.
problem Rational ellipticity of curved manifolds with specific cohomogeneity.
method Proved rationally elliptic for cohomogeneity-three manifolds with positive curvature and no boundary quotient.
result Closed, simply connected, positively curved, cohomogeneity-three manifolds without boundary are rationally elliptic.
Formula conjectured for rational cuspidal curves in projective plane.
problem Counting rational cuspidal curves in projective plane.
method Extending Kontsevich's recursion formula and using geometric input about tangency of curves at nodal points.
result Conjectural formula agrees with earlier computations and extends to rational quartics with E6 singularity.
Generalizes tropical curves by relaxing integrality and rationality requirements.
problem Existence and uniqueness of pseudotropical curves.
method Interpretation as critical points of a quadratic functional, dual polygons, intersection theory.
result Existence and uniqueness of pseudotropical curves established.
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.
K3 surfaces get a rational curve when a divisor is big and positive enough.
problem Finding rational curves on K3 surfaces with specific conditions.
method Degeneration technique to prove existence of integral nodal rational curves.
result Generic Λ-polarised K3 surface has an integral nodal rational curve in the linear system ∣L∣. Study delta invariant of curves on rational surfaces using topological methods.
problem Calculate delta invariant for curves embedded in rational singularities.
method Use topological techniques and Poincaré series.
result Develop formulae for delta invariant in terms of embedded data.
We consider the optimal stopping of a class of spectrally negative jump diffusions. We state a set of conditions under which the value is shown to have a representation in terms of an ordinary nonlinear programming problem. We establish a connection between the considered problem and a stopping problem of an associated…
Classifies curves up to symplectic isotopy.
problem Classifying rational cuspidal curves up to symplectic isotopy.
method Topological tools, pseudoholomorphic techniques, and birational transformations.
result Classifies rational cuspidal curves of degrees 6 and 7 up to symplectic isotopy.
Study minimal rational curves on complex manifolds with isotropic VMRT.
problem Understanding minimal rational curves tangent to distributions on complex manifolds.
method Partial equivariant compactification of metabelian groups.
result Any isotropic VMRT can be realized as VMRT of minimal rational curves tangent to a distribution.
Complete conjecture on rational curves on K3 surfaces.
problem Existence of infinitely many rational curves on K3 surfaces.
method Two new techniques: regeneration and marked point trick.
result Existence of integral curves of unbounded degree for any projective K3 surface.
We formulate the equivalence problem, in the sense of E. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form …
Complex projective manifolds without rational curves are quotients of Abelian varieties.
problem Characterizing complex projective manifolds without rational curves.
method Using conjectures about rational and entire curves on Calabi-Yau varieties.
result Non-hyperbolic complex projective manifolds contain the image of an Abelian variety.
Formula counts rational curves with a specific singular point in projective space.
problem Counting rational degree d curves with an m-fold point in CP2. method Recursive formula derived from Kontsevich's recursion formula, considering a family version.
result Obtained a recursive formula for the number of curves.
Study shows vast rational cohomology in moduli space of curves with level structures.
problem Understanding the cohomology of moduli spaces with level structures.
method Proved existence of enormous rational cohomology in cohomological dimension.
result Cohomological dimension of moduli space of curves is at least g-2.
By using analytic method, we prove that there exist rational curves on compact Hermitian manifolds with positive holomorphic bisectional curvature. It confirms a question of S.-T. Yau. It is well-known that Mori proved in \cite{Mori79} that every compact complex manifold N with c1(N)>0 contains at least one ration…
We explore a relationship between topological properties of orbits of 2-cycles in the symplectomorphism group Symp(M) and the existence of rational curves in M. Under the absence of rational curves hypothesis, we show that evaluation map vanishies on the second homotopy group and obtain a Gottlieb-type vanishing theore…
We study rational cuspidal curves in Hirzebruch surfaces. We provide two obstructions for the existence of rational cuspidal curves in Hirzebruch surfaces with prescribed types of singular points. The first result comes from Heegaard--Floer theory and is a generalization of a result by Livingston and the first author. …
Study non-negatively curved GKM orbifolds and their cohomology.
problem Characterize non-negatively curved GKM orbifolds and their cohomology.
method Analyze rational cohomology rings and isometric actions of finite groups.
result Rational cohomology rings of GKM orbifolds are isomorphic to model orbifolds.
Study rational cuspidal curves in projective surfaces with topological and algebraic obstructions.
problem Obstructing possible configurations of singular points on rational cuspidal curves.
method Two criteria: one based on Bezout theorem, the other on Ozsvath-Szabo inequalities.
result Explicit calculations show similar obstructions from both approaches.
The present paper attempts to show an alternative approach with regards to rational Pythagorean-hodograph (PH) curves and especially more natural approach for rational PH helices (i.e. rational helices). It exploits geometric features of rational helices to obtain a simpler construction of these curves and apply this t…
In this paper we show that the space of nodal rational curves, which is so called a Severi variety (of rational curves), on any non-singular projective surface is always equipped with a natural Einstein-Weyl structure, if the space is 3-dimensional. This is a generalization of the Einstein-Weyl structure on the space o…
Study proposes pricing mechanism for cryptocurrency options.
problem High speculation, volatility, and discontinuity in cryptocurrency markets.
method Proposes a pricing mechanism based on SVCJ model with co-jumps.
result Shows significant contemporaneous anti-correlation between jumps in price and volatility.
This paper proves an upper limit on rational points on curves.
problem Finding rational points on curves of genus at least two.
method Arithmetic and analytic estimates.
result Explicit upper bounds on rational points.