Classifies rational differential forms on the Riemann sphere based on their isotropy group.
problem Classifying rational differential forms on the Riemann sphere based on their isotropy group.
method Analyzing the isotropy group of rational differential forms with simple poles and zeros.
result Finite subgroups of PSL(2, C) are realizable as isotropy groups for rational 1-forms on the Riemann sphere.
The goal and the main result of the paper is to provide a complete description of the field of rational differential invariants of one class of second order ordinary differential equations with scalar control parameter with respect to Lie pseudo-group of local feedback transformations. In particular, considered class d…
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.
Algorithm constructs integrals of hyperexponential 1-forms.
problem Computing integrals of hyperexponential 1-forms.
method Using Schanuel's conjecture, decomposes Hω into f(F(x))+H(x)R(x). result Algorithm constructs a basis of cohomology of differential 1-forms.
The paper provides a differential form interpretation of a theorem about the dimensions of rational homotopy groups of Diff(D^4).
problem Lower bounds of dimensions of rational homotopy groups of Diff(D^4) in terms of graph homology.
method Differential form interpretation and extension to arbitrary even dimensions.
result The proof can be extended to arbitrary even dimensions and made accessible to more readers.
We characterize primary operations in differential cohomology via stacks, and illustrate by differentially refining Steenrod squares and Steenrod powers explicitly. This requires a delicate interplay between integral, rational, and mod p cohomology, as well as cohomology with U(1) coefficients and differential forms. A…
In this paper we show that every rational cohomology class of type (p,p) on a compact Kähler manifold can be representated as a differential (p,p)-form given by an explicit formula involving a Čech cocycle. First we represent Chern characters of smooth vector bundles by Čech cocycles with values in the sheaf of dif…
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.
Parametric Cartan theory of exterior differential systems, and explicit cohomology of projective manifolds reveal united rationality features of differential algebraic geometry.
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.
Extends Chern character to non-abelian cohomology, linking to physics.
problem Generalizing Chern character to non-abelian cohomology.
method Leveraging dg-algebraic rational homotopy theory and de Rham theorem.
result Generalizes Chern-Dold character, Chern-Weil homomorphism, and Cheeger-Simons homomorphism.
The systems of complex analytic second order ordinary differential equations whose solutions close up to become rational curves (after analytic continuation) are characterized by the vanishing of an explicit differential invariant, and turn out to provide an infinite dimensional family of integrable systems.
The paper converts nonalternating forms of rational links into all-even forms and derives formulas for their braid index and HOMFLY polynomial.
problem Finding formulas for the braid index and HOMFLY polynomial of rational links.
method Algorithm to transform nonalternating forms into all-even forms and derivation of formulas.
result Formulas for the braid index and HOMFLY polynomial of rational links in terms of their reduced alternating form.
For a Morse map f:M→S1 Novikov [11] has introduced an analog of Morse complex, defined over the ring $\ZZZ[[t]][t^{-1}]$ of integer Laurent power series. Novikov conjectured, that generically the matrix entries of the differentials in this complex are of the form ∑iaiti, where ai grow at most exponenti…
Researchers found differential invariants for Kundt spacetimes.
problem Equivalence problem for Lorentzian metrics in Kundt spacetimes.
method Found generators for rational differential invariants for Kundt spacetimes.
result Relating findings to other approaches to the equivalence problem.
Homology of abelian differentials stabilizes with more zeros.
problem Understanding the homology of abelian differentials with many simple zeros.
method Developed an h-principle for these strata, valid in a range of homological degrees.
result Homology stabilizes in a range where the number of simple zeros is large.
Paper classifies rational 3-tangles using normal forms and minimal coordinates.
problem Classifying rational 3-tangles up to isotopy.
method Defined normal form and normal coordinate, investigated minimal coordinates, constructed contractible simplicial complex.
result Simplicial complex of normal forms is contractible, leading to classification of rational 3-tangles.
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.
Study non-commutative function algebras using contact geometry.
problem Quantize non-commutative function algebras in several variables.
method Contact geometry and rational differential operators.
result Generalizes known constructions of classical equivariants.
Equivalence of second order differential operators in vector bundles studied.
problem Equivalence problem for second order linear differential operators in vector bundles.
method Description of rational invariants of symbols, finding connections associated with differential operators.
result Solving problems of local and global equivalency of differential operators.
We use superconnections to define and study some natural differential forms on period domains D that parametrize polarized Hodge structures of given type on a rational quadratic vector space V. These forms depend on a choice of vectors v1,…,vr∈V and have a Gaussian shape that peaks on the locu…
Paper proves triple linking form vanishes under specific conditions.
problem Analyzing rational homology cobordism and linking forms.
method Proves vanishing of triple torsion linking form under specific conditions.
result Triple torsion linking form vanishes on a specific Lagrangian.
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.
Algorithm finds Liouvillian solutions for planar rational vector fields.
problem Finding Liouvillian solutions for planar rational vector fields.
method Algorithm to compute telescoper for specific foliations and rational vector fields.
result Algorithm finds Liouvillian solutions for planar rational vector fields, given a large enough complexity bound.
This paper gives two new combinatorial topological proofs of the classification of rational tangles. Each proof rests on an elegant lemma showing that rational tangles are isotopic to canonical alternating rational tangles. The first proof defines the tangle fraction from the canonical form and uses flyping to prove in…
Geometrically describes the linear and quadratic forms for rational links.
problem Predicting generating functions for colored HOMFLY-PT polynomials of rational links.
method Direct geometric description of linear and quadratic forms in terms of configuration spaces.
result Direct geometric description of forms for rational links.
The study sets constraints on 4-manifold forms linked to specific invariants.
problem Understanding the intersection forms of spin 4-manifolds bounded by Seifert rational homology 3-spheres.
method Analyzes constraints using the μ-bar and κ invariants.
result The difference between κ and -μ-bar for a Seifert rational homology 3-sphere is at most 2, and under certain conditions, it is 0.
We give an introduction to the theory of varieties of minimal rational tangents, emphasizing its aspect as a fusion of algebraic geometry and differential geometry, more specifically, a fusion of Mori geometry of minimal rational curves and Cartan geometry of cone structures.
Study spectral gaps in hyperbolic rational homology spheres.
problem Finding spectral gaps in hyperbolic rational homology spheres.
method Construction of families of hyperbolic rational homology spheres with coexact 1-form spectral gaps.
result Provided intervals containing limit points of spectral gaps, with the rightmost interval being [0.8196, 0.8277].
An ordinary differential field (F,d) of characteristic zero, a subgroup H of affine group GL(n,C)∝Cn with respect to its identical representation in Fn and the following two fields of differential rational functions in x=(x1,x2,...,xn)-column vector, $$C< x, d>^H=\{f^d< x> \in C< x, d> : f^d< hx+…
Consider the moduli space of pairs (C,w) where C is a smooth compact complex curve of a given genus and w is a holomorphic 1-form on C with a given list of multiplicities of zeroes. We describe connected components of this space. This classification is important in the study of dynamics of interval exchange transformat…
Study symplectic forms on manifolds to find Lagrangian pinwheels that can be separated.
problem Determine conditions for symplectic forms to carry disjoint Lagrangian pinwheels.
method Use rational blow-up to analyze Lagrangian pinwheels in symplectic manifolds.
result Conditions for disjunction of Lagrangian pinwheels in specific manifolds.
We establish the equivalence of the Tuynman midpoint area formula for a spherical triangle to the classical area formulas of Euler and of Cagnoli. The derivation also yields a variant of the Cagnoli formula in terms of the medial triangle. We introduce the three barycentric coordinates of a point within the spherical t…
Let X be some Riemann surface, and let omega be a meromorphic quadratic differential form on X, that is, omega can be written in local coordinates as f(z) dz^2, for some meromorphic function f. We say that a curve gamma is part of a horizontal leaf of omega if for each t in I, we have that f(gamma(t)) (gamma'(t))^2 is …
In this paper, We introduce an invariant of rational n-tangles which is obtained from the Kauffman bracket. It forms a vector with Laurent polynomial entries. We prove that the invariant classifies the rational 2-tangles and the reduced alternating rational 3-tangles. We conjecture that it classifies the rational 3-tan…
The paper connects curvature positivity to rational connectedness in complex geometry.
problem Establishing a geometric criterion for rational connectedness.
method Uhlenbeck-Yau's continuity method applied to mean curvature positivity.
result Holomorphic tangent bundle mean curvature positivity is equivalent to rational connectedness of compact Kähler manifolds.
The description of invariants of surfaces with respect to the motion groups is reduced to the description of invariants of parameterized surfaces with respect to the motion groups. Existence of a commuting system of invariant partial differential operators (derivatives) and a finite system of invariants, such that any …
Null Lagrangian-preserving surgeries are a generalization of the Garoufalidis and Rozansky null-moves, that these authors introduced to study the Kricker lift of the Kontsevich integral, in the setting of pairs (M,K) composed of a rational homology sphere M and a null-homologous knot K in M. They are defined as replace…
Complex contact manifolds arise naturally in differential geometry, algebraic geometry and exterior differential systems. Their classification would answer an important question about holonomy groups. The geometry of such manifold X is governed by the contact lines contained in X. These are related to the notion of…
We show that the characteristic series for the greedy normal form of a Coxeter group is always a rational series, and prove a reciprocity formula for this series when the group is right-angled and the nerve is Eulerian. As corollaries we obtain many of the known rationality and reciprocity results for the growth series…
New geometric Joyce structures on moduli spaces of quadratic differentials.
problem Constructing Joyce structures on moduli spaces of quadratic differentials.
method Isomonodromic deformations of second-order linear ODEs with rational potential.
result Construction of Joyce structures on moduli spaces of quadratic differentials.
Identifies a mod-p triple cup product for rational homology 3-spheres with specific first homology.
problem Locally flat embeddings in S4 for rational homology 3-spheres method Using triple torsion linking form and torsion-linking duality
result Identifies the mod-p triple cup product for specific rational homology 3-spheres We give a description of recently introduced Doubrov-Ferapontov general heavenly equation in terms of closed differential Plücker two-form, rationally depending on the spectral parameter. We demonstrate that general heavenly equation is an important generating equation in the context of Takasaki hyper-Kähler hierarchy,…
In general, the product of harmonic forms is not harmonic. We study the top exterior power of harmonic two-forms on compact Kaehler manifolds. Often, it is not harmonic. This phenomenon is related to the geometry of the manifold and to the existence of rational curves in particular. K3 surfaces and hyperkaehler manifol…
We show that the Q/Z-valued linking forms on rational homology spheres are (anti-) symmetric and we compute the linking form of a 3-dimensional rational homology sphere in terms of a Heegaard splitting. Both results have been known to a larger or lesser degree, but it is difficult to find rigorous d…
We study Nakai-Moishezon type question and Donaldson's "tamed to compatible" question for almost complex structures on rational four manifolds. By extending Taubes' subvarieties--current--form technique to J−nef genus 0 classes, we give affirmative answers of these two questions for all tamed almost complex structu…
We prove a global algebraic version of the Lie-Tresse theorem which states that the algebra of differential invariants of an algebraic pseudogroup action on a differential equation is generated by a finite number of rational-polynomial differential invariants and invariant derivations.
In this article, we give a classification of Alexander modules of null-homologous knots in rational homology spheres. We characterize these modules A equipped with their Blanchfield forms φ, and the modules A such that there is a unique isomorphism class of (A,φ), and we prove that for the other modules A, there ar…