The Gronwall conjecture states that a planar 3-web of foliations which admits more than one distinct linearizations is locally equivalent to an algebraic web. We propose an analogue of the Gronwall conjecture for the 3-web of foliations by Legendrian curves in a contact three manifold. The Legendrian Gronwall conjectur…
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48 results for “Gronwall's conjecture”
The paper proves a conjecture about linear webs using invariants.
problem Proving Gronwall's conjecture about linear webs.
method Using Agafonov's results, the paper introduces an invariant to prove the conjecture.
result A diffeomorphism that preserves the characteristic of points in isomorphic webs is a homography.
We study non-flat planar 3-webs with infinitesimal symmetries. Using multi-dimensional Schwarzian derivative we give a criterion for linearization of such webs and present a projective classification thereof. Using this classification we show that the Gronwall conjecture is true for 3-webs admitting infinitesimal symme…
On Gronwall conjecturemath.DG
Gronwall conjecture states that a planar 3-web which admits more than one distinct linearization is locally equivalent to an algebraic web. We give a partial answer to the conjecture in the affirmative for the class of planar 3-webs with the web curvature that vanishes to order three at a point. The differential relati…
We present a projectively invariant description of planar linear 3-webs. For a non-hexagonal 3-web, we introduce family of projective torsion-free Cartan connections, the web leaves being geodesics for each member of the family, and give a web linearization criterion. Finally, we propose an algorithm for resolving the …
We find relative differential invariants of orders eight and nine for a planar nonparallelizable 3-web such that their vanishing is necessary and sufficient for a 3-web to be linearizable. This solves the Blaschke conjecture for 3-webs. As a side result, we show that the number of linearizations in the Gronwall conject…
We prove Gronwall-type estimates for the distance of integral curves of smooth vector fields on a Riemannian manifold. Such estimates are of central importance for all methods of solving ODEs in a verified way, i.e., with full control of roundoff errors. Our results may therefore be seen as a prerequisite for the gener…
In the article "On the linearizability of 3-webs" (Nonlinear analysis 47, (2001) pp.2643-2654), published in 2001, we studied the linearizability problem for 3-webs on a 2-dimensional manifold. Four years after the publication of our article, V.V.Goldberg and V.V.Lychagin in the paper "On linearization of planar three-…
Formalizes integral curves on Banach manifolds in Lean.
problem Existence and uniqueness of integral curves on Banach manifolds.
method Formalized differential equations on Banach spaces, then generalized to Banach manifolds.
result Established theorems for integral curves on Banach manifolds.
Simplified analysis of diffusion models using discrete random variables.
problem Theoretical analysis of diffusion models is complex and requires rigorous proofs.
method Simplified framework for analyzing Euler--Maruyama discretization of VP-SDEs using Grönwall's inequality.
result Standard Gaussian noise can be replaced by discrete random variables without sacrificing convergence guarantee.
In the present paper we study geometric structures associated with webs of hypersurfaces. We prove that with any geodesic (n+2)-web on an n-dimensional manifold there is naturally associated a unique projective structure and, provided that one of web foliations is pointed, there is also associated a unique affine struc…
In this paper, we study the herding phenomena in financial markets arising from the combined effect of (1) non-coordinated collective interactions between the market players and (2) concurrent reactions of market players to dynamic market signals. By interpreting the expected rate of return of an asset and the favorabi…
We give sharp conditions on a local biholomorphism which ensure global injectivity. For , such a map is injective if for each complex line , the pre-image embeds holomorphically as a connected domain into , the embedding bei…
This is an introductory chapter in a series in which we take a systematic study of the Yang-Mills equations on curved space-times. In this first, we provide standard material that consists in writing the proof of the global existence of Yang-Mills fields on arbitrary curved space-times using the Klainerman-Rodnianski p…
Study global solutions for Boussinesq systems on curved manifolds.
problem Global existence and uniqueness of solutions to Boussinesq systems on non-compact Riemannian manifolds with gravitational fields.
method Used dispersive and smoothing estimates of a vectorial matrix semigroup to establish global existence and uniqueness of mild solutions for linear systems. Then, applied fixed point arguments to semilinear systems. Proved exponential stability using Gronwall's inequality.
result Established global existence, uniqueness, and exponential stability of mild solutions to the Boussinesq systems on non-compact Riemannian manifolds with gravitational fields.
Study on Navier-Stokes equations on non-compact manifolds, proving existence and decay of solutions.
problem Existence and asymptotic behavior of solutions to Navier-Stokes equations on non-compact manifolds.
method Used -dispersive and smoothing estimates of the Stokes semigroup, fixed point arguments, and Gronwall's inequality.
result Established existence and exponential decay of almost periodic and asymptotically almost periodic mild solutions.
Bounding the approach to oligarchy in a variant of the yard-sale modelcond-mat.stat-mech
Analytical results bound the approach to oligarchy in a modified asset exchange model.
problem Analyzing economic inequality in a modified asset exchange model.
method Analytical results using Gini coefficient and differential inequalities.
result The Gini coefficient is bounded by a first-order differential inequality.
Polyhedra volume conjecture supports Stoker conjecture weakly.
problem Proving the Stoker conjecture for polyhedra.
method Using an extension of Montcouquiol and Weiss' result on polyhedra angles as local coordinates.
result Volume Conjecture for polyhedra implies a weak version of the Stoker conjecture.
Survey on two non-Kähler geometry conjectures.
problem Constant holomorphic sectional curvature and Fino-Vezzoni conjectures in non-Kähler geometry.
method Survey and discussion of historical and recent developments.
result Discussion of conjectures without new results.
Numerical study confirms Brennan's conjecture for a counterexample to Thurston's conjecture.
problem Thurston's conjecture and Brennan's conjecture in planar domains.
method Numerical analysis of a specific counterexample to Thurston's conjecture.
result The counterexample does not contradict Brennan's conjecture.
Uniform bounds for neural network convergence without strong convexity assumptions.
problem Understanding the convergence of neural networks in the feature-learning regime.
method Establishing uniform-in-time weak propagation-of-chaos via mean-field deterministic Wasserstein-gradient-flow dynamics.
result Uniform bounds on the difference between infinite-width and finite-width neural network outputs, showing that fewer neurons can achieve a desired loss.
The non-vanishing conjecture implies the abundance conjecture in certain cases.
problem Abundance conjecture in algebraic geometry.
method Proof of the abundance conjecture under specific conditions.
result The abundance conjecture holds in dimensions ≤ 5 when κ ≥ 0 and ν ≤ 1.
Proves Gromov's conjecture and answers Stoker's polyhedron conjecture.
problem Gromov's flat corner domination conjecture and Stoker's conjecture for convex polyhedra.
method Same techniques applied to prove conjectures.
result Proves Gromov's conjecture and answers Stoker's polyhedron conjecture.
This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings-Jaco theorem which established a similar result for the Poincare Conjecture. The paper also gives two other algebraic conjectures; one is equi…
Paper discusses conjectures and proves some related inequalities.
problem Unified generalization of BW and DDVV inequalities.
method Unified discussion and proofs of conjectures.
result Proves Conjecture 2 and obtains a new upper bound for Conjecture 3.
Symmetry-breaking in three differential geometry conjectures.
problem Exploring the role of symmetry in three differential geometry conjectures.
method Examining the Carathéodory, Willmore, and Lawson Conjectures through the lens of symmetry in 3D space-forms.
result Symmetry is broken, and more general ambient metrics are considered, leading to the failure of the conjectures.
We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that are known to be counterexamples to the coarse Baum-Connes conjecture. In particula…
Study proves Hecke lifting conjecture for torus knots and verifies it for any framed knots.
problem Integrality structure of framed knots' quantum invariants.
method Explicit formulas of colored HOMFLY-PT invariants of torus knots, verified in a limit form for any framed knots.
result Proves Hecke lifting conjecture for torus knots and verifies it for any framed knots.
New proof shows most thin knots satisfy Cabling Conjecture.
problem Cabling Conjecture for thin knots using Heegaard Floer homology.
method Heegaard Floer homology and immersed curves techniques.
result Almost all thin knots satisfy the Cabling Conjecture.
We review the Burghelea conjecture, which constitutes a full computation of the periodic cyclic homology of complex group rings, and its relation to the algebraic Baum-Connes conjecture. The Burghelea conjecture implies the Bass conjecture. We state two conjectures about groups of finite asymptotic dimension, which tog…
Metric SYZ conjecture proved using non-archimedean geometry.
problem Proving the metric SYZ conjecture in large generality.
method Using non-archimedean geometry to support the conjecture.
result Metric SYZ conjecture can be proved in large generality.
Paper confirms Chen's biharmonic conjecture for hypersurfaces in 5D.
problem Chen's conjecture on biharmonic submanifolds in Euclidean spaces.
method Analyzes hypersurfaces in \(\mathbb{R}^5\) for \(n=4\).
result Chen's conjecture is confirmed for hypersurfaces in \(\mathbb{R}^5\) when \(n=4\).
Proved Farrell-Jones conjecture for free-by-cyclic groups.
problem Proving the Farrell-Jones conjecture for a specific class of groups.
method Geometric methods for establishing the Farrell-Jones Conjecture.
result Proved the Farrell-Jones conjecture for free-by-cyclic groups.
Counterexample disproves recent Penrose conjecture variant.
problem A variant of the Penrose conjecture.
method Provided a counterexample.
result The conjectured variant of the Penrose inequality is false.
Proofs On Arnold Conjecturesmath.SG
In this article, we give proofs on the Arnold Lagrangian intersection conjecture on the cotangent bundles, Arnold-Givental Lagrangian intersection conjecture and the Arnold fixed point conjecture.
Proof outlined for 4D smooth Poincaré conjecture.
problem 4-dimensional smooth Poincaré conjecture.
method Outline of proof.
result Proof of 4D smooth Poincaré conjecture.
In this paper, we generalize the Cosmetic Surgery Conjecture to an -cusped hyperbolic -manifold and prove it under the assumption of another well-known conjecture in number theory, so called the Zilber-Pink Conjecture. For and , we show them without the assumption.
The Novikov Conjecturemath.GT
Survey on Novikov conjecture and its applications.
problem Novikov conjecture in topology.
method Survey and recent developments.
result Applications to topological rigidity and non-rigidity.
Paper connects AJ conjecture and colored Jones polynomial potential function.
problem Relationship between -polynomial and colored Jones polynomial.
method Connects AJ conjecture and colored Jones polynomial potential function.
result Establishes connection between -polynomial and colored Jones polynomial potential function.
Akbulut and Kirby conjectured that two knots with the same -surgery are concordant. In this paper, we prove that if the slice-ribbon conjecture is true, then the modified Akbulut-Kirby's conjecture is false. We also give a fibered potential counterexample to the slice-ribbon conjecture.
Counterexample disproves conjectures about log canonical thresholds.
problem Conjectures about log canonical thresholds were disproved.
method Provided a counterexample to both conjectures.
result Conjectures about log canonical thresholds are false.
Study confirms conjecture on Hermitian manifolds with bounded mass.
problem Morse-type integrals in nef (1,1) classes on compact Hermitian manifolds with bounded mass.
method Analyzes conjecture using bounded mass property on compact Hermitian manifolds.
result Confirms Demailly-Păun and Tosatti-Weinkove's conjectures.
Reformulated Markov's conjecture in combinatorial terms.
problem Markov's uniqueness conjecture in integral necklaces.
method Geometric reformulation and combinatorial description.
result Explicitly described set of lengths on modular torus.
Paper confirms Whitehead's conjecture for aspherical 2-complexes.
problem Whitehead's conjecture about aspherical 2-complexes.
method Argument on ribbon sphere-links, generalized for aspherical 2-complexes.
result Whitehead's conjecture confirmed for aspherical 2-complexes.
We survey the recent results and current issues on the topological rigidity problem for closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. A number of open problems and conjectures are presented during the course of the discussion. We also review the status and app…
Verifies a conjecture for the figure eight knot.
problem Relates A-ideal and recurrence ideal of knots.
method Uses quantum A-ideals, q-holonomicity, and AJ conjecture.
result Strong AJ conjecture verified for figure eight knot.
Review of Yau's conjecture on zero sets of Laplace eigenfunctions.
problem Yau's conjecture on zero sets of Laplace eigenfunctions.
method Discussion of old and new results and methods related to the conjecture, including solutions and new results in smooth settings.
result Discussion of Donnelly and Fefferman's solution of the conjecture in the real-analytic Riemannian manifold case and new results in the smooth setting.
Fukaya-Yamaguchi conjecture holds in 4D manifolds with nonnegative curvature.
problem Fundamental group of nonnegative curvature manifolds.
method Observation in dimension 4.
result Fukaya-Yamaguchi conjecture holds in 4D.