Paper defines doodles on closed surfaces, unifying classical and virtual theories.
problem Classifying doodles on closed surfaces, especially non-orientable ones.
method Introducing twisted virtual doodles, defining twin groups, and proving Alexander- and Markov-type theorems.
result Unified theory of doodles, showing trivial center and residually finite properties.
This paper explores Brunnian twin groups and their properties.
problem Alexander-Markov correspondence for isotopy classes of immersed circles.
method Diagrammatic representations and combinatorial structures.
result Brunnian twin groups are free groups when more than two strands are involved.
The paper studies congruence subgroups and crystallographic quotients of small Coxeter groups.
problem The congruence subgroup property for small Coxeter groups.
method Analyzes the properties of small Coxeter groups, proving the failure of the congruence subgroup property for certain groups.
result Proves the failure of the congruence subgroup property for infinite small Coxeter groups which are not virtually abelian.
Virtual twin groups map to symmetric groups, revealing automorphism structure.
problem Understanding homomorphisms between virtual twin groups and symmetric groups.
method Using irreducible right-angled Coxeter groups and right-angled Artin groups.
result A complete description of homomorphisms between virtual twin groups and symmetric groups, including the structure of the automorphism group of VTn. Study of commutator subgroups and crystallographic quotients of virtual groups.
problem Investigate commutator subgroups and crystallographic quotients of virtual groups.
method Derived explicit finite presentations and proved crystallographic properties.
result Explicit finite presentations of commutator subgroups and crystallographic quotients.
Base of fibered correspondence is arbitrary correspondence. Fibered correspondence is interesting when we consider relationship between different bundles. However composition of fibered correspondences may not always be defined. Reduced fibered correspondence is defined only between fibers over the same point of base. …
Generalizes uniformization to algebraic correspondences.
problem Uniformizing non-homeomorphic genus zero orbifolds.
method Constructs algebraic correspondences to simultaneously uniformize orbifolds.
result Realizes Teichmüller space of a punctured sphere in correspondences.
Study of holomorphic correspondences combining entire maps and Fuchsian groups.
problem Understanding dynamics of entire maps and their interactions with Fuchsian groups.
method Systematic study of (∞:∞) holomorphic correspondences arising from conformal combinations of transcendental entire maps and Fuchsian groups. result The resulting correspondence is the composition of a Möbius involution and the deleted covering correspondence of a meromorphic function with a simple pole.
Introduces non-abelian Hodge correspondence linking algebraic structures to geometry.
problem None explicitly stated, focuses on introduction.
method Pedagogical introduction of concepts linking algebraic structures to geometry.
result Explains the non-abelian Hodge correspondence and its connections.
New geometric proof for rational tangles links-quivers correspondence.
problem Recovering symmetric/antisymmetric colored HOMFLY-PT polynomials from a quiver.
method Geometric approach using winding numbers in punctured plane and its second configuration space.
result Explicit description of quivers for rational tangles.
The paper defines a category of Lagrangian correspondences in super Hilbert spaces and constructs a functorial field theory.
problem Understanding composition of Lagrangian correspondences in Hilbert spaces.
method Study of Lagrangian correspondences, construction of a category, and functorial field theory.
result Well-defined composition law in a category of Lagrangian correspondences.
With respect to the Dolbeault complex over the flat manifold $\C^n$, an explicit description of the inverse correspondence of the twistor correspondence is given.
Constructs correspondences on hyperelliptic surfaces combining orbifold groups and Blaschke products.
problem Combining geometric and dynamical properties on hyperelliptic surfaces.
method Analytic combinations on Riemann sphere, algebraic characterization, and Teichmüller spaces.
result Explicit description of correspondences and injection into Hurwitz spaces.
The calculus correspondence has been known to exist between generic pedal evolutions and generic wave front evolutions. In this paper, we first extend the known results on the calculus correspondence to evolutions with multi-parameters, and then give applications of calculus correspondence. Moreover, we discuss the pos…
Study Kobayashi-Hitchin correspondence for special sheaves on Kähler manifolds.
problem Understanding the Kobayashi-Hitchin correspondence for specific sheaves.
method Using Hermitian-Yang-Mills flow on Kähler manifolds with simple normal crossing divisors.
result Established the correspondence for saturated reflexive parabolic sheaves.
Constructs Lagrangian correspondences for Higgs bundles and holomorphic connections.
problem Realizing geometric Langlands correspondences for Higgs bundles and connections.
method Using transversal Higgs bundles and holomorphic connections, induced divisors and parameters.
result Evidence suggests generic realization of Dolbeault geometric Langlands correspondence.
The paper extends Riemann-Hilbert correspondence to foliations.
problem Understanding representations of Lie algebroids and groupoids in foliated settings.
method Establishing an A∞ de Rham theorem and constructing an integration functor. result An equivalence between ∞-representations of L∞-algebroids and ∞-representations of Lie ∞-groupoids for foliations. ROBOT framework solves regression without correspondence for large data and complex models.
problem Regression without known correspondence in large datasets.
method ROBOT framework reformulates regression as a continuous optimization problem and uses hypergradient approach.
result ROBOT achieves better performance than existing methods in linear and nonlinear regression tasks.
The main result of this paper is a discrete Lawson correspondence between discrete CMC surfaces in R^3 and discrete minimal surfaces in S^3. This is a correspondence between two discrete isothermic surfaces. We show that this correspondence is an isometry in the following sense: it preserves the metric coefficients int…
In this paper, we define the corresponding submanifolds to left-invariant Riemannian metrics on Lie groups, and study the following question: does a distinguished left-invariant Riemannian metric on a Lie group correspond to a distinguished submanifold? As a result, we prove that the solvsolitons on three-dimensional s…
Overview of dynamics in algebraic correspondences and their connections.
problem Understanding dynamics in algebraic correspondences and their connections.
method Focus on matings between rational maps and Kleinian groups, highlighting unifying structures.
result Rich dynamics and connections between moduli spaces of rational maps and Kleinian groups.
Taxicab correspondence analysis visualizes sparse text data sets.
problem Visualization of extremely sparse contingency tables.
method Robust variant of correspondence analysis for sparse data.
result Visualized an 8265-dimensional textual data set.
Proves Giroux Correspondence in 3D using Heegaard splittings.
problem Proving Giroux Correspondence in 3D contact manifolds.
method Uses Heegaard splittings of contact manifolds, extending convex surface theory.
result Accessible proof for low-dimensional audience.
We demonstrate an isomorphism between the homology of the strand algebra of bordered Floer homology, and the category algebra of the contact category introduced by Honda. This isomorphism provides a direct correspondence between various notions of Floer homology and arc diagrams, on the one hand, and contact geometry a…
In this paper, we want to construct a one-to-one correspondence from the set of diffeomorphism classes of spin d-twisted homology $\mc P^3$ to the set of isotopy classes of the embedding from S3 to S6, which is a generalization of the Montgomery-Yang correspondence. Furthermore, we will apply this generalized c…
We show Mckay correspondence of Betti numbers of Chen-Ruan coho- mology for omnioriented quasitoric orbifolds. In previous articles with M. Poddar [8], [9], we proved the correspondence for four dimension and six dimensions. Here we deal with the general case.
Paper extends Kobayashi-Hitchin correspondence to non-Kähler manifolds.
problem Applying Kobayashi-Hitchin correspondence to non-Kähler manifolds.
method Continuity method for vortex equation, Kobayashi-Hitchin correspondence for holomorphic pairs.
result Proved solvability of vortex equation on holomorphic vector bundles over compact Hermitian manifolds.
New correspondence links fluxless to fluxy flag manifolds via T-duality.
problem Understanding fluxes on flag manifolds.
method Defining a new correspondence and using infinitesimal T-duality.
result Infinitesimal T-duality generates nontrivial fluxes.
We generalize Lagrangian Floer cohomology to sequences of Lagrangian correspondences. For sequences related by the geometric composition of Lagrangian correspondences we establish an isomorphism of the Floer cohomologies. We give applications to calculations of Floer cohomology, displaceability of Lagrangian correspond…
A neural architecture learns and refines graph correspondences.
problem Learning structural correspondences between graphs.
method Two-stage neural architecture: node embeddings followed by message passing networks.
result Message passing scheme computes well-founded consensus for corresponding neighborhoods.
For a given lattice, we establish an equivalence involving a closed zone of the corresponding Voronoi polytope, a lamina hyperplane of the corresponding Delaunay partition and a quadratic form of rank 1 being an extreme ray of the corresponding L-type domain.
We construct examples of singular self-dual Zollfrei metrics explicitly, by patching a pair of Petean's self-dual split-signature metrics. We prove that there is a natural one-to-one correspondence between these singular metrics and a certain set of embeddings of RP3 to CP3 which has one singular point. This embe…
An integral hyperbolic lattice is called reflective if its automorphism group is generated by reflections, up to finite index. Since 1981, it is known that their number is essentially finite. We show that K3 surfaces over C with reflective Picard lattices can be characterized in terms of compositions of their self-corr…
The paper explains geometric correspondences for homothetic navigation.
problem Understanding geodesic and Jacobi field correspondences in homothetic navigation.
method Providing conceptual explanations and shortcuts to formulas.
result Directly seeing local correspondence between isoparametric functions or hypersurfaces.
Correspondence found between exponential families and affine Grassmannians.
problem Understanding the relationship between exponential families and geometric structures.
method Established a one-to-one correspondence between exponential families and affine Grassmannians.
result Found a correspondence between minimal exponential families and affine Grassmannians.
Holonomies match for higher local systems and principal 2-bundles.
problem Matching holonomies for higher local systems and principal 2-bundles.
method Higher Riemann-Hilbert correspondence and principal 2-bundles.
result Holonomies coincide for both formalisms.
Study proves correspondence for special bundles on complex surfaces.
problem Proving correspondence for parabolic bundles on complex surfaces.
method Kobayashi-Hitchin correspondence for parabolic bundles over compact non-Kähler surfaces.
result Proved Kobayashi-Hitchin correspondence for specified bundles.
The paper establishes a correspondence between Higgs torsors and connections on curves.
problem Establishing a correspondence between Higgs torsors and connections on curves.
method Introduced a stability condition on filtered Stokes local systems and used it to prove a one-to-one correspondence.
result One-to-one correspondence between stable meromorphic parahoric Higgs torsors and stable meromorphic parahoric connections.
Establishes a correspondence between two mathematical identities.
problem None explicitly stated; focuses on identity correspondence.
method Establishes correspondence between Pestov and Weitzenböck identities.
result Established correspondence between Pestov and Weitzenböck identities.
We generalise the hyper-Kahler/quaternionic Kahler (HK/QK) correspondence to include para-geometries, and present a new concise proof that the target manifold of the HK/QK correspondence is quaternionic Kahler. As an application, we construct one-parameter deformations of the temporal and Euclidean supergravity c-map m…
Recently we introduced T-duality in the study of topological insulators. In this paper, we study the bulk-boundary correspondence for three phenomena in condensed matter physics, namely, the quantum Hall effect, the Chern insulator, and time reversal invariant topological insulators. In all of these cases, we show that…
Establishes Kobayashi-Hitchin correspondence for nef and big classes.
problem Analyzing stability and positivity in algebraic geometry.
method Introducing adapted currents and metrics to establish correspondence.
result Equality cases of Bogomolov-Gieseker and Miyaoka-Yau inequalities.
Using quilted Floer cohomology and relative quilt invariants, we define a composition functor for categories of Lagrangian correspondences in monotone and exact symplectic Floer theory. We show that this functor agrees with geometric composition in the case that the composition is smooth and embedded. As a consequence …
The paper concerns a simple model of bicycle kinematics: a bicycle is represented by an oriented segment of constant length in n-dimensional space that can move in such a way that the velocity of its rear end is aligned with the segment (the rear wheel is fixed on the bicycle frame). Starting with a closed trajectory o…
Hashing has been widely adopted for large-scale data retrieval in many domains, due to its low storage cost and high retrieval speed. Existing cross-modal hashing methods optimistically assume that the correspondence between training samples across modalities are readily available. This assumption is unrealistic in pra…
Established a correspondence for toric fibrations using Delzant polytopes.
problem Existence of extremal Kähler metrics on toric fibrations.
method Using weighted constant scalar curvature Kähler metrics and uniform K-stability.
result Equivalence between extremal metrics and weighted uniform K-stability of Delzant polytopes.
Constructs non-unital monoidal category of contact manifolds and Legendrian correspondence calculus
problem Constructing a non-unital monoidal category of contact manifolds without contact forms
method Developing contact topology without contact forms and defining the star product
result Proving the associativity of the star product and the pentagon axiom
Study presents a twistor correspondence for specific geometric structures.
problem Twistor theory for almost-Grassmannian manifolds.
method Utilizes moduli of curves-with-boundary for global correspondence.
result Foundational results in complex setting, global correspondence for real Grassmannian.