New algebraic structure derived from geometric objects.
problem Understanding algebraic structures from geometric objects.
method Proving existence of line bundles and factorization algebras.
result Factorization algebras derived from Beilinson-Drinfeld Grassmannians.
Study of twisted Kapustin-Witten equations on Riemann surfaces.
problem Dimensional reduction of twisted Kapustin-Witten equations.
method Kobayashi-Hitchin type correspondence between Nahm pole solutions and opers.
result Corroborates prediction of Gaiotto and Witten.
Classifies linear embeddings of grassmannians and ind-grassmannians.
problem Understanding linear embeddings of grassmannians and ind-grassmannians.
method Classification through isomorphism of Picard groups and direct limits.
result Most linear embeddings of grassmannians are equivariant.
The purpose of this note is to present a short elementary proof of a theorem due to Faltings and Laumon, saying that the global nilpotent cone is a Lagrangian substack in the cotangent bundle of the moduli space of G-bundles on a complex compact curve. This result plays a crucial role in the Geometric Langlands program…
Study proves existence of precotangent bundles for Grassmannians.
problem Existence of precotangent bundles for Grassmannians.
method Proof for Grassmannians of reflexive Banach spaces and p-restricted Grassmannians of polarized Hilbert space. result Existence of bundle predual to tangent bundle (precotangent bundle).
The paper finds inequalities in Grassmannian geometry.
problem Understanding geometric properties of Grassmannians.
method Analyzes inequalities for elements in Grassmannians.
result Law of Cosines and geodesic triangle inequalities.
Curious structure of special orthogonal, unitary, and symplectic groups as products of Grassmannians discovered.
problem Understanding the structure of special orthogonal, unitary, and symplectic groups.
method Expressing these groups as products of Grassmannians realized as involution matrices.
result Special orthogonal, special unitary, and symplectic groups can be expressed as products of their corresponding Grassmannians.
The affine Grassmannian is realized as a matrix manifold for optimization.
problem Optimization on noncompact manifolds like the affine Grassmannian.
method Riemannian optimization algorithms extended to the affine Grassmannian.
result Standard numerical linear algebra suffices for optimization on the affine Grassmannian.
The paper explores the geometry of Lagrangian Grassmannians and their connection to PDEs.
problem Understanding the geometric properties of Lagrangian subspaces.
method Thorough review of geometric properties and their relation to PDEs.
result Hypersurfaces in the Lagrangian Grassmannian correspond to second-order PDEs.
Constructs explicit p-harmonic functions on Grassmannians and flag manifolds.
problem Finding proper p-harmonic functions on Grassmannians and flag manifolds. method Using the method of eigenfamilies to construct explicit functions.
result Explicit complex-valued proper p-harmonic functions on compact real Grassmannians and non-descending functions on real flag manifolds. Using the Plucker map between grassmannians, we study basic aspects of classic grassmannian geometries. For `hyperbolic' grassmannian geometries, we prove some facts (for instance, that the Plucker map is a minimal isometric embedding) that were previously known in the `elliptic' case.
The paper finds optimal inequalities for hypersurface curvatures in complex Grassmannians.
problem Optimizing inequalities for hypersurface curvatures in complex Grassmannians.
method Analyzing real hypersurfaces in complex Grassmannians to derive inequalities involving scalar curvature and Casorati curvatures.
result Conditions for equality in inequalities involving normalized scalar curvature and Casorati curvatures.
Grassmannian sigma models extend Gross-Neveu model formulations.
problem Understanding sigma models on Grassmannian targets.
method Chiral Gross-Neveu model formulations for orthogonal and symplectic Grassmannians.
result One-loop β-functions proportional to dual Coxeter numbers. No real hypersurfaces found in complex Grassmannians with specific Jacobi operators.
problem Existence of real hypersurfaces in complex Grassmannians with semi-parallel structure Jacobi operator.
method Proved nonexistence through mathematical analysis.
result No real hypersurfaces exist in complex Grassmannians of rank two with semi-parallel structure Jacobi operator.
Grassmannian packings improve CNN kernels' diversity and reduce sparsity.
problem Kernel sparsity and lack of diversity in CNNs decrease model capacity.
method Initialize CNN kernels with Grassmannian packings to maximize diversity and minimize sparsity.
result Grassmannian packings lead to diverse features and improved classification accuracy.
Study harmonic maps into Hilbert-Schmidt Grassmannian.
problem Harmonic spheres conjecture.
method Twistor description of harmonic maps.
result Motivated by harmonic spheres conjecture.
Study on Hopf hypersurfaces in complex Grassmannians, proving constant Reeb curvature.
problem Proving properties of Hopf hypersurfaces in complex Grassmannians.
method Analyzing real hypersurfaces, proving nonexistence of certain foliations, and classifying hypersurfaces.
result Constant Reeb curvature for Hopf hypersurfaces in complex Grassmannians of rank two.
The affine Grassmannian generalizes Euclidean and linear subspaces with rich geometric properties.
problem Formulating machine learning and statistical problems on the affine Grassmannian.
method Showed the affine Grassmannian has multiple structures and affords an analogue of Schubert calculus.
result The affine Grassmannian serves as a concrete computational platform for various machine learning and statistical problems.
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.
The paper classifies real hypersurfaces with a specific Jacobi operator in complex Grassmannians.
problem Classifying real hypersurfaces with a particular Jacobi operator.
method Introducing and classifying real hypersurfaces with a quadratic Killing structure Jacobi operator.
result A classification theorem for Hopf real hypersurfaces with quadratic Killing structure Jacobi operator.
Totally nonnegative Grassmannian and related spaces are shown to be like closed balls.
problem Understanding the topological structure of certain spaces in combinatorics.
method Proving homeomorphic to closed balls using advanced combinatorial and geometric techniques.
result Three significant spaces in combinatorics are proven to be topologically equivalent to closed balls.
Researchers describe even Clifford structures on specific Grassmannians.
problem Understanding even Clifford structures on Grassmannians.
method Explicit description of structures on real, complex, and quaternionic Grassmannians.
result Explicit description of non-flat parallel even Clifford structures of ranks 8, 6, and 5.
Introduces and studies generalized B-opers with bilinear forms.
problem Understanding higher rank opers with bilinear forms.
method Studies the structure of generalized B-opers using jet bundles and geometric structures on Riemann surfaces.
result Structure of generalized B-opers is studied and related to jet bundles and geometric structures.
Differential structure on partial isometries over Grassmannian constructed.
problem No specific problem stated; abstract focuses on method and result.
method Construction of differential structure on partial isometries over restricted Grassmannian.
result Set of partial isometries over restricted Grassmannian becomes a Banach Lie groupoid.
Constructs a Morse-Bott function on symplectic Grassmannians.
problem Defines a function on symplectic Grassmannians.
method Uses a compatible linear complex structure to construct a quadratic Morse-Bott function.
result Critical loci consist of subspaces splitting into isotropic and complex parts.
Generalizes embedding complex Grassmannians into quadrics.
problem Holomorphic isometric embeddings of complex Grassmannians into quadrics.
method Generalization of do Carmo-Wallach theory for moduli spaces.
result Moduli spaces of embeddings discussed.
Smooth structures on infinite dimensional Grassmannians and non-commutative cross-ratios.
problem Smooth structures on infinite dimensional Grassmannians and non-commutative cross-ratios.
method Analyzing and expanding the notion of non-commutative cross-ratios, proving their smoothness.
result Smoothness of non-commutative cross-ratios.
An elliptic complex is found on a Grassmannian space, with zero index.
problem Analyzing differential operators on a specific geometric space.
method Identifying an elliptic complex of length 3 starting with a 2-Dirac operator on the Grassmannian of oriented 2-planes.
result The index of the elliptic complex is zero.
Researchers compute Hochschild cohomology of Grassmannians.
problem Computing Hochschild cohomology of Grassmannians.
method Explicit description of Gerstenhaber algebra structure, vanishing of higher cohomology.
result Decomposition of Hochschild cohomology concentrated in global sections for certain Grassmannians.
New supergeometric Grassmannians created by gluing ν-domains.
problem No specific problem stated; focuses on a new construction.
method Constructing supergeometric Grassmannians by gluing ν-domains.
result Introduces a novel generalization of Grassmannians to supergeometry.
The classical concept of affine locally symmetric spaces allows a generalization for various geometric structures on a smooth manifold. We remind the notion of symmetry for parabolic geometries and we summarize the known facts for ∣1∣--graded parabolic geometries and for almost Grassmannian structures, in particular.…
This paper proves area-minimizing cones over Grassmannian manifolds.
problem Determine if cones over Grassmannian manifolds are area-minimizing.
method Detailed descriptions of embedding maps using Hermitian orthogonal projectors, re-proving area-minimization using Lawlor's Curvature Criterion.
result All cones over Grassmannian manifolds are area-minimizing except for oriented real Grassmannians.
Develops a correspondence between symplectic orbits and Grassmannians.
problem Understanding the homotopy types of Grassmannians of linear subspaces in symplectic vector spaces.
method Uses orbit fibrations and linear symplectic reduction to compute homotopy types.
result Recover observations from Arnold, Oh-Park, and Lee-Leung in different cases.
We prove that there does not exist any semi-parallel real hypersurface in complex two-plane Grassmannians. With this result, the nonexistence of recurrent real hypersurfaces in complex two-plane Grassmannians can also be proved.
This paper generalizes Grassmannians to hyperfields.
problem Exploring Grassmannians in the context of hyperfields.
method Introducing topological hyperfields and relating R and C to matroid hyperfields.
result Relating hyperfields R and C to matroid hyperfields.
New supergroups created from odd involutions in supergeometry.
problem Creating new supergroups from odd involutions.
method Constructing ν−Grassmannians by gluing ν−domains with an odd involution. result Introduced a supergroup associated with odd involutions.
Complex Cayley Grassmannian is a singular variety with smooth singular locus.
problem Characterize the singularities of the Cayley Grassmannian.
method Defined a torus action and used it to prove singularity and cohomology properties.
result Singular locus is smooth and has the same cohomology as CP5. A nonstandard invariant fourth order operator acting on functions on a manifold equipped with an almost Grassmannian structure with an arbitrary trorsion is found by means of the curved translation principle. This operator can be viewed as a Grassmannian analogue of the Paneitz operator well known from conformal geomet…
Reformulated sigma models for complex Grassmannians using Gross-Neveu formalism.
problem Classical aspects of N=(2,2) supersymmetric sigma models with Hermitian symmetric target spaces. method Reformulation using Gross-Neveu formalism, proposing two types of equivalent Lagrangians.
result Proposed two types of equivalent Lagrangians for maximal isotropic Grassmannians, making either supersymmetry or geometry manifest.
Study cohomology rings of Grassmannians using Clifford algebras and symmetric spaces.
problem Understanding cohomology rings of Grassmannians over different fields.
method Explicit generators and relations for de Rham cohomology rings, filtered deformations related to Clifford algebras.
result Explicit generators and relations for the de Rham cohomology rings of Grassmannians.
Characterizes symplectic and odd-symplectic Grassmannians using VMRT.
problem Characterizing Fano manifolds of Picard number 1.
method Using VMRT and local differential geometric structure.
result Symplectic and odd-symplectic Grassmannians are characterized by their VMRT.
Constructs explicit harmonic functions and morphisms on complex and quaternionic Grassmannians.
problem Creating explicit solutions for p-harmonic functions and harmonic morphisms. method Using joint eigenfunctions of the Laplace-Beltrami and conformality operators.
result Induces solutions on dual non-compact Riemannian symmetric spaces.
The parahoric Hitchin system is completely integrable.
problem Integrability of the parahoric Hitchin system on curves of genus > 1.
method Proof of Hitchin map properties and integrability using isotropy and Poisson structure.
result The Hitchin map is a completely integrable system.
Unified framework for smooth structures on coadjoint orbits.
problem Smooth structures on coadjoint orbits of diffeomorphism groups.
method Decorated and augmented nonlinear Grassmannians, functors, smooth structure.
result Uniform description of coadjoint orbits' smooth structures.
Study on complex Grassmannians' rigidity using Einstein deformations.
problem Characterizing integrable infinitesimal Einstein deformations of complex Grassmannians.
method Analyzing the integrability to second order of infinitesimal deformations using Koiso's obstruction polynomial.
result Characterized integrable deformations as an explicit variety in su(n), showing g is isolated for odd n. Almost para-quaternionic structures on smooth manifolds of dimension 2n are equivalent to almost Grassmannian structures of type (2,n). We remind the equivalence and exhibit some interrelations between subjects that were previously studied independently from the para-quaternionic and the Grassmannian point of view.…
Study classifies hypersurfaces in complex hyperbolic spaces with specific operator conditions.
problem Classifying real hypersurfaces with commuting structure Jacobi operators.
method Used simultaneous diagonalization for commuting symmetric operators to classify hypersurfaces.
result Complete classification of real hypersurfaces with commuting condition.
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.