The affine Grassmannian is a noncompact smooth manifold that parameterizes all affine subspaces of a fixed dimension. It is a natural generalization of Euclidean space, points being zero-dimensional affine subspaces. We will realize the affine Grassmannian as a matrix manifold and extend Riemannian optimization algorit…
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48 results for “affine Grassmannian”
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 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 --graded parabolic geometries and for almost Grassmannian structures, in particular.…
New shape representation for airfoils improves design and manufacturing.
problem Designing and manufacturing airfoils efficiently and accurately.
method Combining physics-based and data-driven techniques on a Grassmannian manifold.
result Rich set of novel 2D airfoil deformations not previously captured.
The paper studies a special Grassmannian space and shows it's an orbit of a unitary group.
problem Investigating a specific Grassmannian space of infinite-dimensional subspaces.
method Analyzing the restricted -Schatten class Grassmannian and showing it's an affine coadjoint orbit of a unitary group.
result The restricted -Schatten class Grassmannian is shown to be an affine coadjoint orbit of an infinite-dimensional restricted unitary group.
We give a representation of canonical vector bundles over Grassmannian manifolds as non-compact affine symmetric spaces as well as their Cartan model in the group of the Euclidean motions.
Researchers create a partial resolution of Coulomb branches for gauge theories.
problem Understanding partial resolutions of Coulomb branches in gauge theories.
method Constructing partial resolutions as variants of generalized slices in geometric contexts.
result Identified partial resolutions with specific geometric objects.
New connections found between knot invariants and Rozansky-Witten theory.
problem Understanding physical interpretations of knot invariants.
method Studying Rozansky-Witten theory with non-compact target spaces.
result New formulations of knot invariants using affine Grassmannians and q-series.
Constructs BGG resolutions for symplectic case.
problem Exactness of BGG resolutions in singular infinitesimal characters.
method Penrose transform over Lagrangian Grassmannian.
result Exactness of constructed complex over big affine cell.
Explains quantum cohomology of Grassmannians using tt* equations.
problem Relates quantum cohomology of complex Grassmannians to projective space.
method Uses tt* equations and Lie-theoretic connections.
result Illustrates relations between tt* equations and quantum cohomology.
This is the second companion paper of arXiv:1601.03586. We consider the morphism from the variety of triples introduced in arXiv:1601.03586 to the affine Grassmannian. The direct image of the dualizing complex is a ring object in the equivariant derived category on the affine Grassmannian (equivariant derived Satake ca…
This is a companion paper of arXiv:1601.03586. We study Coulomb branches of unframed and framed quiver gauge theories of type . In the unframed case they are isomorphic to the moduli space of based rational maps from to the flag variety. In the framed case they are slices in the affine Grassmannia…
The paper introduces exponential-wrapped distributions on symmetric spaces for better data modeling.
problem Challenges in statistical modeling due to curvature of data spaces.
method Construction and use of exponential-wrapped distributions on affine locally symmetric spaces.
result Exponential-wrapped distributions on symmetric spaces have useful properties for practical use.
New spaces at infinity identified for Minkowski spacetime.
problem Characterizing asymptotic infinities of Minkowski spacetime.
method Embedding and describing homogeneous spaces of the Poincaré group.
result Determined new structures on asymptotic infinities.
Let Σbe a complete minimal Lagrangian submanifold of \C^n. We identify regions in the Grassmannian of Lagrangian subspaces so that whenever the image of the Gauss map of Σlies in one of these regions, then Σis an affine space.
Let G be a simple algebraic group. Labelled trivalent graphs called webs can be used to product invariants in tensor products of minuscule representations. For each web, we construct a configuration space of points in the affine Grassmannian. Via the geometric Satake correspondence, we relate these configuration spaces…
We construct a model space $C(\gsp(\bR^{2n}))$ for the variety of Abelian simply transitive groups of affine transformations of type ${\rm Sp}(\bR^{2n})$. The model is stratified and its principal stratum is a Zariski-open subbundle of a natural vector bundle over the Grassmannian of Lagrangian subspaces in $\bR^{2n}$.…
Develops a method to construct entire minimal graphs of odd dimensions.
problem Constructing entire minimal graphs of odd dimensions and arbitrary codimensions.
method Evolving-plane ansatz reducing minimal surface system to geodesic equation on Grassmannian.
result Yields a rich family of explicit entire minimal graphs of odd dimension and arbitrary codimension.
We develop an integral geometry of stationary Euler equations defining some function on the Grassmannian of affine lines in the space. This function depends on a putative compactly supported solution of the system, and we deduce a linear differential equation for . We prove also that the purported annulation…
Grassmannian diffusion maps based dimension reduction and classification for high-dimensional datacs.LG
GDMaps reduces high-dimensional data to lower dimensions for better classification.
problem High-dimensional data classification and representation.
method Grassmannian Diffusion Maps technique for nonlinear dimensionality reduction.
result GDMaps effectively identifies intrinsic subspace structures in high-dimensional data.
Positive configurations of points in the affine building were introduced in \cite{Le} as the basic object needed to define higher laminations. We start by giving a self-contained, elementary definition of positive configurations of points in the affine building and their basic properties. Then we study the geometry of …
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.
-associahedramath.CO
Polytopes for posets compactify spaces of order-preserving maps.
problem Compactifying spaces of order-preserving maps for posets.
method Constructing polytopes and compactifying spaces of order-preserving maps.
result Polytopes correspond to nested collections of subsets of posets.
Study proves existence of precotangent bundles for Grassmannians.
problem Existence of precotangent bundles for Grassmannians.
method Proof for Grassmannians of reflexive Banach spaces and -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 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 -harmonic functions on Grassmannians and flag manifolds.
problem Finding proper -harmonic functions on Grassmannians and flag manifolds.
method Using the method of eigenfamilies to construct explicit functions.
result Explicit complex-valued proper -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.
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 -functions proportional to dual Coxeter numbers.
Real hypersurfaces in complex Grassmannians of rank two with semi-parallel structure Jacobi operatormath.DG
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.
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.
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.
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.
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.
Hyperfield Grassmanniansmath.CO
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.
There is a natural filtration on the space of degree- homogeneous polynomials in independent variables with coefficients in the algebra of smooth functions on the Grassmannian , determined by the tautological bundle. In this paper we show that the space of -dimensional integral elements of a…
New supergroups created from odd involutions in supergeometry.
problem Creating new supergroups from odd involutions.
method Constructing Grassmannians by gluing domains with an odd involution.
result Introduced a supergroup associated with odd involutions.