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. 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.
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…
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.
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 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.
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.
The aim of this paper is a characterization of great antipodal sets of complex Grassmannian manifolds as certain designs with the smallest cardinalities.
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.
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.
We prove an extension of a theorem of A.Ros on a characterization of seven compact Kaehler submanifolds by holomorphic pinching to certain submanifolds of the complex Grassmannian manifolds.
Geometric approach to PDEs using contact manifolds and Lagrangian Grassmannians.
problem Scalar PDEs in n variables of order one and two.
method Underlying (2n+1)-dimensional contact manifold and Lagrangian Grassmannian bundle.
result Introduction of geometric methods to PDEs, including scalar PDEs of order one and two.
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.…
Using the convex functions in Grassmannian manifolds we can carry out interior estimates for mean curvature flow of higher codimension. In this way some of the results of Ecker-Huisken can be generalized to higher codimension
Toric contact manifolds defined in any dimension.
problem No specific problem stated; abstract focuses on definition.
method Description via labelled polytope in grassmannian.
result Toric contact manifolds in arbitrary codimension.
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…
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.…
In this paper we study the action of the symplectic operators which are a perturbation of the identity by a Hilbert-Schmidt operator in the Lagrangian Grassmannian manifold.
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.
This paper gives an example of special Lagrangian manifold obtained from a hypersurface of a complex Grassmannian with vanishing first Chern class. The obtained manifold is a 1-torus bundle over the two dimensional real projective space. Such manifolds are interesting for mirror symmetry theory. Other examples of the s…
Introduces Grassmannian learning to incorporate geometric features in machine learning.
problem Subspace-structured features, orthogonality constraints, and low-rank constraints in machine learning.
method Studies the Grassmann manifold to solve mathematical problems in shallow and deep learning.
result Improvements in performance of classic and deep learning algorithms using Grassmannian learning.
The main result of the paper is the complete classification of the compact connected Lie groups acting coisotropically on complex Grassmannians. This is used to determine the polar actions on the same manifolds.
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.
We derive estimates of the Hessian of two smooth functions defined on Grassmannian manifold. Based on it, we can derive curvature estimates for minimal submanifolds in Euclidean space via Gauss map. In this way, the result for Bernstein type theorem done by Jost and the first author could be improved.
A harmonic map from a Riemannian manifold into a Grassmannian manifold is characterized by a vector bundle, a space of sections of this bundle and a Laplace operator. We apply our main theorem, itself a generalization of a Theorem of Takahashi, to generalize the theory of do Carmo and Wallach and to describe the moduli…
The paper studies the distribution of random degeneracy sets on complex manifolds.
problem Distribution of random degeneracy sets on compact Kähler manifolds.
method Asymptotic expansion of induced Grassmannian Chern forms, meromorphic transforms, and Wishart distribution.
result Normalized currents converge to curvature forms with quantitative estimates.
We present several principal bundles of embeddings of compact manifolds (with or without boundary) whose base manifolds are nonlinear Grassmannians. We study their infinite dimensional differential manifold structure in the Fréchet category. This study is motivated by the occurrence of such objects in the geometric Lag…
Minimal equivariant embedding found for flag manifolds.
problem Finding the smallest possible dimension for equivariant embeddings of flag manifolds.
method Proved the smallest possible dimension (n−1)(n+2)/2 for SOn(R)-equivariant embeddings of Flag(k1,…,kp,Rn). result The smallest possible dimension (n−1)(n+2)/2 is the optimal for SOn(R)-equivariant embeddings of Flag(k1,…,kp,Rn). 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 paper studies connectivity of Schur-Horn map images in real Grassmannians.
problem Connectivity of Schur-Horn map images in real Grassmannians.
method Criterion for pre-images of vectors in \(\mathbb{R}^n\) to be connected.
result Criterion for pre-images of vectors in \(\mathbb{R}^n\) to be connected.
Modeling videos and image-sets as linear subspaces has proven beneficial for many visual recognition tasks. However, it also incurs challenges arising from the fact that linear subspaces do not obey Euclidean geometry, but lie on a special type of Riemannian manifolds known as Grassmannian. To leverage the techniques d…
This paper proves area-minimizing cones over products of Grassmannian manifolds.
problem Proving area-minimizing cones over products of Grassmannian manifolds.
method Using Hermitian orthogonal projectors and carefully computing the Jacobian.
result Cone over minimal products of Grassmannian manifolds are area-minimizing.
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.
The paper classifies Fano distributions on specific Fano manifolds.
problem Investigating Fano distributions on Fano manifolds.
method Classification of Fano distributions on various Fano manifolds.
result Classification of codimension one del Pezzo distributions on Fano manifolds with Picard number one.
Classifies totally geodesic submanifolds and polar actions on Stiefel manifolds.
problem Classifying totally geodesic submanifolds and polar actions on Stiefel manifolds.
method Classification through polar actions and cohomogeneity-one actions.
result Classification of orbits of polar actions on Stiefel manifolds.
Curves in Lagrange Grassmannians appear naturally in the intrinsic study of geometric structures on manifolds. By a smooth geometric structure on a manifold we mean any submanifold of its tangent bundle, transversal to the fibers. One can consider the time-optimal problem naturally associate with a geometric structure.…
The book is devoted to study so-called irregular subsets of the Grassmannian manifold Gkn(V) (this class of sets was introduced by author). In the previous variant of the book we restrict ourself only to the case when V is an n-dimensional vector space under the field R. Now we consider irregular subsets …
New Morse-Bott function defined on Stiefel manifolds, revealing complex critical structures.
problem Defining Morse-Bott functions on non-linear Stiefel manifolds.
method Replacing linear height function with a quadratic one, proving it as a Morse-Bott function.
result Critical submanifolds are fibrations of products of Grassmannians, not Grassmannians themselves.
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.
We calculate Euclidean distance degrees for common manifold optimization types.
problem Optimizing on manifold structures.
method Closed-form expressions for stationary points of Euclidean distance function.
result Closed-form expressions for all stationary points on manifold optimization.
This paper clusters networks with annotated time-series data using kernel-ARMA and Grassmannian geometry.
problem Clustering networks with annotated time-series data, including state, node, and subnetwork clustering.
method Extract features from time-series data using kernel-ARMA, map onto Grassmannian, and cluster using Riemannian geometry.
result The proposed framework outperforms state-of-the-art clustering schemes on brain-network data.
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.
Gauss map of skew mean curvature flow satisfies Schrödinger flow.
problem Understanding the geometry of skew mean curvature flow.
method Exploring the oriented Grassmannian manifold and embedding it into exterior product space.
result Gauss map of SMCF satisfies a Schrödinger flow equation.
New machine learning approach finds Kähler metrics through Grassmannian learning.
problem Finding numerical Kähler metrics through machine learning.
method Gradient descent on the Grassmannian manifold, combining with Donaldson's algorithm.
result Observation of nontrivial local minima in moduli space.
The paper studies special Lagrangian manifolds using algebraic topology.
problem Understanding special Lagrangian submanifolds in calibrated geometries.
method Algebraic topology of Grassmannian spaces, focusing on cohomology rings.
result Various results on the topology of Grassmannian spaces and special Lagrangian embeddings.
Study second order integrability of Einstein deformations on Riemannian and Kähler manifolds.
problem Integrability condition for second order infinitesimal Einstein deformations.
method New expression for integrability condition, Koiso obstruction simplification.
result Complete description of integrable deformations on complex 2-plane Grassmannian.
In his book Mickelsson notices that the infinite-dimensional Grassmannian manifold of Segal and Wilson admits a Spin^c structure and after this he naturally considers the problem of defining a Dirac operator on it. Mickelsson gives a possible candidate for such an operator but unfortunately it proves out to be badly di…
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).