Simple construction for classifying spaces of projections of immersions.
problem Classifying spaces for projections of immersions with controlled singularities.
method Explicit simple construction for classifying spaces of maps obtained as hyperplane projections of immersions.
result Structure theorems for these classifying spaces.
Two constructions of classifying spaces for singular maps are connected.
problem Understanding the classifying spaces of singular maps.
method Homotopy theoretical connection between two constructions.
result Some classifying spaces have a simple product structure.
The paper analyzes how invariant classifiers reduce generalization error compared to non-invariant ones.
problem Understanding the generalization error of invariant classifiers.
method Factoring input space into a base space and transformations, deriving conditions for smaller base space complexity.
result Invariant classifiers have a generalization error proportional to the base space complexity, not the input space complexity.
Linear classifiers in product space forms improve scRNA-seq data classification.
problem Linear classification in products of Euclidean, spherical, and hyperbolic spaces.
method Novel formulations of linear classifiers on Riemannian manifolds, proving expressive power, and formalizing perceptron and SVM classifiers.
result Linear classifiers in product space forms have the same expressive power as in Euclidean space of the same dimension.
New classifying spaces constructed from Ore categories with Garside families.
problem Constructing classifying spaces for fundamental groups.
method Using Ore categories with Garside families, introducing Zappa–Szép product.
result New categories and groups constructed, including Braided T of type F∞. We study tt*-geometry on the classifying space for regular singular TERP-structures, e.g., Fourier-Laplace transformations of Brieskorn lattices of isolated hypersurface singularities. We show that (a part of) this classifying space can be canonically equipped with a hermitian structure. We derive an estimate for the h…
Study confirms ropes and long knots have similar topological properties.
problem Classifying space of long knots and its equivalence to short ropes.
method Used techniques from Galatius and Randal-Williams to model classifying space and constructed a map.
result Space of short ropes is weakly homotopy equivalent to the classifying space of long knots.
Homology theory for qualgebras yields knotted graph and foam invariants.
problem Developing a homology theory for qualgebras.
method Constructing a classifying space from prisms and adding degenerate cells.
result Homotopy classes of maps from spheres to the classifying space of G. Classifies polar actions on 3D homogeneous spaces.
problem Classifying polar isometric actions on 3D homogeneous spaces.
method Orbit equivalence classification and study of cohomogeneity one actions.
result Classification of extrinsically homogeneous surfaces and orbit foliations.
Classifies minimal submanifolds in complex hyperbolic spaces.
problem Identifying minimal submanifolds in complex hyperbolic spaces.
method Classification based on extrinsic homogeneity.
result Classification of minimal extrinsically homogeneous submanifolds.
Study shows k-NN classifier is not universally consistent on (0,1) but consistent on discrete and specific measure spaces.
problem Consistency of k-NN classifier under Wasserstein distance on measure spaces. method Analysis of k-NN classifier properties under Wasserstein distance, use of σ-finite metric dimension, geodesic structures of Wasserstein spaces. result Consistency of k-NN classifier on specific measure spaces (discrete, Gaussian, wavelet series) but not on (0,1). Classifies 3-braids as L-space knots.
problem Identifying L-space knots among 3-braids.
method Classification based on L-space properties.
result Closed 3-braids classified as L-space knots.
Classified spaces in low dimensions.
problem Irreducible homogeneous almost Hermite-Lorentz spaces in low dimensions.
method Classification through complex dimension 3.
result Classification of spaces in low dimensions.
Smooth classifying spaces for groups defined using diffeological spaces.
problem Classifying smooth principal bundles for a smooth group G. method Developed the theory of smooth principal bundles using diffeological spaces, defining D-numerable bundles and proving classification results. result Smooth structures on Milnor's spaces EG and BG classify all D-numerable principal bundles over any diffeological space. Paper proposes a new classifier for hyperbolic spaces using horospherical boundaries.
problem Optimization of large margin classifiers in hyperbolic spaces.
method Horospherical decision boundaries for geodesically convex optimization.
result Geodesically convex optimization leads to globally optimal solutions.
Classifies special types of 4D spaces.
problem Classifying non-reductive 4D spaces.
method Classification based on conformal Einstein manifolds.
result Classification of non-reductive 4D homogeneous conformally Einstein manifolds.
We define a diffeology on Milnor's classifying space and prove existence of connections.
problem Classifying spaces and connections in diffeology.
method Definition of diffeology on Milnor's classifying space and proof of connection existence.
result Existence of diffeological connections on principal bundles.
The paper analyzes kernel classifiers' performance in Sobolev spaces and proves their optimality.
problem Theoretical analysis of kernel classifiers' performance in Sobolev spaces.
method Deriving upper and lower bounds on classification excess risk using kernel regression theory and estimating interpolation smoothness.
result The proposed kernel classifier is optimal in Sobolev spaces, with theoretical bounds confirmed by real data.
Classifies foliations on specific symmetric spaces.
problem Classifying foliations on symmetric spaces of rank one.
method Orbit equivalence classification.
result Polar homogeneous foliations classified.
Study of vector bundles over classifying spaces for infinite discrete groups.
problem Understanding vector bundles over classifying spaces for infinite discrete groups.
method Homotopy theoretical framework for infinite discrete groups, relating to Novikov conjecture.
result Established a connection between representation spaces and vector bundles over classifying spaces.
Classifies compact spaces by shape, finite spaces by weak homotopy.
problem Classifying compact Hausdorff spaces and finite topological spaces.
method Constructs a category that classifies spaces by shape and weak homotopy.
result Classifies compact spaces by shape, finite spaces by weak homotopy.
Given a group action on a simplicial complex such that each simplex stabiliser admits a cocompact model of classifying space for proper actions, we give conditions implying the existence of a cocompact model of classifying space for proper actions for the whole group. This is used to generalise previous combination res…
Novel method decomposes configuration space for improved collision checking.
problem Improving collision checking in high-degree-of-freedom robot motion planning.
method Proposes a configuration space decomposition method to build a composite classifier.
result Composite classifier outperforms state-of-the-art single classifier methods.
Classifies g.o. spaces and studies geodesics in generalized Wallach spaces.
problem Classifying and studying geodesics in generalized Wallach spaces.
method Investigates homogeneous geodesics in generalized Wallach spaces for any invariant Riemannian metric.
result Provides classifications and examples of g.o. spaces and geodesics.
This paper classifies Blaschke para-umbilical hypersurfaces in conformal space.
problem Classifying Blaschke para-umbilical hypersurfaces in conformal space.
method Classification through mathematical analysis and extension of previous classifications.
result Classification of Blaschke para-umbilical hypersurfaces in conformal space.
Classifies R-spaces with a specific symmetric structure.
problem Classifying R-spaces with a natural Γ-symmetric structure.
method Classification and determination of maximal antipodal sets.
result Classification of R-spaces with a natural Γ-symmetric structure.
Classifies contact metric spaces using hyperquadric bundles.
problem Classifying contact metric spaces with specific invariant.
method Using tangent hyperquadric bundles of Lorentzian space forms.
result Locally classify spaces with Boeckx invariant ≤ -1.
The first Betti number for a lattice in a classifying space for variations of Hodge structures vanishes.
Classifies CR submanifolds in complex hyperbolic spaces.
problem Understanding CR submanifolds in complex hyperbolic spaces.
method Classifying orbits of a subgroup of the solvable part of the Iwasawa decomposition.
result Classification of homogeneous CR submanifolds.
Research proves the connectedness of classifying spaces for Haefliger structures.
problem Existence and properties of foliations and diffeomorphism groups.
method Analyzes the connectedness of classifying spaces for Haefliger structures.
result Proves (2q−1)-connectedness for framed Haefliger structures. Classifies 4D spaces with circle symmetry and positive curvature.
problem Classifying spaces with specific geometric properties.
method Equivariant homeomorphism classification, infinitesimal geometry analysis.
result Classification of 4D Alexandrov spaces and orbifolds with circle symmetry.
Paper provides first theoretical guarantees for hyperbolic space learning.
problem Learning a classifier in hyperbolic space for hierarchical data.
method Efficient algorithm for large-margin hyperplane learning in hyperbolic space.
result The low embedding dimension in hyperbolic space leads to superior classifier learning guarantees.
Geometric framework for Milnor classifying spaces in diffeological spaces.
problem Milnor classifying spaces in diffeological spaces.
method Developed spherical and projective models with natural diffeological structures, constructed Riemannian metrics, defined differential forms, and introduced Clifford structures.
result Established a coherent geometric setting combining classifying spaces, diffeology, and higher geometric structures.
The main result of this paper is a new classification theorem for links (smooth embeddings in codimension 2). The classifying space is the rack space (defined in [Trunks and classifying spaces, Applied Categorical Structures, 3 (1995) 321--356]) and the classifying bundle is the first James bundle (defined in "James bu…
Paper classifies special slant surfaces with varying curvature.
problem Classifying surfaces with non-constant mean curvature.
method Analyzing special slant surfaces in complex space forms.
result Complete classification of surfaces with non-constant mean curvature.
New groups without finite classifying spaces found in Kodaira fibrations.
problem Finding groups without finite classifying spaces.
method Constructing Kähler groups as fundamental groups of Kodaira fibrations.
result Found Kähler groups that are not commensurable to surface groups and do not have finite classifying spaces.
A new method for one-class classifiers using spanning trees and sub-spaces.
problem Designing efficient one-class classifiers for various datasets.
method Ensemble-of-classifiers approach, spanning trees, sub-spaces, minimum distance.
result Comparable and state-of-the-art performance on benchmark datasets.
New method classifies HCMU surfaces in 3D space forms as Weingarten surfaces.
problem Classifying HCMU surfaces in 3D space forms as Weingarten surfaces.
method Totally different method from previous work.
result Criteria for Weingarten surfaces that are also HCMU surfaces.
Classifies Legendrian Hopf links in lens spaces.
problem Classifying Legendrian Hopf links in lens spaces.
method Classification up to coarse equivalence.
result Legendrian Hopf links in L(p,1) are classified.
The article calculates the minimal model dimensions for classifying spaces of surface braid groups.
problem Classifying spaces for families of virtually abelian subgroups of surface braid groups.
method Analyzes the pure and full braid groups of surfaces with at least one boundary component or one puncture, proving analogous results for amenable subgroups.
result Minimal dimensions of models for classifying spaces are equal to the virtual cohomological dimension plus the rank of virtually abelian subgroups.
Study fractional structures on bundle gerbe modules using rational homotopy theory.
problem Understanding twisted Chern classes of torsion bundle gerbe modules.
method Sullivan's rational homotopy theory to realize twisted Chern classes at the level of classifying spaces.
result Introduction of fractional U-structures as a universal framework.
Enhances classifier performance through feature space transformations and model selection.
problem Improving the accuracy of classifiers by reducing complexity.
method Combining feature mapping, prototype selection, and kernel function transformations to transform data into a more convenient distribution.
result Our methods produce competitive classifiers and are statistically different among them.
Solves a problem related to classifying spaces for proper actions and Nielsen Realization.
problem Whether a cocompact proper topological Γ-manifold is equivariantly homotopy equivalent to the classifying space for proper actions.
method Using Poincaré models and assuming a zero-dimensional singular set, the problem is solved in the Poincaré category.
result New results about Brown's problem are obtained under certain conditions on the underlying group.
New structural result classifies actions on symmetric spaces.
problem Classify cohomogeneity one actions on symmetric spaces.
method Developed new structural result and applied to specific cases.
result Classified cohomogeneity one actions on SL(n,R)/SO(n), n>1.
Classifies actions on complex space forms with Lagrangian orbits.
problem Classifying actions on complex space forms with Lagrangian orbits.
method Classifies holomorphic isometric actions on complex space forms.
result Only examples are Lagrangian affine subspace foliations of complex Euclidean spaces and Lagrangian horocycle foliations of complex hyperbolic spaces.
Paper classifies surfaces with a special direction in Minkowski 3-space.
problem Characterizing surfaces with a canonical principal direction.
method Characterization and classification of surfaces in Minkowski 3-space.
result All surfaces with a canonical principal direction are classified.
Classifies surfaces with special curvature properties.
problem Rotational surfaces with specific curvature conditions.
method Classifies surfaces with rotationally symmetric norms and linear curvature relations.
result Rotational surfaces with linearly related curvatures are classified.
The study classifies biharmonic real hypersurfaces in complex projective spaces.
problem Classifying biharmonic real hypersurfaces in complex projective spaces.
method Analyzing proper biharmonic Hopf and ruled real hypersurfaces with distinct principal curvatures.
result Biharmonic ruled real hypersurfaces in complex projective spaces are minimal.