New examples show non-integer Hausdorff dimensions in collapsing spaces.
problem Understanding Hausdorff dimensions in collapsing Ricci limit spaces.
method Provided examples of spaces with irregular Hausdorff dimensions.
result Hausdorff dimension of singular set exceeds regular set's dimension.
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.
New property: polygons have a fixed dimension regardless of ambient space dimensions.
problem Understanding the dimension of polygon moduli spaces.
method Generalizing the square bending example to polygons of arbitrary edge lengths.
result There are only finitely many moduli spaces of polygons with given edge lengths, even as ambient dimension increases.
The paper examines the capacity dimension of boundaries in CAT(0) spaces.
problem Understanding the capacity dimension of boundaries in CAT(0) spaces.
method Comparison of metrics and study of buildings, with a method for proving asymptotic dimension finiteness.
result Visual and conical metrics on the boundary of hyperbolic CAT(0) spaces give the same capacity dimension.
Short note shows unbounded dimensions in Fano K-moduli spaces.
problem Unboundedness of Fano K-moduli space dimensions.
method Analyzes Fano varieties and their K-moduli spaces.
result Dimension of K-moduli space can be unbounded.
The paper constructs Gromov-Hausdorff metrics for Lorentzian spaces and calculates dimensions.
problem Understanding the geometry and dimensions of Lorentzian spaces.
method Construction of Gromov-Hausdorff metrics, calculation of dimensions, and analysis of Lorentzian spaces.
result Dushnik-Miller dimension of Minkowski spaces is countably infinite.
Paper relates asymptotic dimension to cofinal dimension using coarse proximities.
problem Relating asymptotic dimension to cofinal dimension in metric spaces.
method Introducing coarse proximities and inverse limit constructions.
result Asymptotic dimension is bounded by coarse cofinal dimension and cofinal dimension of Higson corona.
We introduce a new quasi-isometry invariant of metric spaces called the hyperbolic dimension, hypdim, which is a version of the Gromov's asymptotic dimension, asdim. The hyperbolic dimension is at most the asymptotic dimension, however, unlike the asymptotic dimension, the hyperbolic dimension of any Euclidean space R^…
We construct a universal space for the class of proper metric spaces of bounded geometry and of given asymptotic dimension. As a consequence of this result, we establish coincidence of the asymptotic dimension with the asymptotic inductive dimensions.
Given a metric space X of finite asymptotic dimension, we consider a quasi-isometric invariant of the space called dimension function. The space is said to have asymptotic Assouad-Nagata dimension less or equal n if there is a linear dimension function in this dimension. We prove that if X is a tree-graded space …
We consider the notion of dimension in four categories: the category of (unbounded) separable metric spaces and (metrically proper) Lipschitz maps, and the category of (unbounded) separable metric spaces and (metrically proper) uniform maps. A unified treatment is given to the large scale dimension and the small scale …
We discuss a variation of Gromov's notion of asymptotic dimension that was introduced and named Nagata dimension by Assouad. The Nagata dimension turns out to be a quasisymmetry invariant of metric spaces. The class of metric spaces with finite Nagata dimension includes in particular all doubling spaces, metric trees, …
We introduce a quasi-symmetry invariant of a metric space Z called the capacity dimension. Our main result says that for a visual Gromov hyperbolic space X the asymptotic dimension of X is at most the capacity dimension of its boundary at infinity plus 1.
In higher dimensions, Schottky spaces have unique topological properties.
problem Characterize the topology of Schottky spaces in higher dimensions.
method Analyzing the fundamental group and homotopy properties of Schottky spaces in the borderline dimension.
result In the borderline dimension, the space is simply connected but has a dense open part with fundamental group a product of cyclic groups of order two.
Perimeter minimizers in curved spaces have a singular set no more than 5 dimensions.
problem Understanding the structure of minimizers in spaces with bounded Ricci curvature.
method Analysis of non-collapsed Ricci limit spaces with two-sided curvature bounds.
result The Hausdorff dimension of the singular set is at most \(N-5\).
New findings on metric spaces with finite Nagata dimension.
problem Understanding isoperimetric properties in subsets of metric spaces.
method Analyzing quasiconvex subsets with finite Nagata dimension and applying isoperimetric inequalities.
result Quasiconvex subsets of metric spaces with finite Nagata dimension are isoperimetrically undistorted up to a certain dimension.
Researchers calculate cohomological dimensions of manifold configuration spaces, proving arithmeticity and providing bounds.
problem Calculating the cohomological dimensions of configuration spaces of manifolds.
method Defined a reduced Chevalley Eilenberg complex and provided precise formulas and bounds.
result Arithmeticity of cohomological dimensions in configuration spaces of manifolds with non-trivial co-dimension one cohomology groups.
We study the Assouad dimension and the Nagata dimension of metric spaces. As a general result, we prove that the Nagata dimension of a metric space is always bounded from above by the Assouad dimension. Most of the paper is devoted to the study of when these metric dimensions of a metric space are locally given by the …
In the first part of the paper we show how to relate several dimension theories (asymptotic dimension with Higson property, asymptotic dimension of Gromov, and capacity dimension of Buyalo \cite{Buyalo1}) to Nagata-Assouad dimension. This is done by applying two functors on the Lipschitz category of metric spaces: micr…
New geometric flow equations describe how space-time dimensions change.
problem Understanding how the number of space-time dimensions affects geometry.
method Developed D-flow equations to model the variation of space-time geometries.
result Solutions for D-flow equations on D-dimensional spheres and Freund-Rubin Compactification.
Corrects a 1-off error in Harer's spine dimension calculation for decorated Teichmüller spaces.
problem Incorrect dimension calculation in Harer's spine for decorated Teichmüller spaces.
method Identifies and corrects the dimension discrepancy in Harer's spine construction.
result Corrects the dimension of Harer's spine by 1 for decorated Teichmüller spaces.
The study classifies stable submanifolds in product spaces of projective spaces.
problem Classifying stable submanifolds in product spaces of projective spaces.
method Provided a classification theorem for compact stable minimal immersions in product spaces of projective spaces.
result Characterized complex minimal immersions in the product of two complex projective spaces.
Study stability of curvature-dimension condition for negative dimensions.
problem Stability of curvature-dimension condition with negative dimension parameters.
method Introduced CD(K, N)-condition for N < 0, defined distance d_{\mathsf{iKRW}}, proved convergence stability.
result Limit structure of converging metric measure spaces remains CD(K, N) for N < 0.
Functional dimension varies in ReLU networks, with implications for symmetry and connectivity.
problem Understanding the functional dimension of ReLU neural networks.
method Careful definition and analysis of functional dimension, study of quotient space and fibers.
result Functional dimension is inhomogeneous and can be non-constant, with implications for symmetry and connectivity.
We study discrete groups from the view point of a dimension gap in connection to CAT(0) geometry. Developing studies by Brady-Crisp and Bridson, we show that there exist finitely presented groups of geometric dimension 2 which do not act properly on any proper CAT(0) spaces of dimension 2 by isometries, although such a…
We obtain two in a sense dual to each other results: First, that the capacity dimension of every compact, locally self-similar metric space coincides with the topological dimension, and second, that the asymptotic dimension of a metric space, which is asymptotically similar to its compact subspace coincides with the to…
Study of self-dual polygons in higher dimensions, including explicit constructions and dimension calculations.
problem Understanding self-dual polygons in projective spaces of higher dimensions.
method Explicit construction and dimension calculation of moduli spaces of self-dual polygons.
result Provides the dimension of the moduli space for specific cases of n and m.
Proposes a new metric space example showing non-constant topological dimension.
problem Non-constant topological dimension in metric measure spaces.
method Refines Ketterer and Rajala's example to satisfy CD(0,∞) condition.
result Shows non-constancy of topological dimension for CD spaces.
Eluder dimension and information gain are equivalent for reproducing kernel Hilbert spaces.
problem Complexity measures in bandit and reinforcement learning.
method Equivalence of eluder dimension and information gain for reproducing kernel Hilbert spaces.
result Eluder dimension and information gain are equivalent for reproducing kernel Hilbert spaces.
We introduce the notion of large scale inductive dimension for asymptotic resemblance spaces. We prove that the large scale inductive dimension and the asymptotic dimensiongrad are equal in the class of r-convex metric spaces. This class contains the class of all geodesic metric spaces and all finitely generated groups…
New proof shows no negative curvature Einstein metrics in specific dimensions.
problem Proving nonexistence of certain Einstein metrics in 9 and 10 dimensions.
method Cohomogeneity-one approach to show nonexistence of negative curvature Einstein metrics.
result Noncompact homogeneous spaces not diffeomorphic to Euclidean space of dimension 9 or 10 admit no homogeneous Einstein metrics of negative Ricci curvature, with only three potential exceptions.
We consider an enlarged dimension reduction space in functional inverse regression. Our operator and functional analysis based approach facilitates a compact and rigorous formulation of the functional inverse regression problem. It also enables us to expand the possible space where the dimension reduction functions bel…
New theorems link graph properties to metric space dimensions.
problem Understanding large-scale properties of metric spaces and their subsets.
method Intersection graphs and asymptotic dimension analysis.
result Optimal bounds on asymptotic dimension of intersection graphs.
Investigates secant dimensions and identifiability in flag varieties.
problem Secant dimensions and identifiability of flag varieties.
method Numerical conditions ensuring secant varieties have expected dimension and points are identifiable.
result Secant varieties of flag varieties have expected dimension and points are identifiable under certain conditions.
Asymptotic dimension of planes and graphs is at most three.
problem Understanding the geometric complexity of planes and graphs.
method Analyzing geodesic spaces and their homeomorphisms to subsets in the plane.
result The asymptotic dimension of the plane and any planar graph is at most three.
Classifies 7- and 8-dimensional naturally reductive spaces.
problem Classifying naturally reductive spaces in 7 and 8 dimensions.
method Combines structure theory and new construction methods.
result Complete classification of 7- and 8-dimensional naturally reductive spaces.
We compute many dimensions of spaces of finite type invariants of virtual knots (of several kinds) and the dimensions of the corresponding spaces of "weight systems", finding everything to be in agreement with the conjecture that "every weight system integrates".
Thurston's spine dimension exceeds virtual cohomological dimension.
problem Understanding the dimension of Thurston's spine in moduli space.
method Analyzing closed hyperbolic surfaces and their systoles.
result The dimension of the set of surfaces is at least \(5-\varepsilon\) for large genus.
In this article we calculate the dimension of the Hilbert space of Kahler quantization of the moduli space of vortices on a Riemann surface. This dimension is given by the holomorphic Euler characteristic of the quantum line bundle.
Study shows infinite families of manifolds with nonnegative curvature.
problem Finding nonnegatively curved metrics on manifolds.
method Exhibited infinite families of manifolds with specific properties.
result Moduli space of nonnegatively curved metrics has infinitely many components.
A large class of vacuum space-times is constructed in dimension 4+1 from hyperboloidal initial data sets which are not small perturbations of empty space data. These space-times are future geodesically complete, smooth up to their future null infinity, and extend as vacuum space-times through their Cauchy horizon. Dime…
It is well-known that a paracompact space X is of covering dimension n if and only if any map f from X to a simplicial complex K can be pushed into its n-skeleton. We use the same idea to define dimension in the coarse category. It turns out the analog of maps f from X to K is related to asymptotically Lipschitz maps, …
Paper determines Assouad-Nagata dimension for all minor-closed metrics.
problem Understanding the Assouad-Nagata dimension of minor-closed metrics.
method Using edge-weighted graphs and edge-deletion/contraction to model minor-closed metrics, determining their Assouad-Nagata dimension.
result Determined the Assouad-Nagata dimension for every minor-closed metric.
We study the mean dimension of the moduli space of Brody curves. We introduce the notion of "mean energy" and show that this can be used to estimate the mean dimension.
New theorem for nonlocal minimal surfaces in any dimension.
problem Classical half-space theorems for minimal surfaces.
method Established a new half-space theorem.
result Result holds in any dimension.
This paper classifies 13D manifolds based on Bazaikin spaces.
problem Classifying 13-dimensional manifolds based on Bazaikin spaces.
method Topological classification modelled on Aloff-Wallach spaces and Eschenburg spaces.
result Complete topological classification for 13D manifolds based on Bazaikin spaces.
DSNE visualizes data velocity in lower dimensions.
problem Understanding movement patterns in high-dimensional data.
method DSNE is a variation of Stochastic Neighbor Embedding that learns velocity embeddings using Euclidean distances on a unit sphere.
result DSNE enables visualization of data movement in lower dimensions.
The study extends weak connection space to high dimensions, overcoming a key obstacle.
problem Obstacle in studying Yang-Mills Lagrangian in high dimensions.
method Generalized space of weak connections and proved weak sequential closure.
result Established strong approximation property of weak connections by smooth connections.