Improved semialgebraic choices with linear complexity.
problem Finding semialgebraic choices in projections with exponential complexity.
method Allowing approximate selections in Hausdorff sense.
result Constructed an approximate selection with linear degree in complexity.
Study approximates product of spheres using Laplacian eigenvalues.
problem Approximating product of spheres using Laplacian eigenvalues.
method Gromov-Hausdorff approximation with pinching condition on eigenvalues.
result Convergence to product of spheres achieved.
For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spherical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of the unit ball in the nilpotent approximation and it is always a continuous function. We then prove that up to dimension 4 it is smooth, whil…
Solves approximation problems for zonoids and neural networks, closing gaps in dimensions 2 and 3.
problem Approximating zonoids and shallow neural networks in uniform norm.
method Combines techniques to solve both problems, closing gaps in dimensions 2 and 3.
result Completes the solution for zonoid approximation in all dimensions and improves neural network approximation rates.
We introduce a natural definition of Lp-convergence of maps, p≥1, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the Lp-convergence, we establish a theory of …
The study establishes equivalence of conditions on metric manifolds with finite volume.
problem Characterizing metric spaces with a metric fundamental class.
method Analyzing three conditions on metric manifolds with finite volume.
result Conditions (1), (2), and (3) are equivalent for metric manifolds with finite Nagata dimension.
Study spectral distances on compact RCD spaces.
problem Understanding spectral convergence in RCD spaces.
method Established relationships between different spectral convergences and constructed a spectral approximation map.
result Found canonical spectral approximation map for RCD spaces.
Study Poincaré inequality in metric spaces via separating sets.
problem Geometric characterization of Poincaré inequality in metric spaces.
method Properties of separating sets and various notions of energy.
result Equivalence of conditions for 1-Poincaré inequality.
Study shows intrinsic timed Hausdorff convergence leads to Gromov-Hausdorff and big bang convergence.
problem Distance between Lorentzian manifolds.
method Intrinsic timed Hausdorff convergence.
result Intrinsic timed Hausdorff convergence implies Gromov-Hausdorff and big bang convergence.
The paper studies the convergence of SAA for systemic risk measures.
problem Theoretical convergence of SAA for set-valued systemic risk measures.
method General theory and specific case study with mixed-integer programming formulations.
result Theoretical convergence results for SAA under Wijsman and Hausdorff topologies.
Paper introduces a new convergence for Lorentzian spaces using causal diamonds.
problem No specific problem stated; focuses on a new geometric convergence.
method Uses causal diamonds to define a new convergence for Lorentzian spaces.
result Proves a pre-compactness theorem for Lorentzian spaces.
Flexible framework for modeling predictive distributions of time series
problem Modeling predictive distributions of nonlinear time series
method Generative adversarial networks
result Direct simulation-based approximation to predictive distributions
Study shows stability of Schrödinger operator spectral data on a manifold.
problem Determining a manifold and potential function from spectral data.
method Approximation of spectral data on a subset to determine manifold and potential.
result Quantitative stability estimate for Schrödinger operator inverse problem.
The paper extends vector bundle theory to non-Hausdorff manifolds.
problem Generalizing vector bundle theory to non-Hausdorff manifolds.
method Using Čech cohomology to classify real non-Hausdorff line bundles.
result Vector bundles over non-Hausdorff manifolds can be constructed as colimits of standard vector bundles.
Researchers reconstruct simple Riemannian manifolds from boundary wave arrival times.
problem Reconstructing Riemannian manifolds from unknown interior sources and arrival times.
method Discrete metric approximation using labeled Gromov--Hausdorff distance.
result Finite-time approximations converge to the true Riemannian manifold.
Upper bound on singular set dimension for area-minimizing currents.
problem Bounding the dimension of singular points in area-minimizing currents.
method Using upper Minkowski dimension and properties of blow-up scales.
result Upper Minkowski bound of m−2 for the interior singular set. This paper introduces Hausdorff measure and its applications in fractal geometry.
problem Defining and applying Hausdorff measure to fractal geometry.
method Definition of Hausdorff outer measure, Caratheodory's criterion, construction of Hausdorff measure, and introduction of Hausdorff dimension.
result Demonstrates the Hausdorff dimension of the Cantor ternary set.
We investigate compact Hausdorff foliations on compact Riemannian manifolds in the context of the Gromov-Hausdorff distance theory. We give some sufficient conditions for such foliations to be separated in the Gromov-Hausdorff topology.
In dimension 4, we show that a nontrivial flat cone cannot be approximated by smooth Ricci shrinkers with bounded scalar curvature and Harnack inequality, under the pointed-Gromov-Hausdorff topology. As applications, we obtain uniform positive lower bounds of scalar curvature and potential functions on Ricci shrinker…
Study of one-dimensional non-Hausdorff manifolds and their quotient to CW complexes.
problem Understanding and characterizing one-dimensional non-Hausdorff manifolds.
method Analyzing properties of connected non-Hausdorff manifolds and their quotient spaces to CW complexes.
result Existence of a quotient map from a connected non-Hausdorff manifold to an open one-dimensional CW complex.
Hausdorff reflection keeps space shape intact.
problem Preserving shape type in spaces.
method Hausdorff reflection method.
result Hausdorff reflection preserves shape type.
We construct for every finite-dimensional Alexandrov space A and every point p∈A a 2-convex function fp in a small neighborhood around p, which approximates distp2 up to second order. Moreover, the function fp can be lifted to Gromov-Hausdorff close Alexandrov spaces of the same dim…
Study of de Rham cohomology on non-Hausdorff manifolds.
problem De Rham cohomology on non-Hausdorff manifolds.
method Careful discussion of non-Hausdorff differential forms, Mayer-Vietoris sequences.
result Proved de Rham's Theorem and Gauss-Bonnet theorem for non-Hausdorff manifolds, including counterterms.
The preservation of ambient isotopic equivalence under piecewise linear (PL) approximation for smooth knots are prominent in molecular modeling and simulation. Sufficient conditions are given regarding: (1) Hausdorff distance, and (2) a sum of total curvature and derivative. High degree Bezier curves are often used as …
Geodesics with bounded angles have zero Hausdorff dimension.
problem Understanding the geometric properties of geodesics with bounded angles.
method Analyzing the Hausdorff dimension of geodesics with specific angle constraints.
result The set of geodesics with bounded self-intersection angles has a Hausdorff dimension of zero.
The paper proves stability in compact finite dimensional Alexandrov spaces using equivariant Gromov--Hausdorff convergence.
problem Stability in compact finite dimensional Alexandrov spaces.
method Equivariant Gromov--Hausdorff convergence and almost commutative diagrams.
result Stability result in compact finite dimensional Alexandrov spaces.
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.
Proves compactness for timed-metric spaces using new distance and maps.
problem Weak convergence of space-times using timed-Hausdorff distance.
method Uses Gromov's original compactness theorem and introduces addresses.
result Establishes compactness theorem for intrinsic timed-Hausdorff convergence.
Study of Calabi-Yau metrics on converging manifolds, resolving conjectures.
problem Understanding Calabi-Yau metrics on converging manifolds.
method Analysis of Gromov-Hausdorff limits of metrics on Calabi-Yau fibrations.
result Gromov-Hausdorff limit is homeomorphic to the base of the fibration and discriminant locus has high Hausdorff codimension.
We study the geometric Whitney problem on how a Riemannian manifold (M,g) can be constructed to approximate a metric space (X,dX). This problem is closely related to manifold reconstruction where a smooth n-dimensional submanifold S⊂Rm, m>n needs to be constructed to approximate a point clo…
Proves generalization bounds for SGD using Feller processes and Hausdorff dimension.
problem Characterizing generalization properties of SGD in deep learning.
method Proves generalization bounds for SGD under Feller process approximation, linking generalization error to the Hausdorff dimension of trajectories.
result Generalization error controlled by the Hausdorff dimension of trajectories, which is linked to the tail behavior of the driving process.
The paper classifies Kleinian groups with Hausdorff dimension less than 1.
problem Classifying Kleinian groups with specific Hausdorff dimensions.
method Using Hou's result, the paper proves that all convex cocompact Kleinian groups of Hausdorff dimension less than 1 are Schottky groups.
result The classification of convex cocompact Kleinian groups of Hausdorff dimensions less than 1.
The paper proves a conjecture about manifold limits and characterizes their structure.
problem Characterizing limits of manifolds with a uniform contractibility function.
method Short proof using Gromov-Hausdorff distance and ANR properties.
result Obstruction vanishes if and only if the manifold can be approximated by PL-manifolds.
Upper bound for Hausdorff distance between hyperbolic space and its medianization.
problem Calculating the Hausdorff distance between hyperbolic space and its medianization.
method Using de Sitter space to model finite-dimensional hyperbolic space and its medianization, calculating the Hausdorff distance.
result An upper bound for the Hausdorff distance between hyperbolic space and its medianization is calculated.
Strongly convex bodies can be approximated by smooth ones.
problem Approximating strongly convex bodies with smooth ones.
method Using C2 locally strongly convex bodies. result Smooth approximations of strongly convex bodies exist and can be controlled in terms of Hausdorff distance.
Study Gromov-Hausdorff convergence of metric pairs and tuples.
problem Understanding convergence in metric spaces.
method Prove equivalence of definitions, embedding, completeness, and compactness theorems.
result Relative version of Fukaya's theorem and finiteness theorem for stratified spaces.
New Hausdorff integrations for Lie algebroids and symplectic groupoids.
problem Integrating Lie algebroids and symplectic groupoids.
method Hausdorff versions of Lie Integration Theorems 1 and 2, Lie equivalences, and algebraic approach to holonomy.
result Generalization of integration of subalgebroids to non-wide cases and detailed exploration of foliation groupoids.
We investigate the approximate j-dimensionality of the singularity sets of minimal surfaces prescribed by Simon. This leads to the clasification of 8 variations of approximately j-dimensional surfacs in terms of dimension and locally finite Hausdorff measure. We show that the singularity sets must either be well behave…
Proof of Reifenberg theorem in metric spaces, expanding on Cheeger and Colding's work.
problem Proving the Reifenberg theorem in metric spaces using Gromov-Hausdorff distance.
method Detailed proof of Cheeger and Colding's result, expanding on their arguments.
result BiLipschitz version of the Reifenberg theorem in metric spaces.
We prove that for a compact subgroup H of a locally compact Hausdorff group G, the following properties are mutually equivalent: (1) G/H is a manifold, (2) G/H is finite-dimensional and locally connected, (3) G/H is locally contractible, (4) G/H is an ANE for paracompact spaces, (5) G/H is a metrizable $G…
Research shows surfaces close to planes in Hausdorff distance.
problem Understanding submanifolds with entropy close to one.
method Analyzing entropy and Hausdorff distance to prove rigidity.
result Submanifolds with entropy close to one are close to planes.
Study describes limits of surfaces in a mathematical space.
problem Understanding limits of surfaces in mathematical spaces.
method Completely described Gromov-Hausdorff closure of surfaces.
result Completely described the closure of surfaces.
The paper strengthens a theorem on crossings under linear perturbations with Hausdorff measure estimates.
problem Understanding multiple-point crossings under linear perturbations.
method Establishes a transversality theorem with Hausdorff measure estimates for exceptional parameter sets.
result Explicit upper bounds on the Hausdorff dimension of the exceptional set.
Study uses equivariant topology to measure distances between G metric spaces.
problem Measuring distances between G metric spaces.
method Equivariant topology methods to derive lower bounds.
result Sharp bounds on Gromov Hausdorff distance between spheres.
Compactness theorem for timed-metric spaces established.
problem Compactness of timed-metric spaces and causality.
method Timed-Gromov--Hausdorff distance and intrinsic timed-Hausdorff distance.
result Induces same notion of convergence as intrinsic timed-Hausdorff distance.
Study on Hausdorff dimension of Anosov subgroup limit sets under specific affine complexity.
problem Investigating the Hausdorff dimension of Anosov subgroup limit sets with self-affine complexity.
method Analyzing the Hausdorff dimension of projective limit sets Λ1(Γ) of Anosov subgroups Γ under specific assumptions about their affine complexity. result The Hausdorff dimension of Λ1(Γ) is determined by the critical exponent of the first simple root under partial quasi-self-similarity. Maps continuous Riemann surfaces to complex space with specific properties.
problem Embedding Riemann surfaces into complex space with controlled poles and boundaries.
method Continuous map with specified properties, including effective poles and Hausdorff dimension constraints.
result Existence of Jordan curves in the image of the map, each of Hausdorff dimension one.
Study extends Hausdorff dimension Hessian results to new hyperconvex representations.
problem Extending classical results on Hausdorff dimension Hessian.
method Analyzes (1,1,2)-hyperconvex representations and small complex deformations.
result Positive definiteness of Hessian of Hausdorff dimension for co-compact Γ in PO(n,1).