We revisit Spakula's uniform K-homology, construct the external product for it and use this to deduce homotopy invariance of uniform K-homology. We define uniform K-theory and on manifolds of bounded geometry we give an interpretation of it via vector bundles of bounded geometry. We further construct a cap product with…
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48 results for “Uniformization”
In "Rips complexes and covers in the uniform category" \cite{Rips} the authors define, following James \cite{J}, covering maps of uniform spaces and introduce the concept of generalized uniform covering maps. Conditions for the existence of universal uniform covering maps and generalized uniform covering maps are given…
Paper proves discrete uniformizations converge to continuous for surfaces of genus ≥1.
problem Computing uniformizations for surfaces of genus >1.
method Discrete conformality and uniformization on triangle meshes.
result Discrete uniformizations approximate continuous uniformization for closed surfaces of genus ≥1.
This paper refines homotopy theory for cubical sets and uniform spaces.
problem Classical homotopy theory limitations in cubical sets and uniform spaces.
method Develops a uniform-theoretic refinement for cubical sets and uniform spaces, lifting to a full and faithful embedding.
result Lifts classical homotopy categories to new uniform homotopy categories, generalizing cohomology theories.
Survey on uniformization of metric surfaces, including fractal and topological manifolds.
problem Uniformization of metric surfaces homeomorphic to 2D topological manifolds.
method Various uniformization theorems, including quasisymmetric and quasiconformal approaches.
result Uniformization results for metric spheres and arbitrary metric surfaces.
Berestovskii and Plaut introduced the concept of a coverable uniform space when developing their theory of generalized universal covering maps for uniform spaces. Brodskiy, Dydak, LaBuz, and Mitra introduced the concept of a locally uniformly joinable uniform space when developing their theory of generalized uniform co…
Study shows gap between uniform convergence and test error in random feature models.
problem Understanding the gap between uniform convergence and test error in random feature models.
method Analytical expressions for uniform convergence over norm balls, interpolators, and minimum norm interpolator risk derived and proved.
result Uniform convergence over interpolators still gives a non-trivial bound of test error even when classical uniform convergence is vacuous.
The study explores maps of 2- and 3-uniform tilings on the torus.
problem Understanding the number of vertex orbits in quotient maps of 2- and 3-uniform tilings.
method Analyzing the quotient maps of 2- and 3-uniform tilings on the torus.
result Bounds on the number of vertex orbits in quotient maps of 2- and 3-uniform tilings.
Surface Nematic Quasi-Uniformitycond-mat.soft
New approach to nematic fields on surfaces, relaxing uniformity to quasi-uniformity.
problem Identifying least distorted nematic fields on generic surfaces.
method Relaxing the notion of uniformity into quasi-uniformity and proving parallel transport by geodesics.
result All quasi-uniform fields are parallel transported by the geodesics of the surface.
New uniformity tester ensures consistent results across different samples.
problem Non-replicable behavior of uniformity testing algorithms.
method Develops a replicable uniformity tester with improved sample complexity.
result Achieves nearly linear dependence on replicability factor .
Uniform estimates for Kaehler metrics' diameters and volumes.
problem Estimating diameters and volumes of Kaehler metrics.
method Proving uniform diameter and volume estimates for a family of Kaehler metrics.
result Uniform estimates for diameters and volumes of Kaehler metrics.
We generalize Roe's index theorem for graded generalized Dirac operators on amenable manifolds to multigraded elliptic uniform pseudodifferential operators. This generalization will follow as a corollary from a local index theorem that is valid on any manifold of bounded geometry. This local formula incorporates the un…
Generalizes uniformization to algebraic correspondences.
problem Uniformizing non-homeomorphic genus zero orbifolds.
method Constructs algebraic correspondences to simultaneously uniformize orbifolds.
result Realizes Teichmüller space of a punctured sphere in correspondences.
Uniform foliations with Reeb components on 3-manifolds.
problem Characterizing uniform foliations with Reeb components.
method Study of foliations on 3-manifolds with infinite fundamental group.
result Examples and results on behavior of foliations.
Uniform Lie algebras are combinatorially defined two-step nilpotent Lie algebras which can be used to define Einstein solvmanifolds. These Einstein spaces often have nontrivial isotropy groups. We derive basic properties of uniform Lie algebras and we classify uniform Lie algebras with five or fewer generators. We defi…
Uniformizes surfaces with boundaries, focusing on triple junctions.
problem Uniformization of surfaces with boundaries, especially triple junctions.
method Extends conformal structure results to triple junction surfaces.
result Weak uniformization results for triple junction surfaces.
Uniform K-stability ensures existence of special metrics on toric manifolds.
problem Existence of conformally Kähler, Einstein-Maxwell metrics on toric manifolds.
method Introducing uniform K-stability and showing its equivalence to properness of relative K-energy.
result Uniform K-stability is necessary and sufficient for the existence of -extremal metrics on toric manifolds.
The paper solves a uniform Yau-Tian-Donaldson conjecture for toric manifolds.
problem Uniform Yau-Tian-Donaldson conjecture for polarized toric manifolds.
method Combinatorial sufficient condition for relative K-polystability.
result Uniform relative K-polystability condition established.
In the product space H^n \times R; we obtain uniform a priori C^0 horizontal length estimates, uniform a priori C^1 boundary gradient estimates, as well as uniform modulus of continuity, for a class of horizontal minimal equations. In two independent variables, we derive a certain uniform global a priori C^1 estimates …
Uniform estimates for Calabi-Yau degenerations proved.
problem Calabi-Yau degenerations of polarised algebraic manifolds.
method Uniform Skoda and -estimates for Kähler potentials.
result Uniform Skoda type estimate and -estimate for Calabi-Yau Kähler potentials proved.
Study proves uniform ellipticity implies uniform polyconvexity for anisotropic energy functionals.
problem Investigating uniform ellipticity and polyconvexity for anisotropic geometric energy functionals.
method Proves a variant of a recent result using real polyhedral chains.
result Uniform ellipticity of an anisotropic energy functional implies uniform polyconvexity of the integrand.
We generalize Roe's Index Theorem for operators of Dirac type on open manifolds to elliptic pseudodifferential operators. To this end we introduce a class of pseudodifferential operators on manifolds of bounded geometry which is more general than similar classes defined by other authors. We revisit Spakula's uniform K-…
Uniform K-homology theory applied to elliptic operators on manifolds with boundary.
problem Developing a theory to study boundary conditions for elliptic operators on non-compact manifolds.
method Theory of relative uniform K-homology, developing a relative index map.
result Uniform K-homology classes of boundary conditions and their connection to the higher ρ-invariant.
Improved uniform convergence bound with fat-shattering dimension reduces sample complexity gap.
problem Gap between upper and lower bounds on sample complexity for fat-shattering dimension.
method Provided an improved uniform convergence bound.
result Closed the gap between existing upper and lower bounds on sample complexity.
In this paper we deduce a local deformation lemma for uniform embeddings in a metric covering space over a compact manifold from the deformation lemma for embeddings of a compact subspace in a manifold. This implies the local contractibility of the group of uniform homeomorphisms of such a metric covering space under t…
New approach for uniform estimates in complex equations.
problem Uniform estimates for solutions to complex Monge-Ampere equations.
method Efficient new approach to uniform estimates.
result Efficient method for uniform estimates in geometric PDEs.
A local deformation property for uniform embeddings in metric manifolds (LD) is formulated and its behaviour is studied in a formal view point. It is shown that any metric manifold with a geometric group action, typical metric spaces (Euclidean space, hyperbolic space and cylinders) and for κ\leq 0 the κ-cone ends over…
Uniform waist inequalities proven for manifolds with Kazhdan groups in codimension two.
problem Proving uniform waist inequalities for manifolds with specific group properties.
method Using finite covers and Cheeger inequality for manifolds with Kazhdan fundamental groups.
result Finite covers of manifolds with Kazhdan groups satisfy uniform waist inequalities in codimension two.
Metrizable uniform spacesmath.GT
Three themes of general topology: quotient spaces; absolute retracts; and inverse limits - are reapproached here in the setting of metrizable uniform spaces, with an eye to applications in geometric and algebraic topology. The results include: 1) If f: A -> Y is a uniformly continuous map, where X and Y are metric spac…
We prove some criteria for uniform K-stability of log Fano pairs. In particular, we show that uniform K-stability is equivalent to -invariant having a positive lower bound. Then we study the relation between optimal destabilization conjecture and the conjectural equivalence between uniform K-stability and K-stabilit…
Uniform bounds derived for fully non-linear equations.
problem Bounding fully non-linear equations uniformly in background metrics.
method Auxiliary Monge-Ampère equations and entropy-like quantities.
result Uniform bounds for systems coupling fully non-linear equations to their linearizations.
Proves uniform K-stability is open in Kähler cone.
problem Stability of Kähler metrics in complex geometry.
method Introduced new norm on test configurations and estimates for non-archimedean energy functionals.
result Uniform K-stability is an open condition in the Kähler cone.
Uniform measures have played a fundamental role in geometric measure theory since they naturally appear as tangent objects. For instance, they were essential in the groundbreaking work of Preiss on the rectifiability of Radon measures. However, relatively little is understood about the structure of general uniform meas…
A new method achieves optimal uniformity in designs with minimal flexibility.
problem Achieving optimal uniformity in designs with minimal flexibility.
method Derive a lower bound on the uniformity constant and use a greedy construction to achieve this bound, then extend the scheme for more flexibility.
result A simple greedy construction achieves the optimal uniformity constant.
Study infinite genus surfaces and Schottky groups for uniformization.
problem Investigate infinite genus surfaces and Schottky groups for uniformization.
method Definitions and proofs for infinite genus surfaces and Schottky groups, showing uniformization by Schottky groups.
result Infinite genus surfaces and handlebodies can be topologically and quasiconformally uniformized by Schottky groups.
Uniform bounds for complex equations using Monge-Ampère method.
problem Bounding solutions to complex equations.
method Auxiliary Monge-Ampère equation method.
result Uniform bounds remain valid even as background metrics degenerate.
Logistic regression gets a new, simpler uniform bound.
problem Finding a uniform bound for logistic regression's empirical risk.
method PAC-Bayes approach with second-order expansion and Rademacher-complexity bounds.
result Provides a dimension-free uniform concentration bound.
Notions of uniform manifoldsmath.DG
New uniformity definition on noncompact manifolds without metrics.
problem No Riemannian metric for uniformity on noncompact manifolds.
method Definition of uniformity without metrics, equivalent to bounded geometry.
result Equivalent definition of uniformity on noncompact manifolds.
Paper generalizes discrete uniformization for genus-zero surfaces.
problem Discrete uniformization for surfaces of genus zero.
method Reduction to planar cases via stereographic projections.
result Generalization of discrete uniformization to genus-zero surfaces.
New approach to adversarial robustness with non-uniform perturbations.
problem Real-world adversaries craft adversarial examples with non-uniform perturbations.
method Proposes non-uniform perturbations based on feature dependencies and data distribution.
result Shows improved robustness to real-world attacks compared to uniform perturbations.
Uniform entropy bound for Ricci shrinkers with bounded curvature.
problem Bounding entropy for Ricci shrinkers with specific curvature constraints.
method Establishing uniform entropy bounds for simply connected Ricci shrinkers with a finite second homotopy group and uniform curvature bounds.
result Uniform entropy bound for simply connected Ricci shrinkers with a finite second homotopy group and uniform curvature bounds.
New approach finds minima of geodesic lengths for non-uniform fillings.
problem Finding minima of geodesic length functions for non-uniform fillings.
method Elementary optimization for 4-regular topological fillings, analysis of fat graphs and optimization techniques.
result Minima of geodesic length functions are found to be at triangle surfaces in both analyzed classes of non-uniform fillings.
Uniformizes compact Sasakian manifolds into circle bundles.
problem Deforming compact Sasakian manifolds to locally isomorphic forms.
method Criterion based on circle bundles of anti-canonical bundles over Hermitian symmetric spaces.
result Sasakian manifolds can be deformed to locally isomorphic forms.
We show that an elliptic uniform pseudodifferential operator over a manifold of bounded geometry defines a class in uniform K-homology, and that this class only depends on the principal symbol of the operator.
Simpler, faster algorithm for uniformity testing in the shuffle model.
problem Testing uniformity of data in the shuffle model with privacy constraints.
method Simplified analysis and use of privacy amplification via shuffling.
result An algorithm with the same guarantees but simpler and more streamlined.
Equivalence proven between uniformizing varieties and tensors, generalizing uniformization results.
problem Characterizing complex-projective varieties with klt singularities and ample canonical divisors.
method Constructing a uniformizing variation of Hodge structure from slope zero tensors and vice versa.
result Generalization of uniformization results to singular settings, including quotients of tube domains.
We show that the sets in a family with finite VC dimension can be uniformly approximated within a given error by a finite partition. Immediate corollaries include the fact that VC classes have finite bracketing numbers, satisfy uniform laws of averages under strong dependence, and exhibit uniform mixing. Our results ar…
Classifies 2-uniform maps on torus with formulas and asymptotic bounds.
problem Classifying 2-uniform maps on torus.
method Classification through arithmetic functions and asymptotic analysis.
result Explicit formulas and asymptotic bounds for 2-uniform maps on torus.