The paper studies medial and conflict sets under continuous deformations in Riemannian manifolds.
problem Understanding the behavior of medial and conflict sets under continuous deformations in Riemannian manifolds.
method A new approach using Kuratowski convergence to express the deformation process as a continuous process.
result Main `medial axis inner semi-continuity' result proved useful in computing tangent cones of the medial axis.
The abstract discusses convergence properties of Lipschitz functions and sets defined by equations.
problem Convergence of Lipschitz functions and sets defined by equations.
method Painlevé-Kuratowski convergence applied to Lipschitz functions and sets defined by equations.
result Generalizations and reverses of classical theorems on convergence of functions and sets.
Characterizes 3D embeddability of certain 2D complexes via excluded minors.
problem Characterizing embeddability of specific 2D complexes in 3-space.
method Using Kuratowski-type characterisation via excluded minors.
result Answers Lovász, Pardon, and Wagner's questions about embeddability.
Paper proves how 3D shapes can be embedded without obstructions.
problem Embedding 2D shapes in 3D space without crossing.
method Analogue of Kuratowski's graph planarity theorem for 3D.
result Finite list of obstructions except one infinite family.
Study gives bounds on filling radius for Riemannian manifolds.
problem Finding bounds on the filling radius of Riemannian manifolds.
method Curvature-dependent bounds for all closed manifolds and upper bounds for submersion and submetry cases.
result Upper and lower bounds on the filling radius for specific types of manifolds.
This paper is purely expositional. The statement of the Kuratowski graph planarity criterion is simple and well-known. However, its classical proof is not easy. In this paper we present the Makarychev proof (with further simplifications by Prasolov, Telishev, Zaslavski and the author) which is possibly the simplest. In…
Study large mass G2 and Calabi--Yau monopoles, proving convergence and identifying key sets.
problem Large mass limits of G2 and Calabi--Yau monopoles on specific manifolds. method Common Θ-monopole framework, variational compactness theory, and finer analysis. result Identifies currents and shows saturation of calibration inequalities; defines sets S, Z, and C. Classifies critical complexes for embedding in 3-sphere.
problem Classifying minimal obstructions to embedding in 3-sphere.
method Combinatorial approach using reduction graphs and forest attachments.
result Exactly seven critical complexes identified.
The study characterizes embeddable 2-complexes in 3-space.
problem Characterizing embeddable 2-dimensional simplicial complexes in 3-space.
method Characterization through excluded minors and extensions.
result Characterized embeddable 2-complexes in 3-space, including cones over K5 and K3,3, and related constructions. Researchers found all embeddings of Kuratowski graphs on a double torus.
problem Characterizing embeddings of Kuratowski graphs K3,3 and K5 on the double torus. method Constructive approach using Burnside's Lemma and automorphism groups.
result 14 orientable and 17 non-orientable 2-cell embeddings of K5 on the double torus. We determine the set of Busemann points of an arbitrary finite-dimensional normed space. These are the points of the horofunction boundary that are the limits of "almost-geodesics". We prove that all points in the horofunction boundary are Busemann points if and only if the set of extreme sets of the dual unit ball is …
We show that the Kuratowski imbedding of a Riemannian manifold in L^\infty, exploited in Gromov's proof of the systolic inequality for essential manifolds, admits an approximation by a (1+C)-bi-Lipschitz (onto its image), finite-dimensional imbedding for every C>0. Our key tool is the first variation formula thought of…
Dual matroids help embed 2-complexes in 3-space.
problem Characterizing embeddability of 2-dimensional simplicial complexes in 3-space.
method Introducing dual matroids and using them to extend Kuratowski's theorem.
result Dual matroids provide a new way to characterize embeddability.
With any (open or closed) cover of a space T we associate certain homotopy classes of maps T into n-spheres. These homotopy invariants can be considered as obstructions for extensions of covers of a subspace A to a space X. We using these obstructions for generalizations of the classic KKM (Knaster-Kuratowski-Mazurkiew…
For any collection of graphs we find the minimal dimension d such that the product of these graphs is embeddable into the d-dimensional Euclidean space. In particular, we prove that the n-th powers of the Kuratowsky graphs are not embeddable into the 2n-dimensional Euclidean space. This is a solution of a problem of Me…
Unified small and large scale geometry concepts.
problem Combining topology and geometry for large and small scale spaces.
method Introducing orthogonality relations to unify topology and geometry.
result Optimal orthogonality relations for compactifications.
Criteria for embedding simplicial complexes into manifolds, reducing a topological problem to algebra.
problem Embedding simplicial complexes into manifolds.
method Interplay between geometric topology, combinatorics, and linear algebra; calculation of generators in configuration space homology.
result Criteria for Z2-embeddability of certain simplicial complexes to 2k-dimensional manifolds. We offer the following explanation of the statement of the Kuratowski graph planarity criterion and of 6/7 of the statement of the Robertson-Seymour-Thomas intrinsic linking criterion. Let us call a cell complex 'dichotomial' if to every cell there corresponds a unique cell with the complementary set of vertices. Then …
NOs can learn any finite collection of classes in functional data.
problem Learning finite collections of classes in infinite-dimensional spaces.
method Proved sample-based neural operators can learn any finite collection of classes in an infinite-dimensional reproducing kernel Hilbert space.
result NOs can learn any finite collection of classes in an infinite-dimensional reproducing kernel Hilbert space, even when the classes are not convex or connected.
Paper shows convergence types match for almost minimal sets.
problem Matching convergence types for almost minimal sets.
method Hausdorff and varifold convergence comparison on almost minimal sets.
result Hausdorff and varifold convergence coincide on almost minimal sets.
Proves weak convergence equals mean convergence in GGC.
problem Proving convergence in GGC distributions.
method Using generalized gamma convolution (GGC) and expected utility maximization.
result Weak convergence implies mean convergence in GGC.
Introduces generalized almost statistical convergence and its properties.
problem Developing a new convergence concept for sequences.
method Introducing generalized almost statistical convergence and proving its properties.
result Existence of a GAS convergent sequence that is neither statistical nor almost convergent.
This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We begin with a description of three extrinsic notions which have been applied to study sequences of submanifolds in Euclidean…
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.
Benjamini-Schramm convergence equals spectral convergence for lattices.
problem Equivalence of convergence notions for lattices.
method Extending conditions to locally compact groups and using relative L2-theory.
result Equivalence of Benjamini-Schramm and spectral convergence under mild conditions.
Studied SGD convergence under weak conditions.
problem Convergence of SGD in nonconvex optimization.
method Analyzed biased nonconvex SGD under mild conditions.
result Provided convergence rates and complexities.
Study on convergence rate of Q-curvature flow in 6 dimensions.
problem Analyzing the convergence rate of Q-curvature flow in 6 dimensions. method Provided an example of a slowly converging Q6-curvature flow in dimension 6. result The Q-curvature flow in 6 dimensions does not always converge exponentially, unlike in 2 dimensions. Establishes geometric convergence of iterative optimization algorithms.
problem Analyzes convergence of iterative optimization algorithms under general assumptions.
method General framework for iterative optimization algorithms, proving asymptotic geometric convergence and providing convergence rates.
result Asymptotic geometric convergence of iterative optimization algorithms with exact rate.
Uniform counting formulas for orthogeodesics in Kleinian groups converge.
problem Counting orthogeodesics in Kleinian groups converging to a limit.
method Spectral gap of the limit manifold and geodesic flow mixing property.
result Asymptotically uniform counting formulas for orthogeodesics.
The paper connects different convergence concepts in geometric analysis.
problem Comparing convergence concepts in geometric analysis.
method Relating Lp convergence and volume convergence to Intrinsic Flat and Gromov-Hausdorff convergence. result Conditions for convergence of Riemannian manifolds under specific conditions.
The paper explores null distance convergence for warped product spacetimes.
problem Defining convergence for sequences of spacetimes as metric spaces.
method Using the null distance to define convergence of spacetimes.
result Optimal convergence theorem for warped product spacetimes.
New quasi-Newton method guarantees global superlinear convergence.
problem Global convergence and superlinear convergence of quasi-Newton methods.
method Hybrid proximal extragradient method with online learning for Hessian approximation.
result First globally convergent quasi-Newton method with explicit superlinear convergence rate.
We investigate finite-time decoupled convergence in nonlinear two-time-scale stochastic approximation.
problem Achieving decoupled convergence in nonlinear two-time-scale stochastic approximation.
method Nested local linearity assumption, suitable step size selection, convergence analysis of matrix cross term, fourth-order moment convergence rates.
result Finite-time decoupled convergence rates can be achieved in nonlinear two-time-scale stochastic approximation with proper step size selection.
The article introduces a new convergence concept for Lorentzian spaces and applies it to generalized cones.
problem Stability of curvature bounds in generalized Lorentzian cones.
method Introduces ℓ-convergence for Lorentzian pre-length spaces, applies it to generalized cones, and proves stability of curvature bounds. result Sharp timelike curvature and curvature-dimension bounds for generalized cones are established.
AdaBoost's classifier and margins converge to a known value.
problem Convergence properties of AdaBoost algorithm.
method Formal proofs of convergence properties of AdaBoost's classifier and margins.
result AdaBoost's classifier and margins converge to a known value.
Conditions for local spectral convergence in RCD*(K,N) spaces are identified.
problem Conditions for local spectral convergence in RCD*(K,N) spaces.
method Identifying necessary and sufficient conditions for local spectral convergence.
result Necessary and sufficient conditions for local spectral convergence in balls of RCD*(K,N) spaces are established.
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 paper contrasts different convergence notions in geometric analysis.
problem Exploring discrepancies between various convergence concepts in geometric analysis.
method Examples and proof of a theorem requiring specific bounds on warping functions.
result Warped product manifolds can have different convergence limits even if the warping functions converge in Lp. The Sinkhorn-Knopp derivatives converge with linear rate.
problem Optimal transport problem with entropic regularization.
method Iterative proportional fitting procedure.
result Derivatives converge with linear rate.
The paper examines convergence of distances in Lipschitz structures on manifolds.
problem Convergence of distances in Lipschitz vector fields and norms on manifolds.
method Analysis of convergence of distances associated to converging structures of Lipschitz vector fields and norms.
result Under mild controllability assumption, distances converge locally uniformly to the limit Carnot-Carathéodory distance.
Momentum improves federated learning convergence.
problem Accelerating convergence in federated learning.
method Integrates momentum into federated learning via momentum gradient descent.
result Momentum federated learning accelerates convergence compared to standard federated learning.
New approach to geometric quantization for symplectic manifolds.
problem Quantization of symplectic manifolds with non-singular Lagrangian fibrations.
method Using spectral convergence of metric measure spaces, the authors develop a new geometric quantization approach.
result Spectral and quantum Hilbert space convergence results for Kähler and almost Kähler quantizations.
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 …
We prove a criterion of convergence in the augmented Teichmueller space that can be phrased in terms of convergence of the hyperbolic metrics or of quasiconformal convergence away from the nodes.
Random walks on convergence groups are studied, extending properties from hyperbolic groups.
problem Properties of random walks on hyperbolic groups are extended to convergence groups.
method Extending properties of random walks from hyperbolic groups to convergence groups with specific conditions.
result Random walks on convergence groups can be analyzed with a compact topology, leading to new insights into the Poisson boundary.
Sharp convergence theorem for sphere submanifolds proved.
problem Sphere submanifolds in spheres.
method Proved a sharp convergence theorem.
result New differentiable sphere theorem for submanifolds in spheres.
DCDC calculates convergence rates for Markov chains using neural networks.
problem Computing precise convergence rates for Markov chains is hard.
method Developed a neural network-based algorithm (DCDC) to bound convergence rates in Wasserstein distance.
result Demonstrated effective convergence bounds for real-world Markov chains.
Study on normalized Betti numbers in non-positively curved manifolds.
problem Convergence of normalized Betti numbers in non-positively curved manifolds.
method Benjamini-Schramm convergence, volume-normalization, irreducible symmetric spaces of noncompact type.
result Normalized Betti numbers converge for non-compact manifolds.