Study different topologies on locally homogeneous spaces, finding new convergence examples.
problem Understanding convergence of locally homogeneous spaces under various topologies.
method Examined three topologies on moduli space of equivariant isometry classes of locally homogeneous Riemannian spaces.
result Found first examples of locally homogeneous spaces converging in one topology but not in another.
Topological manifolds can be embedded flatly in high-dimensional Euclidean space and are locally retracts.
problem Embedding and retraction of topological manifolds in Euclidean spaces.
method Locally flat embedding and retraction of manifolds in high-dimensional Euclidean space.
result Every topological n-manifold can be embedded locally flatly in R2n+1 and is a retract of some neighborhood in R2n+1. A groupoid is a small category in which each morphism has an inverse. A topological groupoid is a groupoid in which both sets of objects and morphisms have topologies such that all groupoid structure maps are continuous. The notion of monodromy groupoid of a topological groupoid generalises those of fundamental groupoi…
The boundary of hyperbolic groups is locally simply connected.
problem Topology of hyperbolic group boundaries
method Proving local simple connectedness in terms of global topology
result Boundary is locally simply connected if and only if complement of any point is simply connected
The study explores how different Grothendieck topologies and functors between categories preserve locality.
problem Exploring relationships between different Grothendieck topologies and functors.
method Using Grothendieck topologies and functors to relate categories and geometric objects.
result Objects like sheaves, groupoids, and functors are invariant under equivalences of Grothendieck topologies and certain functors.
Locally flat submanifolds have finite CW complex complements.
problem Understanding the structure of manifold complements.
method Direct proof using homotopy equivalence and CW complexes.
result Complements of locally flat submanifolds are finite CW complexes.
Extends topological results for nonpositive curvature spaces.
problem Characterize and understand topological properties of BNPC spaces.
method Extends results from CAT(0) spaces to BNPC spaces, using links and geodesic bicombings.
result Locally BNPC topological manifolds have discrete singular sets and are homeomorphic to Euclidean space.
Study shows local topologies of certain geometric spaces.
problem Local topological properties of geometric spaces.
method Analysis of Gromov-Hausdorff limits of manifolds with bounded Ricci curvature.
result Local b1 vanishes for regular loci in limits of non-collapsed manifolds. Global and local blowups of manifolds are proven equivalent.
problem Equivalence of global and local blowups in differential topology.
method Proof of equivalence between global and local constructions of blowups.
result Global and local constructions of blowups are shown to be equivalent.
Study on median algebra structures on Euclidean spaces and manifolds with local CAT(0) cubulation.
problem Understanding median algebra structures on Euclidean spaces and manifolds.
method Showed local CAT(0) cubulation for median structures on ER homology manifolds.
result Median structures on ER homology manifolds have a local CAT(0) cubulation structure.
FibeRed reduces complex data dimensions while preserving topology.
problem Hard embedding of topologically complex datasets in low-dimensional Euclidean space.
method Modeling datasets with vector bundles, reducing fibers while preserving topology.
result FibeRed learns topologically faithful embeddings in lower dimensions than existing methods.
Study local limits of subgroups in SL3(R).
problem Understanding subgroups of SL3(R) in a specific topology. method Conjugation and Chabauty topology analysis.
result Described local limits of all connected subgroups.
We present a simple-to-apply criterion for recognizing topological groups that are (locally) homeomorphic to LF-spaces.
Every locally compact local group is locally isomorphic to a topological group.
Smoothly embed 3-manifolds in 5-manifolds, simplifying topological to smooth.
problem Embedding 3-manifolds smoothly in 5-manifolds.
method Homotopy and small homotopy to achieve smooth embeddings.
result Locally flat embeddings are homotopic to smooth ones.
We introduce the notion of a local torus action modeled on the standard representation (for simplicity, we call it a local torus action). It is a generalization of a locally standard torus action and also an underlying structure of a locally toric Lagrangian fibration. For a local torus action, we define two invariants…
Expands Bredon's trick for applications in geometry and topology.
problem Local-to-global extension principles in geometric and topological contexts.
method Novel applications and frameworks for stratified pseudomanifolds, Ricci flow, and persistent homology.
result Establishes Bredon's trick as a unifying framework.
We present short proofs of all known topological properties of general Busemann G-spaces (at present no other property is known for dimensions more than four). We prove that all small metric spheres in locally G-homogeneous Busemann G-spaces are homeomorphic and strongly topologically homogeneous. This is a key r…
The paper proves finite topological type theorems for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume.
problem Proving finite topological type theorems for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume.
method The approach removes constraints of sectional curvature or conjugate radius and extends to previous related studies.
result Theorems are proven for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume, without the need for triangle comparison of Toponogov type.
Bredon's trick helps extend local properties to global topological spaces.
problem Extending local properties to global topological spaces.
method Bredon's trick for local properties to global spaces.
result Bredon's trick allows for natural alternative demonstrations of classic results.
New topological complexity measures for neural networks.
problem Measuring complexity of neural network functions.
method Generalized piecewise-linear Morse theory applied to ReLU networks.
result Local complexity can be arbitrarily high.
GNNs may be limited by graph topology, affecting their learning outcomes.
problem Understanding how graph topology influences GNN behavior and performance.
method Investigating the interaction between local topological features and GNN message-passing schemes.
result Locally similar neighborhoods can lead to consistent node representations, affecting GNN performance.
Study local and global aspects of complex plane curve embeddings.
problem Local and global problems of complex plane curve embeddings.
method Braid monodromy, local and global analysis.
result Historical progress in understanding complex plane curve embeddings.
The Chabauty space of a topological group is the set of its closed subgroups, endowed with a natural topology. As soon as n>2, the Chabauty space of Rn has a rather intricate topology and is not a manifold. By an investigation of its local structure, we fit it into a wider, but too wild, class of topological space…
Connected sums defined for codimension two locally flat submanifolds in higher dimensions.
problem Defining connected sums for codimension two locally flat submanifolds in various dimensions.
method Using results from higher dimensional topological manifolds and four-manifolds, defining connected sums for codimension two locally flat submanifolds.
result A well-defined connected sum exists up to orientation preserving homeomorphism.
It is shown that the hyperspace of all nonempty closed subsets $\Cld_{AW}(X)$ of a separable metric space X endowed with the Attouch-Wets topology is homeomorphic to a separable Hilbert space if and only if the completion of X is proper, locally connected and contains no bounded connected component, X is topologi…
The paper examines the topology of quaternionic toric actions on manifolds.
problem Understanding the global topology of manifolds with quaternionic toric actions.
method Established toric, differential, and tetraplectic foundations. Constructed spectral sequences for the orbit projection to describe cohomology and K-theory.
result Explicit descriptions of cohomology and K-theory for manifolds with quaternionic toric actions, extending complex toric topology.
A new clustering algorithm GDT improves on HDBSCAN for uneven data.
problem Data clustering with uneven distribution and high noise.
method GDT combines local and global structures, forming local clusters and estimating a global topological graph based on connectivity between clusters.
result GDT achieves SOTA performance on various datasets with low time complexity.
Authors classify 3D locally standard T-pseudomanifolds under weaker conditions.
problem Classifying equivariant homeomorphism types of 3D locally standard T-pseudomanifolds.
method Introduced and classified by characteristic data under homotopy equivalence condition.
result Condition for classification can be removed when dimension is at most three.
Invariants for colored links found, with topological protection of certain knots.
problem Finding invariants for colored links and understanding their topological stability.
method Equivariant bordism groups of Conner and Floyd, topological surgeries.
result Topological protection of certain tricolored knots, with specific transformations.
Ricci flow smooths locally collapsing manifolds with controlled curvature.
problem Locally collapsing manifolds with controlled Ricci curvature.
method Ricci flow for a definite period of time, detecting collapsing infranil fiber bundles.
result Topological conditions detect collapsing infranil fiber bundles.
Paper shows Ricci flow with surgery converging to Taub-NUT metric.
problem Existence of Ricci flow with surgery and topology change.
method Ricci flow with surgery and local topology change.
result Flow converges to Taub-NUT metric on \mathbb{R}^4.
We use barcodes to analyze neural networks' loss surfaces, revealing important properties.
problem Understanding the topology of neural networks' loss surfaces.
method Topological data analysis using Morse complexes and barcodes.
result Barcodes of local minima are located in a small part of the loss function's range and decrease with network depth and width.
Study of affine transformations on topological manifolds, focusing on local freeness and solvability.
problem Understanding the action of affine transformations on topological manifolds.
method Analyzing the subgroup of homeomorphisms that lift to affine transformations and studying the resulting foliation.
result The connected component of the subgroup acts locally freely and is solvable, with additional properties for polynomial manifolds.
This paper discusses topological and locally linear actions of finite groups on S4. Local linearity of the orientation preserving actions on S4 forces the group to be a subgroup of SO(5). On the other hand, orientation reversing topological actions of "exotic" groups G (i.e. G⊂O(5)) on S4 are …
The main purpose of this paper is to give a topological and symplectic classification of completely integrable Hamiltonian systems in terms of characteristic classes and other local and global invariants.
Let f:M→M be an expansive homeomorphism with dense topologically hyperbolic periodic points, M a compact manifold. Then there is a local product structure in an open and dense subset of M. Moreover, if some topologically hyperbolic periodic point has codimension one, then this local product structure is …
Topological attention improves forecasting of univariate time series.
problem Forecasting univariate time series using local topological features.
method Topological attention mechanism that integrates local topological properties into forecasting models.
result Topological attention leads to state-of-the-art performance on the M4 benchmark.
We examine the topology of various spaces of locally homogeneous affine manifolds which arise from the classification result of Opozda [B. Opozda, A classification of locally homogeneous connections on 2-dimensional manifolds, Differential Geom. Appl. 21 (2004), 173-198.] as orbits of the action of GL(2,R) (…
Paper classifies pseudomanifolds over stratified spaces.
problem Classifying pseudomanifolds over stratified spaces.
method Introducing locally standard T-pseudomanifolds and using characteristic data. result Locally standard T-pseudomanifolds over topological stratified pseudomanifolds are classified by their characteristic data. We show that in any triangulated 3-manifold, every index n topologically minimal surface can be transformed to a surface which has local indices (as computed in each tetrahedron) that sum to at most n. This generalizes classical theorems of Kneser and Haken, and more recent theorems of Rubinstein and Stocking, and is t…
We connect topological changes that can occur in 3-space via surgery, with black hole formation, the formation of wormholes and new generalizations of these phenomena, including relationships between quantum entanglement and wormhole formation. By considering the initial manifold as the 3-dimensional spatial sectio…
We use an idea of Wang and Yau to give a new definition of quasi-local mass for a topological sphere in an initial date set. The new definition modifies Brown-York's definition by using certain spinor norm as lapse function. And it requires mean curvature of the topological sphere satisfies apparent horizon conditions,…
Topologically protected vortex knots and links are proposed and proven.
problem Decaying of tangled vortex structures through local reconnections and strand crossings.
method Proposed and proven topologically protected vortex structures using non-Abelian topological vortices.
result Existence of topologically protected Q8-colored links and classification using the Q-invariant. v2: An additional assumption was added in Theorem 4.8. In order to show that a connected abelian group is admissible on the site of locally compact spaces we must in addition assume that it is locally topologically divisible. This condition is used in the proof of Lemma 4.62.
We show that the topological classification and the smooth classification are generically the same for certain families of plane curves in a semi-local case(the double local case). Especially we give the normal form of transversely jointed two families of plane curves with second order contact at the envelope.
In this study, we present and analyze a framework for geometric and topological estimation for mapping of unknown environments. We consider agents mimicking motion behaviors of cyborg insects, known as biobots, and exploit coordinate-free local interactions among them to infer geometric and topological information abou…
Unified analysis for decentralized SGD across various topologies and updates.
problem Analysis of decentralized SGD methods with changing topologies and local updates.
method Unified convergence analysis covering local SGD updates and adaptive network topology.
result Universal convergence rates for smooth problems, interpolating between heterogeneous and iid-data settings.