The aim of this paper is to provide a gentle introduction to Chabauty topology, while very little background knowledge is assumed. As an example, we provide pictures for the Chabauty space of C*. Note that the description of this space is not new; however the pictures are novel.
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
168,742 papers · 148 categories
Trend · papers per month
48 results for “Chabauty topology”
We describe an explicit metric that induces the Chabauty topology on the space of closed subsets of a proper metric space M.
The space of closed subgroups of math.GT
The Chabauty space of a topological group is the set of its closed subgroups, endowed with a natural topology. As soon as , the Chabauty space of 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…
The space of closed subgroups of a locally compact topological group is endowed with a natural topology, called the Chabauty topology. Let X be a symmetric space of noncompact type, and G be its group of isometries. The space X identifies with the subspace of maximal compact subgroups of G : taking the closure gives ri…
Space of hyperbolic surfaces is path-connected.
problem Topology of hyperbolic surfaces and their subspaces.
method Constructing paths using Fenchel-Nielsen coordinates and shrinking curves.
result Path-connectivity of the space of hyperbolic surfaces.
Let G be a real semisimple Lie group with finite center, with a finite number of connected components and without compact factor. We are interested in the homogeneous space of Cartan subgroups of G, which can be also seen as the space of maximal flats of the symmetric space of G. We define its Chabauty compactification…
The space of closed subgroups of a locally compact topological group is endowed with a natural topology, called the Chabauty topology. We completely describe the space of closed sugroups of the group RxZ, which is not trivial : for example, its fundamental group is uncountable.
Study continuous paths in discrete subgroups of hyperbolic space, proving combination and decomposition theorems.
problem Understanding continuous paths in discrete subgroups of hyperbolic space.
method Combination theorem and chromatography technique.
result Construction of an exotic path of discrete subgroups with no isomorphic subgroups.
The paper finds free semigroups in dense subgroups of Lie groups with critical exponents arbitrarily close to the subgroup's.
problem Finding free semigroups with critical exponents arbitrarily close to a subgroup's in dense subgroups of Lie groups.
method Analyzing Zariski dense discrete subgroups of Lie groups, showing the existence of free semigroups with critical exponents arbitrarily close to the subgroup's.
result The existence of free semigroups with critical exponents arbitrarily close to the subgroup's in dense subgroups of Lie groups.
In this paper we describe the local limits under conjugation of all closed connected subgroups of in the Chabauty topology.
Chabauty Limits of Subgroups of math.GT
We study the Chabauty compactification of two families of closed subgroups of . The first family is the set of all parahoric subgroups of . Although the Chabauty compactification of parahoric subgroups is well studied, we give a different and more geometric proof using various Le…
Classifies limits of groups of involutions in SL(2,F) over local fields.
problem Classifying limits of groups of involutions in SL(2,F) over local fields.
method Classifying involutions, proving polar decompositions, classifying limits.
result Classification of Chabauty limits of various groups of involutions.
Study of lattices and subgroups in PSL2(R) with grafting continuity.
problem Topology of subgroups in PSL2(R) and their properties.
method Identifying spaces of lattices and elementary subgroups, proving continuity of conformal grafting.
result Spaces of lattices are fiber orbibundles over moduli space, and closures have specific topological properties.
We give a complete description of the closure of the space of one-generator closed subgroups of PSL2(R) for the Chabauty topology, by computing explicitly the matrices associated with elements of Aut(D) = PSL2(R), and finding quantities parametrizing the limit cases. Along the way, we investigate under what conditions …
This paper gives a detailed analysis of the Cannon--Thurston maps associated to a general class of hyperbolic free group extensions. Let denote a free groups of finite rank and consider a \emph{convex cocompact} subgroup , i.e. one for which the orbit map from into the free factor comp…
The paper develops algorithms and topological invariants for distinguishing dynamic systems.
problem Distinguishing the topological type of surfaces and functions in dynamic systems.
method Construction of algorithms and topological invariants using discrete topological structures.
result The development of discrete topological structures for topological equivalence of dynamic systems.
Study of digital topology concepts like hyperspaces and function graphs.
problem Adapting classical topology concepts to digital topology.
method Define digital hyperspaces and function graphs, study their properties.
result Some relationships and graphical properties of digital hyperspaces and function graphs.
The paper explores conditions for topological rigidity in quotients of the Davis complex.
problem Understanding when quotients of the Davis complex are topologically rigid.
method Analyzing quotients of the Davis complex of right-angled Coxeter groups and conditions on defining graphs.
result Introduction of infinitely many infinite topologically rigid subclasses.
Aggregation and structural phase transitions of semiflexible polymer bundles: a braided circuit topology approachcond-mat.soft
A new approach uses circuit topology to study complex polymer interactions.
problem Understanding structural phase transitions in entangled polymer systems.
method Braided circuit topology framework for multiple-chain systems.
result Circuit topological motif fractions are effective order parameters for structural transitions.
Novel theory combines combinatorial and topological elements.
problem Understanding combinatorial phenomena at the intersection of topology.
method Synthesizes combinatorial and topological approaches with a new framing concept.
result Framed combinatorial spaces exhibit better behavior than classical spaces.
Topological Data Analysis and Topological Deep Learning Beyond Persistent Homology -- A Reviewmath.HO
This review explores TDA and TDL beyond persistent homology.
problem Limitations of persistent homology in capturing topological invariants and homotopic evolution.
method Spectral representations, sheaf theory, Mayer topology, interaction topology, differential topology, geometric topology.
result Review of topological tools for various data types.
New proof for some knots being topologically slice.
problem Understanding which knots are topologically slice.
method Equivariant topological slice disks for strongly negative amphichiral knots.
result Strongly negative amphichiral knots with trivial Alexander polynomial are equivariantly topologically slice.
Constructs algorithms to recognize and classify 2D surfaces.
problem Recognizing and classifying 2D surfaces in dynamic systems.
method Discrete topological structures and algorithms for simplicial and CW-complexes.
result Determines the topological type of 2-manifolds.
Paper constructs continuous families of topological Morse functions.
problem Existence and deformability of topological Morse functions.
method Simple construction of continuous families of topological Morse functions.
result Gives a construction of continuous families of topological Morse functions.
Novel M-theory approach classifies topological phases of matter.
problem Classifying and understanding topological phases of matter.
method Establishing a correspondence between (2+1)d topological field theories and non-hyperbolic 3-manifolds, identifying topological phases from internal wrapped 3-manifolds.
result Paves a new route toward the classification of topological phases of matter, including fermionic and non-unitary phases.
New smoothing techniques for topological surfaces in 4-manifolds.
problem Topological isotopy of surfaces in smooth 4-manifolds.
method Combining Quinn's smoothing theory with Gabai's light bulb theorem and other developments.
result Proves topological = smooth results for certain disks and spheres.
Diffeological spaces are generalizations of smooth manifolds which include singular spaces and function spaces. For each diffeological space, Iglesias-Zemmour introduced a natural topology called the -topology. However, the -topology has not yet been studied seriously in the existing literature. In this paper, we…
Critiques incorrect fixed point assertions in digital topology.
problem Incorrect or incorrectly proven fixed point assertions in digital topology.
method Critical review of existing assertions.
result Identifies and critiques incorrect fixed point assertions.
We give a foundational account on topological racks and quandles. Specifically, we define the notions of ideals, kernels, units, and inner automorphism group in the context of topological racks. Further, we investigate topological rack modules and principal rack bundles. Central extensions of topological racks are then…
The compact-open topology on the homeomorphism group of a surface without boundary is minimalmath.GT
Minimal topology on surface homeomorphisms proven.
problem Proving the compact-open topology is minimal for surface homeomorphisms.
method Combining Hausdorff group topology properties and automatic continuity results.
result Compact-open topology is unique Hausdorff separable group topology on surface homeomorphisms.
Study topological constraints for minimal hyperbolic surface laminations.
problem Understanding which topological surfaces can be leaves in minimal hyperbolic surface laminations.
method Analyzes topological obstructions and embeddings of Cantor tree-like surfaces.
result All possible topological types of leaves can be simultaneously embedded in a lamination.
This paper proposes a new method for learning covers of geometric datasets to improve topological inference and visualization.
problem Improving topological inference and visualization of large-scale geometric datasets.
method Proposes a method for learning topologically-faithful covers of geometric datasets using optimization.
result Simplicial complexes obtained from learned covers outperform standard methods in terms of size and representation of large-scale topology.
Proposes TSBP for matching topological signal distributions.
problem Matching signal distributions on topological domains.
method Topological Schrödinger Bridge (TSBP) with linear topology-aware stochastic dynamics.
result Derives closed-form topological SB (TSB) for Gaussian boundary distributions.
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…
Unified framework for T-duality in both trivial and non-trivial topologies.
problem Unified description of T-duality for metrics and B-fields in non-trivial topology.
method Developed a new unifying framework for T-duality.
result Unified description of T-duality for metrics and B-fields in non-trivial topology.
Paper introduces a new topological loss for better convergence.
problem Optimizing topological losses for model's desired topological behavior.
method Introduces a new regularized topology-aware loss function.
result Guarantees efficient optimization of the new loss function.
Survey on hyperplane arrangements and their topology.
problem Topology of hyperplane arrangements.
method Focus on the relationship between topology and real structure.
result Relationship between topology and real structure of hyperplane arrangements.
Concepts of polymer statistical topologycond-mat.stat-mech
I review few conceptual steps in analytic description of topological interactions, which constitute the basis of a new interdisciplinary branch in mathematical physics, "Statistical Topology", emerged at the edge of topology and statistical physics of fluctuating non-phantom rope-like objects. This new branch is called…
Lectures on topological field theories and differential cohomology.
problem Exploring topological field theories and their connections to differential cohomology.
method Introduction to topological field theory and generalized Abelian gauge theories.
result Explains the relationship between topological field theories and differential cohomology.
A novel method extracts topological features from word embeddings for text classification.
problem High dimensional and noisy text representations in natural language processing.
method Persistent homology for topological data analysis on word embeddings.
result Topological features outperform conventional text mining features on long textual documents.
Quantum physics model uses knot theory for fragile topology.
problem Modeling quantum physics' fragile topology.
method Knot theoretic algorithm.
result Quantum physics' fragile topology modeled.
Study topological hyperbolicity of moduli spaces of elliptic surfaces.
problem Characterize the largeness of the topological fundamental group of complex varieties.
method Introduce topological hyperbolicity and provide supporting evidence for moduli spaces of elliptic surfaces.
result Establish a weak form of topological hyperbolicity for moduli spaces of elliptic surfaces of Kodaira dimension one.
Topology aids in solving machine learning classification problems.
problem Machine learning classification problems.
method Classical topology applied to neural networks.
result Topology guides neural network architecture and training.
The paper explores the topology of polygonal meshes and their properties.
problem Understanding the topological properties of polygonal meshes.
method Overview of topological concepts, definitions of intrinsic and extrinsic topology, proofs of Euler and Euler-Poincaré formulas, and discussion on cutting meshes.
result Detailed understanding and definitions of polygonal mesh topology, including intrinsic and extrinsic properties.
New method constructs Stein surfaces using topological isotopy.
problem Creating Stein surfaces in complex surfaces.
method Combining Freedman's topology with Eliashberg's holomorphic theory.
result Every tame 2-complex can be isotoped to a Stein surface.
Topology-GS improves 3D GS for better structural and feature integrity.
problem Compromised pixel-level and feature-level integrity in 3D GS.
method Incorporates Local Persistent Voronoi Interpolation (LPVI) and PersLoss based on persistent homology.
result Topology-GS outperforms existing methods in PSNR, SSIM, and LPIPS metrics.
Novel tRSA combines geometry and topology for brain and model analysis.
problem Traditional RSA overlooks topological information in neural representations.
method Topological RSA (tRSA) using nonlinear monotonic transforms.
result Robust model comparisons and novel insights into neural computation.
Extends topological groupoids and studies their properties.
problem Understanding topological groupoid extensions and their properties.
method Introduces topological groupoid extensions and relates them to gerbes over topological stacks.
result Properties of gerbes over Serre, Hurewicz stacks are studied.