Study on large scale surjections and their properties.
problem Properties of large scale surjections and their invariants.
method Coarse disjointness and large scale n-to-1 functions.
result Characterization of asymptotic dimension for coarse structures.
This paper is devoted to dualization of paracompactness to the coarse category via the concept of R-disjointness. Property A of G.Yu can be seen as a coarse variant of amenability via partitions of unity and leads to a dualization of paracompactness via partitions of unity. On the other hand, finite decomposition com…
Decomposes ultrametric spaces into scaled simplices.
problem Understanding the structure of ultrametric spaces.
method Introducing metric resolutions and coarse disjoint union.
result Constructs universal spaces for asymptotic dimension 0.
Study measures complexity of surfaces using a new graph to prove group properties.
problem Understanding the complexity and structure of mapping class groups.
method Introduces a non-peripheral curve graph and uses it to analyze the structure of mapping class groups.
result Proves properties of the mapping class group based on the complexity measure.
Defines a new free product for coarse spaces.
problem No specific problem stated; focuses on a new mathematical concept.
method Defines and analyzes free products for coarse spaces.
result Free products preserve coarse properties and have a dimension bound.
Study on coarse homotopy groups, proving equivalence and matching with usual homotopy groups.
problem Understanding coarse homotopy groups in abstract coarse structures.
method Developed geometric triangulation techniques for cones to prove the equivalence and matching of coarse homotopy groups with usual homotopy groups.
result Coarse homotopy groups of the cone of a compact simplicial complex coincide with the usual homotopy groups of the underlying compact simplicial complex.
Study semi-coarse spaces' homotopy and homology, extending coarse geometry.
problem Extend homotopy and homology concepts to semi-coarse spaces.
method Analyze homotopy and construct homology groups invariant under semi-coarse homotopy equivalence.
result Show semi-coarse homology is isomorphic to Vietoris-Rips homology for graphs.
Coarse assembly maps generalize known results for K-homology.
problem Generalizing known results for K-homology to other coarse homology theories.
method Constructing coarse assembly maps as natural transformations between coarse homology theories.
result Explicit calculation of the domain of the coarse assembly map in terms of locally finite homology theory.
Defines coarse versions of property C and decomposition complexity.
problem Tackles coarse versions of property C and decomposition complexity.
method Uses coarse spaces to define coarse versions of asymptotic property C and decomposition complexity.
result Proves coarse property C implies coarse property A and shares features with metric analogs.
Defines coarse cohomology of space complements, proving new duality results.
problem Defining and studying coarse cohomology of space complements.
method Introducing a model space, new approach to PD spaces, homological criterion.
result Proves new versions of coarse Poincaré duality and Alexander duality.
Research preserves coarse property C and related dimensions through direct products.
problem Preserving properties in coarse geometry through direct products.
method Demonstrates preservation of coarse property C and related dimensions through finite coarse direct products.
result Coarse property C and related dimensions are preserved by direct products.
In this article, we introduce the notion of a functor on coarse spaces being coarsely excisive- a coarse analogue of the notion of a functor on topological spaces being excisive. Further, taking cones, a coarsely excisive functor yields a topologically excisive functor, and for coarse topological spaces there is an ass…
Interprets coarse symbol and index classes for Callias type operators.
problem Understanding coarse geometry and index classes for Callias type operators.
method Interprets coarse symbol and index classes in terms of K-theory classes of coarse corona.
result Local positivity and invertibility conditions are incorporated into support conditions in K-theory.
New maps for large-scale geometry factorize into monotone and light.
problem Large-scale analogues of topological monotone and light maps.
method Introducing coarsely monotone and coarsely light maps, showing factorization system, and proving stability.
result Coarsely monotone maps are stable under pullbacks in the coarse category.
We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that are known to be counterexamples to the coarse Baum-Connes conjecture. In particula…
New concept of coarse medians for higher rank symmetric spaces.
problem Understanding medians in higher rank symmetric spaces.
method Introducing coarse r-median spaces and proving their existence. result Existence of coarse higher medians on divisible and quasi-homogeneous convex domains.
We introduce a notion of fibred coarse embedding into Hilbert space for metric spaces, which is a generalization of Gromov's notion of coarse embedding into Hilbert space. It turns out that a large class of expander graphs admit such an embedding. We show that the maximal coarse Baum-Connes conjecture holds for metric …
Following Roe and others (see, e.g., [MR1451755]), we (re)develop coarse geometry from the foundations, taking a categorical point of view. In this paper, we concentrate on the discrete case in which topology plays no role. Our theory is particularly suited to the development of the_Roe (C*-)algebras_ C*(X) and their K…
The study explores ends in coarse homotopy of proper geodesic spaces.
problem Understanding ends in coarse homotopy of proper geodesic spaces.
method Recontextualizing ends as a functor and proving properties of coarse path components.
result Existence of a natural surjection from coarse path components to ends, not always an injection.
Paper develops coarse homotopy theory for geometric group theory.
problem Problems in geometric group theory and higher index theory.
method Proves a Coarse Lifting Lemma for certain surjective maps.
result Obtains results analogous to classical topological results for quotients.
Classifies homeomorphism groups of countable Stone spaces up to coarse equivalence.
problem Classifying non-locally compact topological groups using geometric group theory.
method Classification based on coarsely bounded sets and quasi-isometry.
result Groups in the second class are quasi-isometric to the Hamming cube.
The paper studies properties of group relations induced by compatible coarse structures.
problem Properties of asymptotic resemblance relations on groups.
method Generalization of asymptotic dimension and introduction of set theoretic coupling.
result Groups with compatible coarse structures that admit a set theoretic coupling are asymptotic equivalent.
We investigate the coarse homology of leaves in foliations of compact manifolds. This is motivated by the observation that the non-leaves constructed by Schweitzer and by Zeghib all have non-finitely generated coarse homology. This led us to ask whether the coarse homology of leaves in a compact manifold always has to …
Paper relates asymptotic dimension to cofinal dimension using coarse proximities.
problem Relating asymptotic dimension to cofinal dimension in metric spaces.
method Introducing coarse proximities and inverse limit constructions.
result Asymptotic dimension is bounded by coarse cofinal dimension and cofinal dimension of Higson corona.
Coarse homotopy theory connects Euclidean cones to shape theory of compact spaces.
problem Establishing connections between coarse homotopy theory and shape theory.
method Using pointed shape invariants and inverse mapping telescopes.
result Proving two compact spaces are strong shape equivalent if their Euclidean cones are coarsely homotopy equivalent.
Unique median structures found in hyperbolic spaces.
problem Uniqueness of median structures in hyperbolic spaces.
method Analyzing product of hyperbolic spaces and properties of relative hyperbolicity.
result Non-hyperbolic pants graphs can have unique median structures.
This paper extends boundary embedding results to coarsely convex spaces.
problem Generalizing boundary embedding results to coarsely convex spaces.
method Generalizing Dydak and Virk's work on Gromov hyperbolic spaces to coarsely convex spaces.
result Maps between coarsely convex spaces induce continuous maps between their boundaries.
Constructs index class for manifolds with group action.
problem Index class construction for manifolds with group action.
method Constructs coarse index class with support condition using equivariant Dirac operator.
result Shows coarse relative index theorem and compatibility with suspension isomorphism.
The paper shows how coarse embeddings affect homological Dehn functions.
problem Characterizing groups with coarse embeddings into hyperbolic groups.
method Demonstrates a coarse embedding theorem for homological filling functions.
result Characterizes groups with coarse embeddings into hyperbolic groups of geometric dimension 2.
The study of smoothing arcs and curves on surfaces, proving tautness and arc length spectrum properties.
problem Analyzing the geometric and combinatorial effects of smoothing intersections in arcs or curves.
method Geometric and combinatorial analysis, proving tautness and arc length spectrum properties.
result Shortest arcs with self-intersections have exactly or at most one more self-intersection than the self-intersection number.
Machine learning improves coarse-graining of molecular dynamics models.
problem Creating accurate coarse-grained models for molecular dynamics simulations.
method Reformulated coarse-graining as a supervised machine learning problem using statistical learning theory and deep learning (CGnets).
result CGnets can capture multi-body terms and all-atom explicit-solvent free energy surfaces with fewer coarse-grained beads.
Generalizes Bestvina's Z-boundaries to coarse Z-boundaries.
problem Establishing properties of Z-boundaries for groups. method Introducing a new concept of a 'coarse Z-boundary' and proving theorems about it. result Admitting a coarse Z-boundary is a pure quasi-isometry invariant. Recent research in coarse geometry revealed similarities between certain concepts of analysis, large scale geometry, and topology. Property A of G.Yu is the coarse analog of amenability for groups and its generalization (exact spaces) was later strengthened to be the large scale analog of paracompact spaces using parti…
Study on rigidity of warped cones and expanders.
problem Coarse geometry of warped cones and expanders.
method Use of coarse topology for warped cones, including computation of coarse fundamental groups.
result Rigidity theorem for coarse geometry of warped cones.
Groups with specific properties have similar cubulations and coarse median structures.
problem Understanding the structure of certain groups through cubical coarsening.
method Analyzing right-angled Artin and Coxeter groups, focusing on automorphisms and cubulations.
result Automorphisms of specific groups preserve coarse median structures and have nice fixed subgroups.
We investigate the fixed point property of the group actions on a coarse space and its Higson corona. We deduce the coarse version of Brouwer's fixed point theorem.
The paper characterizes arithmetic metrics in coarsely geometric settings.
problem Characterizing arithmetic metrics in coarsely geometric settings.
method Using coarse-geometric commensurators and under the Hilbert-Smith conjecture.
result Positive answer in general and unconditional for specific cases.
Paper proves a relative version of coarse Alexander duality and applies it to Jordan cycles.
problem Proving a relative version of coarse Alexander duality.
method Introduced a relative Čech homology satisfying Eilenberg-Steenrod Exactness Axiom.
result Jordan cycle invariant in proving existence of certain 3-manifold groups.
Develops Patterson-Sullivan theory for coarse cocycles.
problem None explicitly stated in the abstract.
method Theory of Patterson--Sullivan measures for coarse cocycles of convergence groups.
result Existence, uniqueness, and ergodicity results for Patterson-Sullivan measures under geometric assumptions.
Proposes LUCBRank for efficient coarse ranking with adaptive sampling.
problem Active coarse ranking of items into clusters with approximate rank.
method PAC algorithm LUCBRank for adaptive sampling from reward distributions.
result LUCBRank outperforms baseline methods in synthetic and real-world data.
Machine learning generates coarse-grained force fields for molecular dynamics.
problem Creating thermodynamically consistent coarse-grained models for larger systems.
method Hybrid architecture using graph neural networks to learn molecular features.
result Framework reproduces thermodynamics for small biomolecular systems.
Characterizes quasi-isometric embeddings in coarsely Lipschitz category.
problem Understanding quasi-isometric embeddings in geometric terms.
method Formalizes quasi-isometric embeddings as regular monomorphisms in coarsely Lipschitz category.
result Quasi-isometric embeddings are equivalently characterised as effective, strong, or extremal monomorphisms.
Efficient algorithms learn from coarse labels instead of fine grained ones.
problem Learning from coarse labels when fine labels are unavailable.
method Formalized coarse label settings, used a reduction for SQs, and provided efficient algorithms.
result Any problem learnable from fine labels can be learned efficiently from coarse labels.
Abstract: Formalizes metric spaces with coarse properties, generalizing finite decomposition complexity.
problem Understanding metric spaces with coarse properties.
method Formalizing and generalizing finite decomposition complexity.
result Determines sufficient conditions for metric spaces to satisfy Property A.
Study groups admitting unbounded quasimorphisms to R with coarsely-connected quasikernel.
problem Understanding PD3 groups and their properties. method Coarse generalization of Shapiro's lemma, homological isoperimetric inequalities, and Margolis's coarse homological algebra.
result Groups admitting unbounded quasimorphisms to R with coarsely-connected quasikernel are either torus or Klein-bottle bundles over S^1, or quasiisometric to Riemannian manifolds.
Quantum cellular automata form a homology theory.
problem Understanding the topological structure of quantum cellular automata.
method Formal properties of coarse homology theories.
result Quantum cellular automata naturally form the degree-zero part of a coarse homology theory.
GDML learns effective CG models from all-atom data.
problem Learning effective coarse-grained force fields efficiently.
method Ensemble learning with stratified sampling and GDML.
result GDML yields smaller free energy error than neural networks.
New coarse LS-category introduced for groups and spaces.
problem Large-scale topological properties of groups and spaces.
method Introducing a coarse analog of Lusternik-Schnirelmann category for metric spaces.
result Established lower and upper bounds for geometrically finite and bicombable groups.