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.
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.
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…
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.
Maps between metric spaces preserve asymptotic dimension up to a constant.
problem Understanding how the asymptotic dimension changes under coarse maps.
method Using Higson coronas and coarsely open coarsely n-to-1 maps. result The asymptotic dimension of Y is at most the asymptotic dimension of X plus a constant. Study of mapping class groups on infinite graphs, focusing on their large-scale geometry.
problem Understanding the large-scale geometry of mapping class groups on infinite graphs.
method Using coarse geometry techniques, classify coarsely bounded groups and compute asymptotic dimension.
result Identify conditions for global and local coarsely bounded pure mapping class groups of infinite rank graphs.
Study large-scale geometry of infinite type surface mapping class groups.
problem Classify surfaces based on mapping class group properties.
method Coarse geometry, using Rosendal's framework.
result Classification of surfaces based on group properties.
Study of CB generating sets for infinite-type surfaces.
problem Understanding CB generating sets for infinite-type surfaces.
method Constructing CB generating sets for specific infinite-type surfaces.
result Examples of surfaces with and without CB generating sets.
It is well-known that a paracompact space X is of covering dimension at most n if and only if any map f:X→K from X to a simplicial complex K can be pushed into its n-skeleton K(n). We use the same idea to characterize asymptotic dimension in the coarse category of arbitrary coarse spaces. Cont…
Harmonic maps prove quasi-isometric embeddings are close to unique.
problem Understanding quasi-isometric embeddings between pinched Hadamard manifolds.
method Proving quasi-isometric maps are close to harmonic maps.
result Quasi-isometric maps are within bounded distance from a unique harmonic map.
Study on embedding tree products into groups, distinguishing them.
problem Quasi-isometric embedding of tree products into various groups.
method Using coarse embeddings of products of bushy trees into hierarchically hyperbolic spaces.
result Quasi-isometrically distinguish and rule out embeddings between groups.
Auto-encoders learn atomistic to coarse-grained mappings for molecular dynamics.
problem Simulating large systems in molecular dynamics is computationally expensive.
method Auto-encoders learn both atomistic to coarse-grained mappings and the coarse-grained potential energy function.
result Auto-encoders enable efficient simulation of larger systems in molecular dynamics.
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.
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…
Abstract machinery finds obstructions to uniform positive scalar curvature.
problem Finding obstructions to uniform positive scalar curvature.
method Coarse index theory and embedding submanifolds.
result Abstract machinery constructs wrong way maps on K-theory. Graph neural network predicts optimal coarse-grained mapping operators.
problem Optimal coarse-grained mapping operators selection for molecular dynamics simulations.
method Graph Neural Network (DSGPM) trained on expert-annotated data.
result DSGPM outperforms state-of-the-art methods in graph segmentation.
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.
This study extends a result on quasi-isometry of hyperbolic groups to relatively hyperbolic groups.
problem Classifying groups up to quasi-isometry, focusing on relatively hyperbolic groups.
method Defining quasiconformal maps and showing their equivalence to coarsely cusp-preserving quasi-isometries between Bowditch boundaries.
result Quasiconformal maps between Bowditch boundaries of relatively hyperbolic groups are equivalent to coarsely cusp-preserving quasi-isometries.
Let M be a closed 3-manifold with a given Heegaard splitting. We show that after a single stabilization, some core of the stabilized splitting has arbitrarily high distance with respect to the splitting surface. This generalizes a result of Minsky, Moriah, and Schleimer for knots in S^3. We also show that in the comple…
The paper explores non-amenability in infinite-type surfaces and graphs.
problem Determining non-amenability in mapping class groups of infinite-type surfaces and graphs.
method Analyzes mapping class groups of infinite-type surfaces and graphs, provides examples and exhibits classes of groups.
result Completely determines non-amenability of mapping class groups of infinite-type surfaces and graphs.
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.
Study of pure mapping class groups on infinite graphs.
problem Classifying graphs with specific mapping class groups.
method Completely classified graphs with pure mapping class groups.
result Established semidirect product decomposition and computed first integral cohomology.
Growth rate of Dehn twist lattice points in Teichmüller space is slower than mapping class group lattice points.
problem Analyzing the growth rate of Dehn twist lattice points in Teichmüller space.
method Comparing growth rates of Dehn twist, mapping class group, and multi-twist lattice points.
result The growth rate of Dehn twist lattice points is coarsely asymptotic to $e^{ rac{h}{2}R}$, slower than the mapping class group.
Big mapping class groups of infinite type surfaces have infinite asymptotic dimension.
problem Understanding asymptotic dimension of big mapping class groups of infinite type surfaces.
method Analyzing big mapping class groups with coarsely bounded generating sets and essential shifts.
result Big mapping class groups of infinite type surfaces have infinite asymptotic dimension.
Stable cubulations and bicombings in mapping class groups and Teichmüller spaces.
problem Understanding geometric structures in mapping class groups and Teichmüller spaces.
method Proving stably approximated by CAT(0) cube complexes, applying to broader colorable hierarchically hyperbolic spaces and groups.
result Stable cubulations and bicombings in mapping class groups and Teichmüller spaces, with stable coarse barycenters.
Proposes a new coarse-graining method for statistical mechanics.
problem Uncertainty in coarse-graining and information loss.
method Directed probabilistic model with latent variables, Bayesian approach, MC-EM scheme.
result Predictive posterior distributions provide confidence in coarse-grained models.
Study mapping class group actions on infinite-type surfaces.
problem When does the mapping class group of an infinite-type surface act on a graph with unbounded orbits?
method Introduced a topological invariant to determine the existence of such actions.
result Many big mapping class groups have nontrivial coarse geometry.
Improved CG force-field learning from all-atom data.
problem Training accurate coarse-grained models from all-atom simulations is challenging.
method Optimized force mapping to improve statistical efficiency of force-field learning.
result Substantially improved CG force-fields can be learned from the same simulation data.
Survey on Thurston metric on Teichmüller space, focusing on extremal maps.
problem Constructing extremal Lipschitz maps between hyperbolic surfaces.
method Review of constructions including Thurston's original work.
result Coarse geometry and isometry rigidity of Thurston metric discussed.
Using ideas from shape theory we embed the coarse category of metric spaces into the category of direct sequences of simplicial complexes with bonding maps being simplicial. Two direct sequences of simplicial complexes are equivalent if one of them can be transformed to the other by contiguous factorizations of bonding…
The paper constructs coronas for combable spaces under specific conditions.
problem Constructing boundaries for combable spaces.
method Introducing properness, coherence, and expandingness for combings; constructing coronas using Rips complexes.
result Bijectivity of transgression maps, injectivity of the coarse assembly map, and surjectivity of the coarse co-assembly map for groups.
Using methods from coarse topology we show that fundamental classes of closed enlargeable manifolds map non-trivially both to the rational homology of their fundamental groups and to the K-theory of the corresponding reduced C*-algebras. Our proofs do not depend on the Baum--Connes conjecture and provide independent co…
Applying the DPW version of the theory developed by Burstall and Guest for harmonic maps of finite uniton type, we derive a coarse classification of Willmore two-spheres in Sn+2 in terms of the normalized potential of their (harmonic) conformal Gauss maps. Moreover, for the case of S6, some geometric properties…
Between the category of exact metric spaces with bounded geometry (about which much is known) and the larger category of arbitrary exact metric spaces (about which little is known) lies the intermediate category of asymptotically exact metric spaces. We show that the coarse Baum-Connes assembly map is naturally split s…
Extends Paulin's result to relatively hyperbolic groups.
problem Proving quasi-isometric equivalence between relatively hyperbolic groups.
method Introducing relative quasi-Mobius maps and using coarsely cusp-preserving quasi-isometries.
result Establishes a homeomorphism between Bowditch boundaries inducing quasi-Mobius maps.
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.
Maps and embeddings between hyperbolic spaces and their boundaries studied.
problem Understanding relations between maps and embeddings between relatively hyperbolic spaces and their boundaries.
method Establishing correspondences between quasi-isometric embeddings and quasisymmetric embeddings, using polynomial distortion.
result Characterization of hyperbolic relative groups with polynomial distortion embeddings.
Paper explores coarse embeddings between symmetric spaces and Euclidean buildings, answering open questions.
problem Understanding coarse embeddings between symmetric spaces and Euclidean buildings.
method Generalization of quasi-isometric embeddings, focusing on coarse embeddings without Euclidean factors.
result Rank is monotonous under coarse embeddings when the domain does not contain a Euclidean factor.
The aim of this paper is to investigate properties preserved and co-preserved by coarsely n-to-1 functions, in particular by the quotient maps X→X/∼ induced by a finite group G acting by isometries on a metric space X. The coarse properties we are mainly interested in are related to asymptotic dimension a…
The paper counts mapping classes by Nielsen-Thurston type, finding growth rates for different subsets.
problem Counting mapping classes in Teichmüller space with different subsets.
method Introduced complexity length to measure negative curvature of curve complexes.
result Growth rates for finite-order, reducible, and multitwists subsets.
Study on stable mixed commutator length in coarse group theory.
problem Understanding the large scale behavior of stable mixed commutator length in group theory.
method Introducing a bi-invariant metric function and connecting it to coarse group theoretic structures and invariant quasimorphisms.
result Proved that the coarse kernel of the coarse homomorphism is isomorphic to Z^ℓ as a coarse group.
Geometric models help classify infinite-type surface mapping class groups.
problem Classifying the asymptotic dimension of infinite-type surface mapping class groups.
method Constructing metric graphs of simple arcs and curves preserved by the action of the group, showing coarse and quasi-isometric properties.
result The asymptotic dimension of stable boundedly generated infinite-type surface mapping class groups is infinite.
Uniform lattices in certain semi-simple groups contain Anosov surface subgroups.
problem Understanding surface subgroups in uniform lattices of semi-simple groups.
method Introducing K-Sullivan maps and using coarse geometry of flag manifolds. result Quantitative version of surface subgroup theorem, showing closeness to smooth round circles.
There is a well-known correspondence between infinite trees and ultrametric spaces which can be interpreted as an equivalence of categories and comes from considering the end space of the tree. In this equivalence, uniformly continuous maps between the end spaces are translated to some classes of coarse maps (or even c…
In this paper, the first of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs is an expander and the girth of the graphs tends to infinity, then the coarse Baum-Connes assembly map is injective, but not surjective, for the associated metric space X. Exp…
We consider the notion of dimension in four categories: the category of (unbounded) separable metric spaces and (metrically proper) Lipschitz maps, and the category of (unbounded) separable metric spaces and (metrically proper) uniform maps. A unified treatment is given to the large scale dimension and the small scale …
Embeds persistence diagrams into Hilbert spaces to use kernel methods.
problem No inner product structure on persistence diagrams.
method Shows non-embeddability of persistence diagrams into Hilbert spaces.
result Persistence diagrams with bottleneck distance do not coarse embed into Hilbert spaces.
Advances data-driven coarse-graining for complex systems.
problem Extracting governing equations from high-dimensional, time-scale disparity problems.
method Probabilistic state-space model with Stochastic Variational Inference for sparse Bayesian learning.
result Quantifies predictive uncertainty and reconstructs fine-scale system evolution.