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.
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 n n -to- 1 1 1 maps. result The asymptotic dimension of Y Y Y is at most the asymptotic dimension of X X X plus a constant. The paper explores properties preserved by coarsely n-to-1 functions in metric spaces.
problem Investigating properties preserved by coarsely n-to-1 functions in metric spaces.
method Analyzing properties related to asymptotic dimension and its generalizations.
result The class of spaces with straight finite decomposition complexity coincides with the class of spaces of countable asymptotic dimension.
Study shows how to approximate functions using neural networks.
problem Approximating measurable functions on hypercube.
method Using affine neural networks to approximate measurable functions.
result Any measurable function can be approximated by a bounded number of neurons.
Learn low-degree functions with few random queries.
problem Learning low-degree functions from limited random queries.
method Learn bounded functions f : { − 1 , 1 } n o [ − 1 , 1 ] f:\{-1,1\}^n o[-1,1] f : { − 1 , 1 } n o [ − 1 , 1 ] of degree at most d d d with L 2 L_2 L 2 -accuracy ε \varepsilon ε and confidence 1 − δ 1-δ 1 − δ from log ( f r a c n δ ) ε − d − 1 C d 3 / 2 log d \log( frac{n}δ)\,\varepsilon^{-d-1} C^{d^{3/2}\sqrt{\log d}} log ( f r a c n δ ) ε − d − 1 C d 3/2 l o g d random queries. result Learn low-degree functions efficiently with logarithmic number of random queries.
Improved decision tree learning guarantees for complex functions.
problem Achieving provable guarantees for decision tree induction with complex target functions.
method Introduces a new splitting criterion that considers correlations between target function and subsets of attributes.
result Proves provable guarantees for all target functions with respect to the uniform distribution, circumventing previous impossibility results.
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.
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…
A neural network predicts coarse-scale basis functions for efficient uncertainty quantification.
problem Efficiently estimating coarse-scale basis functions for multiscale methods.
method Data-driven approach using neural networks fitted to solution samples.
result Significant computational savings for uncertainty quantification tasks.
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.
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.
Defines coarse Ricci curvature on metric measure spaces.
problem Defines a new curvature measure for metric measure spaces.
method Uses Bakry-Emery framework to define coarse Ricci curvature.
result Recover Ricci tensor on smooth Riemannian manifolds.
BCGD algorithm improves training of quantized neural networks.
problem Training quantized deep neural networks at low bit-widths.
method Introduces coarse gradient descent and blended correction for training.
result BCGD achieves high accuracy in quantized neural networks.
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.
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.
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.
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.
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.
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…
New framework embeds physics in coarse-grained models without big data.
problem Lack of big data and computational demand in data-driven coarse-graining.
method Proposes a novel objective based on reverse Kullback-Leibler divergence that incorporates physics in the form of force fields.
result Generative coarse-grained model predicts atomistic configurations and reveals physicochemical CVs.
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 propose a new approach for analyzing price fluctuations in their strongly correlated regime ranging from minutes to months. This is done by employing a self-similarity assumption for the magnitude of coarse-grained price fluctuation or volatility. The existence of a Cramer function, the characteristic function for s…
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 r 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 …
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.
Combines coarse learners for nonparametric probabilistic regression.
problem Predicting full probability density functions without strong assumptions.
method Combining gradient boosted forests trained on coarsened target values.
result Prediction intervals have high fidelity and provide valuable insights.
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 …
We are concerned about the coarse and precise aspects of a priori estimates for Green's function of a regular domain for the Laplacian-Betrami operator on any 3 ≤ n 3\le n 3 ≤ n -dimensional complete non-compact boundary-free Riemannian manifold through the square Sobolev/Nash/logarithmic-Sobolev inequalities plus the rough and s…
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.
Abstract: Unifies small and large scale geometries using linear algebra concepts.
problem Tackles unification of small and large scale geometries.
method Uses analog of multilinear forms from Linear Algebra to compactify and unify various compactifications.
result Simple proofs of generalized theorems in coarse topology, including a new result about Higson coronas.
The free factor complex of rank 4+ fails a combinatorial isoperimetric inequality.
problem Failure of combinatorial isoperimetric inequality in the free factor complex.
method Construction of a coarsely Lipschitz function from the upward link of a free factor to integers.
result A loop in the free factor complex requires linearly growing number of 2-simplices to fill.
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.
Extends coarse index theory to locally compact groups for Callias operators.
problem Developing an equivariant coarse index theory for non-cocompact actions.
method Using admissible modules and localised K K K -theory of group C ∗ C^* C ∗ -algebras. result Equivariant index for Callias operators is a special case of the localised index.
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.
Generalizes Bestvina's Z \mathcal{Z} Z -boundaries to coarse Z \mathcal{Z} Z -boundaries.
problem Establishing properties of Z \mathcal{Z} Z -boundaries for groups. method Introducing a new concept of a 'coarse Z \mathcal{Z} Z -boundary' and proving theorems about it. result Admitting a coarse Z \mathcal{Z} 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…