arXiv research

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 …

This paper extends boundary embedding results to coarsely convex spaces.

The paper shows how coarse embeddings affect homological Dehn functions.

Characterizes quasi-isometric embeddings in coarsely Lipschitz category.

Study on embedding tree products into groups, distinguishing them.

Paper explores coarse embeddings between symmetric spaces and Euclidean buildings, answering open questions.

Estimates for graph embeddings into symmetric spaces derived from coarse geometry.

Harmonic maps prove quasi-isometric embeddings are close to unique.

Novel coarse extrinsic curvature for Riemannian submanifolds.

Research shows persistence diagrams embed into Hilbert space, aiding in their analysis.

Paper proves a relative version of coarse Alexander duality and applies it to Jordan cycles.

Maps and embeddings between hyperbolic spaces and their boundaries studied.

For an embedded submanifold $Σ\subset\mathbb{R}^{N}$, Belkin and Niyogi showed that one can approximate the Laplacian operator using heat kernels. Using a definition of coarse Ricci curvature derived by iterating Laplacians, we approximate the coarse Ricci curvature of submanifolds $Σ$ in the same way. For this purpose…

Improves few-shot learning for hierarchical data using hyperbolic space.

Embeds persistence diagrams into Hilbert spaces to use kernel methods.

Abstract machinery finds obstructions to uniform positive scalar curvature.

Constructs free semigroups with critical exponents close to but less than ambient groups.

There is a word metric $d$ on countably generated free group $Γ$ such that $(Γ,d)$ does not admit a coarse uniform embedding into a Hilbert space.

Paper tackles cross-granularity few-shot learning with meta-embedder.

Contextual bandit learning is an increasingly popular approach to optimizing recommender systems via user feedback, but can be slow to converge in practice due to the need for exploring a large feature space. In this paper, we propose a coarse-to-fine hierarchical approach for encoding prior knowledge that drastically …

New framework embeds physics in coarse-grained models without big data.

We prove that two countable locally finite-by-abelian groups G,H endowed with proper left-invariant metrics are coarsely equivalent if and only if their asymptotic dimensions coincide and the groups are either both finitely-generated or both are infinitely generated. On the other hand, we show that each countable group…

BoostNE learns multiple network embeddings from coarse to fine.

This paper is a survey of some of the developments in coarse extrinsic geometry since its inception in the work of Gromov. Distortion, as measured by comparing the diameter of balls relative to different metrics, can be regarded as one of the simplist extrinsic notions. Results and examples concerning distorted subgrou…

We prove that a metric space does not coarsely embed into a Hilbert space if and only if it satisfies a sequence of Poincaré inequalities, which can be formulated in terms of (generalized) expanders. We also give quantitative statements, relative to the compression. In the equivariant context, our result says that a gr…

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…

This study uses persistent homology to analyze complex transitional networks from time series data.

The paper introduces Poincaré profiles for metric spaces and groups, linking them to conformal dimension.

We prove that if $G$ is a non-uniform lattice in a rank-one semi-simple Lie group $\ne Isom(\H^2_\R)$ then $G$ is quasi-isometrically co-Hopf. This means that every quasi-isometric embedding $G\to G$ is coarsely onto and thus is a quasi-isometry.

Defines a new free product for coarse spaces.

Study on coarse homotopy groups, proving equivalence and matching with usual homotopy groups.

Study semi-coarse spaces' homotopy and homology, extending coarse geometry.

Coarse assembly maps generalize known results for K-homology.

Defines coarse cohomology of space complements, proving new duality results.

Research preserves coarse property C and related dimensions through 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…

We introduce large scale analogues of topological monotone and light maps, which we call coarsely monotone and coarsely light maps respectively. We show that these two classes of maps constitute a factorization system on the coarse category. We also show how coarsely monotone maps arise from a reflection in a similar w…

Interprets coarse symbol and index classes for Callias type operators.

The coarse category was established by Roe to distill the salient features of the large-scale approach to metric spaces and groups that was started by Gromov. In this paper, we use the language of coarse spaces to define coarse versions of asymptotic property C and decomposition complexity. We prove that coarse propert…

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.

The complex of domains $D(S)$ is a geometric tool with a very rich simplicial structure, it contains the curve complex $C(S)$ as a simplicial subcomplex. In this paper we shall regard it as a metric space, endowed with the metric which makes each simplex Euclidean with edges of length 1, and we shall discuss its coarse…

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.

Paper develops coarse homotopy theory for geometric group theory.

The study quantifies topological expansion properties of complexes and their embeddings.

Classifies homeomorphism groups of countable Stone spaces up to coarse equivalence.

The paper studies properties of group relations induced by compatible coarse structures.