Related papers: The Shadow of Vietoris--Rips Complexes in Limits

Any quasi-isometry of the complex of curves is bounded distance from a simplicial automorphism. As a consequence, the quasi-isometry type of the curve complex determines the homeomorphism type of the surface.

A metric thickening of a given metric space $X$ is any metric space admitting an isometric embedding of $X$. Thickenings have found use in applications of topology to data analysis, where one may approximate the shape of a dataset via the…

Recently, new exploratory channels have opened up for the physics of highly compact objects, such as gravitational waves and black hole shadows. Moreover, more precise analysis and observations are now possible in the physics of accretion…

For a path-connected metric space $(X,d)$, the $n$-th homotopy group $\pi_n(X)$ inherits a natural pseudometric from the $n$-th iterated loop space with the uniform metric. This pseudometric gives $\pi_n(X)$ the structure of a topological…

Let $S$ be a complete flat surface, such as the Euclidean plane. We determine the homeomorphism class of the space of all curves on $S$ which start and end at given points in given directions and whose curvatures are constrained to lie in a…

We develop a new metric and inverse-limit framework for Deaconu-Renault systems arising from local homeomorphisms between open subsets of locally compact zero-dimensional spaces. Our starting point is the Paterson-type compactification of…

Let R be a commutative noetherian ring. We give criteria for a complex of cotorsion flat R-modules to be minimal, in the sense that every self homotopy equivalence is an isomorphism. To do this, we exploit Enochs' description of the…

The convexity of a set can be generalized to the two weaker notions of reach and $r$-convexity; both describe the regularity of a set's boundary. For any compact subset of $\mathbb{R}^d$, we provide methods for computing upper bounds on…

We investigate the structure of $\omega$-limit (resp. $\alpha$-limit) sets for a monotone map $f$ on a regular curve $X$. %Let $X$ be a regular curve and let $f: X\longrightarrowX$ be a monotone map. We show that for any $x\in X$ (resp. for…

A curious feature of complex scattering potentials v(x) is the appearance of spectral singularities. We offer a quantitative description of spectral singularities that identifies them with an obstruction to the existence of a complete…

Latschev's theorem provides sufficient conditions on a metric space $M$ and $\delta > 0$ for the homotopy type of $M$ to agree with that of the Vietoris-Rips complex $\mathcal{R}^{\delta}(N)$ of any nearby space $N$ in the Gromov-Hausdorff…

In this work, we introduce and study the forbidden-vertices problem. Given a polytope P and a subset X of its vertices, we study the complexity of linear optimization over the subset of vertices of P that are not contained in X. This…

This paper is a study of the interaction between the combinatorics of boundaries of convex polytopes in arbitrary dimension and their metric geometry. Let S be the boundary of a convex polytope of dimension d+1, or more generally let S be a…

We present the `Basic S*' algorithm for computing shortest path through a metric simplicial complex. In particular, given a metric graph, $G$, which is constructed as a discrete representation of an underlying configuration space (a larger…

Utilizing the discrete homotopy methods developed for uniform spaces by Berestovskii-Plaut, we define the critical spectrum Cr(X) of a metric space, generalizing to the non-geodesic case the covering spectrum defined by Sormani-Wei and the…

Given an arbitrary spectral space $X$, we consider the set ${\boldsymbol{\mathcal{X}}}(X)$ of all nonempty subsets of $X$ that are closed with respect to the inverse topology. We introduce a Zariski-like topology on…

This paper presents explicit assumptions for the existence of interleaving homotopy equivalences of both Vietoris-Rips and Lesnick complexes associated to an inclusion of data sets. Consequences of these assumptions are investigated on the…

In this work, we show that, for any simply-connected elliptic space $S$ admitting a pure minimal Sullivan model with a differential of constant length, we have ${\rm TC}_0(S)\leq 2{\rm cat}_0(S)+\chi_{\pi}(S)$ where $\chi_{\pi}(S)$ is the…

We develop a theory of $\times$-homotopy, fundamental groupoids and covering spaces that apply to non-simple graphs, generalizing existing results for simple graphs. We prove that $\times$-homotopies from finite graphs can be decomposed…

The collection of all topologies on a set X forms a complete lattice with respect to the inclusion order, which have been investigated by many researchers. Sobriety is one of the core and extensively studied properties in non-Hausdorff…