Related papers: Persistent Sullivan Minimal Models of Metric Space…

In this paper we study variations of the Hopf theorem concerning continuous maps $f$ of a compact Riemannian manifold $M$ of dimension $n$ to $\mathbb{R}^n$. We investigate the case when $M$ is a closed convex $n$-dimensional surface and…

We address the problem of estimating topological features from data in high dimensional Euclidean spaces under the manifold assumption. Our approach is based on the computation of persistent homology of the space of data points endowed with…

We utilize a condition for algebraic curvature operators called surgery stability as suggested by the work of S. Hoelzel to investigate the space of riemannian metrics over closed manifolds satisfying these conditions. Our main result is a…

The design of a metric between probability distributions is a longstanding problem motivated by numerous applications in Machine Learning. Focusing on continuous probability distributions on the Euclidean space $\mathbb{R}^d$, we introduce…

Minimal surfaces in closed 3-manifolds are classically constructed via the Almgren-Pitts approach. The Allen-Cahn approximation has proved to be a powerful alternative, and Chodosh and Mantoulidis (in Ann. Math. 2020) used it to give a new…

In this work, a metric is presented on the set of boundedly-compact pointed metric spaces that generates the Gromov-Hausdorff topology. A similar metric is defined for measured metric spaces that generates the Gromov-Hausdorff-Prokhorov…

We introduce irreducible correspondences that enables us to calculate the Gromov--Hausdorff distances effectively. By means of these correspondences, we show that the set of all metric spaces each consisting of no more than $3$ points is…

Vietoris-Rips metric thickenings have previously been proposed as an alternate approach to understanding Vietoris-Rips simplicial complexes and their persistent homology. Recent work has shown that for totally bounded metric spaces,…

We present a new general framework for metrization of Gromov-Hausdorff-type topologies on non-compact metric spaces. We also give easy-to-check conditions for separability and completeness and hence the measure theoretic requirements are…

Persistent homology is one of the most popular methods in topological data analysis. An initial step in its use involves constructing a nested sequence of simplicial complexes. There is an abundance of different complexes to choose from,…

We present a general method to derive continuity estimates for conditional probabilities of general (possibly continuous) spin models sub jected to local transformations. Such systems arise in the study of a stochastic time-evolution of…

We study distributions of persistent homology barcodes associated to taking subsamples of a fixed size from metric measure spaces. We show that such distributions provide robust invariants of metric measure spaces, and illustrate their use…

We consider the intersection map on the family of non-empty $\omega$-Scott-open sets of the lattice of opens of a topological space. We prove that in a certain class of topological spaces the intersection map forms a continuous retraction…

It is known that parallel manipulators suffer from singular configurations. Evaluating the distance between a given configuration to the closest singular one is of interest for industrial applications (e.g.\ singularity-free path planning).…

The main theorem of this paper asserts that the inclusion of the space of projective Lagrangian planes into the space of Lagrangian submanifolds of complex projective space induces an injective homomorphism of fundamental groups. We…

We consider the question of continuity of limit sets for sequences of geometrically finite subgroups of isometry groups of rank-one symmetric spaces, and prove analogues of classical (Kleinian) theorems in this context. In particular we…

This paper surveys various distance measures for networks and graphs that were introduced in persistent homology. The scope of the paper is limited to network distances that were actually used in brain networks but the methods can be easily…

For Riem(M) the space of Riemannian metrics over a compact 3-manifold without boundary $M$, we study topological properties of the dense open subspace Riem'(M) of metrics which possess no Killing vectors. Given the stratification of…

We construct worst-case examples for the standard reduction algorithm for computing persistent homology. Our constructions are similar to the worst-case examples introduced by Morozov, but we replace the single-triangle arrangement with a…

Time-delay embedding is a fundamental technique in Topological Data Analysis (TDA) for reconstructing the phase space dynamics of time-series data. Persistent homology effectively identifies global topological features, such as loops…