Related papers: Maximally Persistent Cycles in Random Geometric Co…

We construct persistent bundles over configuration spaces of hard spheres and use the characteristic classes of these persistent bundles to give obstructions for embedding problems. The configuration spaces of $k$-hard spheres ${\rm…

The persistent homology of a stationary point process on ${\bf R}^N$ is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops,…

Persistent homology, the study of holes that appear in data as one thickens balls centered around its points over time, has theoretically guaranteed stability. That is, small data perturbations guarantee small changes in the lifetimes of…

In this work we investigate the parallel computation of homology using the Mayer-Vietoris principle. We present a two stage approach for parallelizing persistence. In the first stage, we produce a cover of the input cell complex by…

We extend classical tools from rational homotopy theory to topological data analysis by introducing persistent Sullivan minimal models of persistent topological spaces. Our main result establishes that the interleaving distance between such…

Persistent homology is an important methodology in topological data analysis which adapts theory from algebraic topology to data settings. Computing persistent homology produces persistence diagrams, which have been successfully used in…

Given a finite set in a metric space, the topological analysis generalizes hierarchical clustering using a 1-parameter family of homology groups to quantify connectivity in all dimensions. The connectivity is compactly described by the…

In this note, we study the emergence of Hamiltonian Berge cycles in random $r$-uniform hypergraphs. For $r\geq 3$, we prove an optimal stopping-time result that if edges are sequently added to an initially empty $r$-graph, then as soon as…

A tight Hamilton cycle in a $k$-uniform hypergraph ($k$-graph) $G$ is a cyclic ordering of the vertices of $G$ such that every set of $k$ consecutive vertices in the ordering forms an edge. R\"{o}dl, Ruci\'{n}ski, and Szemer\'{e}di proved…

Persistent homology provides a robust methodology to infer topological structures from point cloud data. Here we explore the persistent homology of point clouds embedded into a probabilistic setting, exploiting the theory of point…

We present a tight extremal threshold for the existence of Hamilton cycles in graphs with large minimum degree and without a large ``bipartite hole`` (two disjoint sets of vertices with no edges between them). This result extends Dirac's…

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…

Cycle representatives of persistent homology classes can be used to provide descriptions of topological features in data. However, the non-uniqueness of these representatives creates ambiguity and can lead to many different interpretations…

We introduce $k$-robust clique complexes, a family of simplicial complexes that generalizes the traditional clique complex. Here, a subset of vertices forms a simplex provided it does not contain an independent set of size $k$. We…

We consider random sub-graphs of a fixed graph $G=(V,E)$ with large minimum degree. We fix a positive integer $k$ and let $G_k$ be the random sub-graph where each $v\in V$ independently chooses $k$ random neighbors, making $kn$ edges in…

One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and…

This article has two goals. First, we hope to give an accessible introduction to persistent equivariant cohomology. Given a topological group $G$ acting on a filtered space, persistent Borel equivariant cohomology measures not only the…

We introduce a linear dimensionality reduction technique preserving topological features via persistent homology. The method is designed to find linear projection $L$ which preserves the persistent diagram of a point cloud $\mathbb{X}$ via…

A Hamilton Berge cycle of a hypergraph on $n$ vertices is an alternating sequence $(v_1, e_1, v_2, \ldots, v_n, e_n)$ of distinct vertices $v_1, \ldots, v_n$ and distinct hyperedges $e_1, \ldots, e_n$ such that $\{v_1,v_n\}\subseteq e_n$…

We study sufficient conditions for the existence of Hamilton cycles in uniformly dense $3$-uniform hypergraphs. Problems of this type were first considered by Lenz, Mubayi, and Mycroft for loose Hamilton cycles and Aigner-Horev and Levy…