Related papers: Maximally Persistent Cycles in Random Geometric Co…
Manifold reconstruction has been extensively studied for the last decade or so, especially in two and three dimensions. Recently, significant improvements were made in higher dimensions, leading to new methods to reconstruct large classes…
This paper introduces a new percolation model motivated from polymer materials. The mathematical model is defined over a random cubical set in the $d$-dimensional space $\mathbb{R}^d$ and focuses on generations and percolations of…
Topologically ordered phases of matter display a number of unique characteristics, including ground states that can be interpreted as patterns of closed strings. In this paper, we consider the problem of detecting and distinguishing closed…
We introduce a new invariant defined on the vertices of a given filtered simplicial complex, called codensity, which controls the impact of removing vertices on persistent homology. We achieve this control through the use of an interleaving…
We present a parallelizable algorithm for computing the persistent homology of a filtered chain complex. Our approach differs from the commonly used reduction algorithm by first computing persistence pairs within local chunks, then…
We study Hamiltonicity in random subgraphs of the hypercube $\mathcal{Q}^n$. Our first main theorem is an optimal hitting time result. Consider the random process which includes the edges of $\mathcal{Q}^n$ according to a uniformly chosen…
We develop persistent homology in the setting of filtrations of (Cech) closure spaces. Examples of filtrations of closure spaces include metric spaces, weighted graphs, weighted directed graphs, and filtrations of topological spaces. We use…
We study the distances of edges within cliques in a soft random geometric graph on a torus, where the vertices are points of a homogeneous Poisson point process, and far-away points are less likely to be connected than nearby points. We…
In this paper, we study the persistent homology of the offset filtration of algebraic varieties. We prove the algebraicity of two quantities central to the computation of persistent homology. Moreover, we connect persistent homology and…
We consider the degree-Rips construction from topological data analysis, which provides a density-sensitive, multiparameter hierarchical clustering algorithm. We analyze its stability to perturbations of the input data using the…
Graph-based representations of point-cloud data are widely used in data science and machine learning, including epsilon-graphs that contain edges between pairs of data points that are nearer than epsilon and kNN-graphs that connect each…
We consider random filtered complexes built over marked point processes on Euclidean spaces. Examples of our filtered complexes include a filtration of $\check{\textrm{C}}$ech complexes of a family of sets with various sizes, growths, and…
In this article we propose a novel approach for comparing the persistent homology representations of two spaces (filtrations). Commonly used methods are based on numerical summaries such as persistence diagrams and persistence landscapes,…
Proximity complexes and filtrations are central constructions in topological data analysis. Built using distance functions, or more generally metrics, they are often used to infer connectivity information from point clouds. Here we…
We define and study several new interleaving distances for persistent cohomology which take into account the algebraic structures of the cohomology of a space, for instance the cup product or the action of the Steenrod algebra. In…
We study the emergence of loose Hamilton cycles in subgraphs of random hypergraphs. Our main result states that the minimum $d$-degree threshold for loose Hamiltonicity relative to the random $k$-uniform hypergraph $H_k(n,p)$ coincides with…
Poisson processes in the space of $k$-dimensional totally geodesic subspaces ($k$-flats) in a $d$-dimensional standard space of constant curvature $\kappa\in\{-1,0,1\}$ are studied, whose distributions are invariant under the isometries of…
Recently it has been shown that computing the dimension of the first homology group $H_1(K)$ of a simplicial $2$-complex $K$ embedded linearly in $\mathbb{R}^4$ is as hard as computing the rank of a sparse $0-1$ matrix. This puts a major…
We give several results showing that different discrete structures typically gain certain spanning substructures (in particular, Hamilton cycles) after a modest random perturbation. First, we prove that adding linearly many random edges to…
We introduce a refinement of persistent homology that detects simple-homotopy-theoretic phenomena invisible to homology. Given a filtered simplicial complex, we define the Morse complexity profile as the minimal number of critical simplices…