The persistent homology of a stationary point process on RN is studied in this paper. As a
generalization of continuum percolation theory, we study higher dimensional topological …

We review a collection of models of random simplicial complexes together with some of the
most exciting phenomena related to them. We do not attempt to cover all existing models …

Persistence diagrams are efficient descriptors of the topology of a point cloud. As they do not
naturally belong to a Hilbert space, standard statistical methods cannot be directly applied to …

This paper tackles the problem of coefficient field choice in persistent homology. When we
compute a persistence diagram, we need to select a coefficient field before computation. We …

A fundamental challenge in multiparameter persistent homology is the absence of a
complete and discrete invariant. To address this issue, we propose an enhanced framework …

A weighted $ d-$ complex is a simplicial complex of dimension $ d $ in which each face is
assigned a real-valued weight. We derive three key results here concerning persistence …

We prove that the fractal dimension of a metric space equipped with an Ahlfors regular
measure can be recovered from the persistent homology of random samples. Our main …

We compute the eigenvalues of up-Laplacians on cubical lattices and derive the torsion-
weighted count of spanning acycles in cubical lattices by using the matrix-tree theorem for …

We study the homology of random Čech complexes generated by a homogeneous Poisson
process. We focus on 'homological connectivity'—the stage where the random complex is …

We investigate the topological dynamics of extreme sample clouds generated by a heavy tail
distribution on R^d by establishing various limit theorems for Betti numbers, a basic …