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Nathan Kershaw

Publications and source records attributed to Nathan Kershaw.

6 recordsLinked to original sources
arXiv2026

RedZeD: Computing persistent homology by Reduction to Zero Differentials

We introduce a new algorithm for computing persistent homology of Vietoris-Rips filtrations, which offers a considerable improvement both in terms of time and memory over the existing implementations of the persistence pairing algorithm. The key innovation, called active enumeration, is made possible by a new theoretical framework of Reduction to Zero Differentials (hence RedZeD) in which to view persistent homology.

cs.CG↗
arXiv2026

Categorical foundations of discrete dynamical systems

We develop categorical foundations of discrete dynamical systems, aimed at understanding how the structure of the system affects its dynamics. We introduce the notion of cycle sets to analyze attractors of a system, and use this to generalize multiple decomposition theorems of Kadelka, Veliz-Cuba, Murrugarra, and the last two authors from Boolean networks to arbitrary discrete dynamical systems.

math.DS↗
arXiv2026

Discrete homology computations by reduction to zero differentials

We develop a new algorithm for computing (persistent) discrete homology of graphs using reduction to zero differentials and active enumeration. This allows us to compute the fourth homology group of the Greene sphere, along with several previously unknown groups. We also show that persistent discrete homology computes faster than simplicial homology of Vietoris-Rips complex in the high-noise non-metric settings, making it a better choice for noisy data sets.

cs.CG↗
arXiv2026

Faster computations of discrete homology

Machine computation of the discrete homology of graphs has stopped at degree two. We present an algorithm that reaches degree four. It generates the singular cubes inductively, pairing cubes one degree down instead of filtering all set maps; quotients the chain modules by the hyperoctahedral group action, over a field of sufficiently large characteristic; and shrinks the graph beforehand using homotopy invariance. The fourth homology group of the five-cycle, previously beyond the reach of machine computation, is computed in under two days.

cs.CG↗
arXiv2026

Topological data analysis using persistent discrete homology

We propose persistent discrete homology as a tool for topological data analysis and discuss its advantages over the existing methods. In particular, we provide empirical evidence that persistent discrete homology is more noise-resistant than persistent homology of the Vietoris-Rips complex for data coming from non-metric settings.

math.AT↗