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Strange Random Topology of the Circle

Author Uzu Lim

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Author Details

Uzu Lim
  • Mathematical Institute, University of Oxford, UK

Funding

Uzu Lim is supported by BMS Centre for Innovation and Translational Research Europe and Korea Foundation for Advanced Studies.

Acknowledgements

The author would like to thank Vidit Nanda and Harald Oberhauser for their contributions during initial stages of this research. The author is grateful to Henry Adams and Tadas Temčinas for valuable discussions.

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Uzu Lim. Strange Random Topology of the Circle. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 70:1-70:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.70

Abstract

A paradigm in topological data analysis asserts that persistent homology should be computed to recover the homology of a data manifold. But could there be more to persistent homology? In this paper I bound probabilities that a random m Čech complex built on a circle attains high-dimensional topology. This builds on the known result that any nerve complex of circular arcs has the homotopy type of a bouquet of spheres. We observe a phase transition going from one 1-sphere, bouquet of 2-spheres, one 3-sphere, bouquet of 4-spheres, and so on. Furthermore, the even-dimensional Betti numbers become arbitrarily large over shrinking intervals. Our main tool is an exact computation of the expected Euler characteristic, combined with constraints on homotopy types. The systematic behaviour we observe cannot be regarded as a "topological noise", and calls for deeper investigations from the TDA community.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
Keywords
  • Topological data analysis
  • persistent homology
  • stochastic topology

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