Topological signals of singularities in Ricci flow

Paul M. Alsing, Howard A Blair, Matthew Corne, Gordon Jones, Warner A. Miller, Konstantin Mischaikow, Vidit Nanda

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a discrete sample of times. We analyze the topological signals of geometric criticality obtained numerically from the application of persistent homology to models manifesting singularities under Ricci flow. The results we obtain for these numerical models suggest that the topological signals distinguish global singularity formation (collapse to a round point) from local singularity formation (neckpinch). Finally, we discuss the interpretation and implication of these results and future applications.

Original languageEnglish (US)
Article number24
JournalAxioms
Volume6
Issue number3
DOIs
StatePublished - Sep 1 2017

Keywords

  • Discrete Ricci flow
  • Persistent homology
  • Ricci flow
  • Singularity detection

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Analysis
  • Geometry and Topology
  • Logic
  • Mathematical Physics

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  • Cite this

    Alsing, P. M., Blair, H. A., Corne, M., Jones, G., Miller, W. A., Mischaikow, K., & Nanda, V. (2017). Topological signals of singularities in Ricci flow. Axioms, 6(3), [24]. https://doi.org/10.3390/axioms6030024