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Axioms 2017, 6(3), 24; doi:10.3390/axioms6030024

Topological Signals of Singularities in Ricci Flow

1
Air Force Research Laboratory, Information Directorate, Rome, NY 13441, USA
2
Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY 13244, USA
3
Department of Mathematics, Statistics, and Computer Science, University of Wisconsin-Stout, Menomonie, WI 54751, USA
4
Department of Physics, Florida Atlantic University, Boca Raton, FL 33431, USA
5
Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA
6
Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USA
*
Author to whom correspondence should be addressed.
Received: 13 June 2017 / Revised: 25 July 2017 / Accepted: 27 July 2017 / Published: 1 August 2017
(This article belongs to the Special Issue Discrete Geometry and its Applications)
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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. View Full-Text
Keywords: persistent homology; Ricci flow; discrete Ricci flow; singularity detection persistent homology; Ricci flow; discrete Ricci flow; singularity detection
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Alsing, P.M.; Blair, H.A.; Corne, M.; Jones, G.; Miller, W.A.; Mischaikow, K.; Nanda, V. Topological Signals of Singularities in Ricci Flow. Axioms 2017, 6, 24.

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