Topological Signals of Singularities in Ricci Flow
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...
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| Format: | Article |
| Language: | English |
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MDPI AG
2017-08-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/6/3/24 |
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| author | Paul M. Alsing Howard A. Blair Matthew Corne Gordon Jones Warner A. Miller Konstantin Mischaikow Vidit Nanda |
| author_facet | Paul M. Alsing Howard A. Blair Matthew Corne Gordon Jones Warner A. Miller Konstantin Mischaikow Vidit Nanda |
| author_sort | Paul M. Alsing |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-14T00:25:16Z |
| format | Article |
| id | doaj.art-0cdf03ec82c14e689f8db7e9647cd289 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-04-14T00:25:16Z |
| publishDate | 2017-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-0cdf03ec82c14e689f8db7e9647cd2892022-12-22T02:22:47ZengMDPI AGAxioms2075-16802017-08-01632410.3390/axioms6030024axioms6030024Topological Signals of Singularities in Ricci FlowPaul M. Alsing0Howard A. Blair1Matthew Corne2Gordon Jones3Warner A. Miller4Konstantin Mischaikow5Vidit Nanda6Air Force Research Laboratory, Information Directorate, Rome, NY 13441, USADepartment of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY 13244, USADepartment of Mathematics, Statistics, and Computer Science, University of Wisconsin-Stout, Menomonie, WI 54751, USADepartment of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY 13244, USADepartment of Physics, Florida Atlantic University, Boca Raton, FL 33431, USADepartment of Mathematics, Rutgers University, Piscataway, NJ 08854, USADepartment of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USAWe 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.https://www.mdpi.com/2075-1680/6/3/24persistent homologyRicci flowdiscrete Ricci flowsingularity detection |
| spellingShingle | Paul M. Alsing Howard A. Blair Matthew Corne Gordon Jones Warner A. Miller Konstantin Mischaikow Vidit Nanda Topological Signals of Singularities in Ricci Flow Axioms persistent homology Ricci flow discrete Ricci flow singularity detection |
| title | Topological Signals of Singularities in Ricci Flow |
| title_full | Topological Signals of Singularities in Ricci Flow |
| title_fullStr | Topological Signals of Singularities in Ricci Flow |
| title_full_unstemmed | Topological Signals of Singularities in Ricci Flow |
| title_short | Topological Signals of Singularities in Ricci Flow |
| title_sort | topological signals of singularities in ricci flow |
| topic | persistent homology Ricci flow discrete Ricci flow singularity detection |
| url | https://www.mdpi.com/2075-1680/6/3/24 |
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