@article{e2e65b3291e144c988177b8141e70b57,
title = "Topological signals of singularities in Ricci flow",
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.",
keywords = "Discrete Ricci flow, Persistent homology, Ricci flow, Singularity detection",
author = "Alsing, {Paul M.} and Blair, {Howard A.} and Matthew Corne and Gordon Jones and Miller, {Warner A.} and Konstantin Mischaikow and Vidit Nanda",
note = "Funding Information: Acknowledgments: Howard A. Blair acknowledges support from United States Air Force (USAF) Grant # FA8750-11-12-0275. MC acknowledges support from the University of Wisconsin-Stout Faculty Start-Up Fund and the National Academy of Sciences, National Research Council (NRC) Research Associateship program. We thank Louise Camalier, Rory Conboye and Don Sheehy for suggestions regarding this work. Warner A. Miller acknowledges support from United States Air Force (USAF) Grants # FA8750-11-2-0089 and # FA8750-15-2-0047 and support from the Air Force Research Laboratory/Trusted Systems Branch (AFRL/RITA) through the Summer Faculty Fellowship Program and the Visiting Faculty Research Program (VFRP) administered through the Griffiss Institute. Konstantin Mischaikow acknowledges the National Science Foundation for Grants NSF-DMS-0835621, 0915019, 1125174, 1248071 and contracts from the Air Force Office of Scientific Research (AFOSR) and the Defense Advanced Research Projects Agency (DARPA). Vidit Nanda is grateful to Robert Ghrist for insightful discussions. His work was funded in part by the Defense Advanced Research Projects Agency Grant DARPA DSO-FA9550-12-1-0416. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the Air Force Research Laboratory. Publisher Copyright: {\textcopyright} 2017 by the authors.",
year = "2017",
month = sep,
day = "1",
doi = "10.3390/axioms6030024",
language = "English (US)",
volume = "6",
journal = "Axioms",
issn = "2075-1680",
publisher = "MDPI AG",
number = "3",
}