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
Cite
@article{arxiv.1502.02630,
title = {Topological Signals of Singularities in Ricci Flow},
author = {Paul M. Alsing and Howard A. Blair and Matthew Corne and Gordon Jones and Warner A. Miller and Konstantin Mischaikow and Vidit Nanda},
journal= {arXiv preprint arXiv:1502.02630},
year = {2017}
}
Comments
24 pages, 14 figures