Topological Signals of Singularities in Ricci Flow

TL;DR

This paper uses computational homology and persistent homology to identify and distinguish between different types of singularities in geometries undergoing Ricci flow, providing a new topological perspective.

Contribution

It introduces a novel application of persistent homology to analyze singularity formation in Ricci flow, offering quantitative topological signals that differentiate singularity types.

Findings

Topological signals distinguish global from local singularities.

Persistent homology captures geometric criticality during Ricci flow.

Method provides quantitative measures of singularity behavior.

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.