  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.
