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.
Highlights
Ricci flow (RF) [1] is a system of partial differential equations (PDEs) that has been used in the classification of two- and three-dimensional geometries [2,3] via the uniformization theorem and geometrization conjecture, respectively
In view of the applied problems, solutions to the RF equations are represented as datasets obtained via numerical integration
Suppose that one is given a collection of sparse datasets, without a metric, representing an object altered under RF
Summary
Ricci flow (RF) [1] is a system of partial differential equations (PDEs) that has been used in the classification of two- and three-dimensional geometries [2,3] via the uniformization theorem and geometrization conjecture, respectively. Compact surfaces with two-sphere topology collapse to round points, while in three dimensions, a finite number of pinching singularities can occur in different parts of the geometry, depending on the problem being considered. Such singularity formation has been thoroughly studied (cf [4,5,6,7]). Suppose that one is given a collection of sparse datasets, without a metric, representing an object altered under RF This poses a challenge if one desires to know a metric or the curvature of the object at different times. It is worthwhile to explore whether a common feature is shared in the evolution of such datasets because, if so, it would allow one to identify the type of system without having to know information about the metric
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