Topological Signals of Singularities in Ricci Flow

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.


Introduction
Persistent homology (PH) [7,17,33,40] is an algebraic-topological method for data analysis. It is useful for analyzing data generated by nonlinear processes. Examples include data sets not amenable to a linear fit, a collection of data points representing some surface or manifold, or information for which a suitable metric is unknown or not easily specified.
Hamilton's Ricci flow (RF) [29] has been used in the classification of two and three dimensional geometries [11,47] via the uniformization theorem and geometrization conjecture, respectively. Of interest is the formation of singularities [4,5,27] in the solutions of the RF equations. In two dimensions, compact surfaces with 2-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.
To investigate certain kinds of problems, such as those related to imaging or complex networks that admit only a coarse-grained resolution, and to efficiently characterize geometries with complex topology, it was worthwhile to develop a discrete form of RF that we refer to as simplicial Ricci flow (SRF) [37].
The goal of this work is to use persistent homology to detect critical geometric phenomena in SRF. In this way, we are able numerically to model RF and to see, at various stages of the time evolution, the behavior of geometric objects. If possible, we would like to have a method that quantitatively detects common trends in the evolution of such objects under RF. While we can manually run codes to acquire certain information at specific locations of a data structure, we are also interested in the behavior of the entire collection of data. Even with a complete set of data and a means to measure it, numerical simulations typically limit us to pointwise snapshots at single moments of time of quantities of interest. Also, such simulations are plagued with stiffness that obstructs or distorts the evolution of a system. Persistent homology provides a method for evaluating an entire collection of and tracking trends in data across an entire domain of interest and across all steps of time evolution by sorting legitimate topological signals from noise. In this way, PH can also provide relevant information about the behavior of the data despite artificial obstructions.
In Section 2, we review the mathematical preliminaries needed. This includes brief expositions of Ricci flow and its simplicial counterpart SRF, singularity formation in RF, the models of interest, and persistent homology. In Section 3, we describe our data-generating algorithm, the initial and boundary conditions for the different models, the assignment of the data to a triangulation, and the implementation of the Perseus code for the persistent homology analysis. In Section 4, we present the analysis of our models under the machinery of PH. Our results show that PH is able to assess when uniformization occurs, when collapse occurs, or when a singularity is encountered by analyzing those topological features which persist or vanish. We conclude with Section 5, where we review the results and discuss their interpretation and implication. Finally, we present future applications.

Preliminaries and Background
We begin with a brief outline of both simplicial Ricci flow and persistent homology. Since our treatment of these topics is cursory, we refer the interested reader to [36,37] for various technical details pertaining to simplicial Ricci flow and to [7,17,40] for overviews of persistent homology and its manifold applications.

Ricci Flow
RF is a diffusive flow of curvature across a compact manifold and was introduced in the early 1980's by R. Hamilton [6]. RF was instrumental in Perelman's proof of Thurston's geometrization conjecture and continues to yield new insights into problems in pure and applied mathematics. It has solved a broad spectrum of difficult problems in medical science, computer science and engineering [6,10,11,12]. In three and higher dimensions, RF can develop singularities in curvature. For 3dimensional manifolds, such flows can admit a finite number of these singularities at distinct times in the interval t ∈ [0, ∞). Identifying, understanding, and surgically removing these singularities was an essential step in Hamilton's program and Perelman's proof to decompose a manifold into its prime and simple pieces. Efficiently characterizing these singularities for three and higher dimensional geometries that undergo RF is the primary interest of this manuscript.
Since we will be dealing with complex topologies and geometries we are primarily interested in a piecewise linear (PL) formulation of Hamilton's RF. To this end it will be convenient for us to examine the mixed index form of Hamilton's RF equations. In particular, if we let M be a smooth compact Riemannian manifold with metric g, then for a geometry undergoing RF the fractional change in time of the metric is proportional to the mixed index Ricci tensor, where the · indicates differentiation with respect to an external time parameter, t. The Ricci tensor is a contraction of the Riemann tensor, Rc a b := Rm ac bc , and provides a certain average of the sectional curvatures of all 2-planes (each identified by indices c and b) along a given direction (given by index b). As mentioned above, this RF equation (1) yields a forced diffusion equation for the curvature, i.e., the scalar curvature, R := Rc a a , evolves under this flow aṡ and tends to uniformly distribute curvature over the manifold. Here, g is the usual Beltrami-Laplace operator with respect to the metric. Hamilton showed that on any compact 3-manifold with positive Ricci curvature, the normalized RF (scaled so as to preserve total volume of the manifold under evolution) converges to a metric of positive constant sectional curvature [29]. As mentioned, on general 3-manifolds, the RF will ordinarily develop a finite number of singularities [30]. The understanding of these singularities under RF by Hamilton and Perelman has led to the geometrization of 3-manifolds and to advances in the geometric analysis of non-linear PDE's [42]. We outline below a recent program to develop the RF equations on piecewise manifolds in an attempt to provide an efficient algorithm to characterize 3-geometries.

Simplicial Ricci Flow
Many applications of this curvature flow have utilized the numerical evolution of piecewise-flat simplicial 2-surfaces and collectively are called combinatorial Ricci flow (CRF) [13,26]. It is a widely accepted verity in computational science that a geometry with complex topology is most naturally represented in a coordinate-free way by an unstructured mesh. This is apparent in the engineering applications utilizing finite-volume [41] and finite-element [32] algorithms, and it is equally true in physics within the field of general relativity through Regge calculus [22,43] and in electrodynamics by discrete exterior calculus [15]. One expects a wealth of exciting new applications for discrete formulations of RF in three and higher dimensions. In particular, canonical metrics on 3-dimensional manifolds have proven to be valuable for the analysis of topological structure, and we believe that discrete Ricci flow in 3dimensions can be used for shape matching and volumetric parameterization. Our presumption is based on the uniformization theorem in 2-dimensions and the finer topological taxonomy in higher dimensions by Thurston's Geometrization Theorem [47]. We envision that applications in these higher-dimensional geometries will involve geometries with complex topologies with boundaries, e.g. modeling of the structure of the human heart and grey and white matter of the brain, as well as the analysis of embedded networks in three dimensions to examine critical points, congestion, and load balance.
A higher dimensional generalization of CRF called SRF was introduced in [37], for piecewise-linear or flat (PL or PF) simplicial manifolds in three and higher dimensions. PL manifolds are topological manifolds possessing a piecewise-linear structure such that, given an atlas on the manifold, maps between charts are piecewise-linear functions. SRF is founded on Regge calculus and the work of Alexandrov [1,2,9,20,34,35,43]. The SRF equations converge to their continuum counterparts, and have been shown to be equivalent for an interesting class of singular models in three dimensions [36].
To each d-dimensional PL-submanifold K ⊂ R n of Euclidean space, one can associate its dual circumcentric graph K * . This is a graph embedded in R n whose vertices are the circumcenters of the d-simplices which comprise K, and a straight edge between two such vertices is present if and only if the underlying d-simplices share a common (d − 1)-dimensional facet. Figure 1 illustrates a piece of the dual graph when d = 2. A non-zero curvature is concentrated at this central co-dimension 2 vertex, v, when the sum of the internal angles of the triangles (six in this case) is different from 2π. This angle deficit is Alexandrov's integrated curvature at hinge v, and is a measure of the conic singularity structure at vertex v. Both in Regge calculus and in discrete exterior calculus, the Gaussian curvature K v at vertex v is defined to be distributed over the circumcentric polygon v * as shown in Fig. 1, In three dimensions the PL cells are tetrahedra. Curvature of this simplicial 3-geometry is concentrated on each circumcentric dual area * . The vertices of this polygon are the circumcenters of each of the tetrahedra sharing edge . This co-dimension 2 edge in the simplicial 3-geometry is circumcentrically dual to * . In particular, if is shared by n ≥ 3 tetrahedra. The circumcenters of these tetrahedra form an n-gon in a similar way to the 2-dimensional example above. To reemphasize, we refer to this n-gon as * , and it is dual to the hinge (edge) . Ordinarily the sum of the dihedral angles, θ i , of the n tetrahedra along edge does not sum to 2π. In this case we define the Gaussian curvature at hinge , This procedure to define the Gaussian curvature at the co-dimension 2 facets of a d-dimensional simplicial complex can be extended to arbitrary dimension. Here the hinge h is a co-dimensional 2 simplex, and the dual circumcentric area is an n-gon we refer to as h * . And in general, where the hinge h is a co-dimension 2 facet of the simplicial lattice. The orthogonality of the simplicial complex and its dual circumcentric lattice gives rise to curvature in planes perpendicular to the co-dimension 2 hinges. No matter how one parallel transports a vector around a hinge h, the vector will return rotated by the deficit angle in a plane perpendicular to the hinge. In 3-dimensions this rotation bi-vector is orthogonal to hinge (edge) . In this sense, and within the region spanned by the dual n-gon, h * and the hinge h, the space is an Einstein space, i.e. the Riemann tensor (Rm), Ricci tensor, and scalar curvature are proportional to the Gaussian curvature. We refer to these ultra-local measures of curvatures as hybrid curvatures, The domain for these hybrid curvatures is the direct product of the hinge h and its circumcentric dual area h * , which we refer to as the hybrid cell as shown in Fig. 2. For convenience, we will restrict ourselves to a 3-dimensional PL geometry for the remainder of this section. Here the hinges are the edges, . Exploded around the perimeter is each of these six tetrahedra. We form the dual circumcentric hexagon, * by connecting the circumcenters of each of these tetrahedra together. Here * is perpendicular to hinge as illustrated in the upper right diagram. We form the hybrid block in 3-dimensions by connecting the co-dimension 2 hinge (edge) = ab with its dual circumcentric n-gon * . This hybrid block is shown in the lower right part of the figure. This hybrid block and all others provide a proper tiling of the 3-geometry. The geometry within each hybrid polygon is strictly an Einstein space.
The construction of the PL-manifold's Rc tensor on the simplicial geometry requires a circumcentrically based weighted sum over the hybrid curvature tensors (scalars) in Eqs. 7-9 analogous to the trace over tensors for continuum manifolds. We refer the reader to [37] for the details of this construction. In particular, we recently showed that each dual edge λ in the circumcentric lattice is the meeting place of exactly three dual n-gons and is the natural place to construct the Rc λ tensor. The dual edge λ is dual to a triangle in the simplicial geometry (Fig. 3). The edges of this triangle are the hinges where the curvature is concentrated. In this sense, by summing the hybrid block rotations for each of these three edges, we are taking the trace of the Riemann tensor.  Figure 3: We illustrate here the lattice geometry used to define the Rc tensor for a 3-dimensional simplicial lattice. The Rc tensor is naturally defined on an edge, λ = σ * 1 ∈ S * . Along this edge we define one of the unit triad vectors, e λ , of the λ-basis, where e λ = λê λ as usual. In this illustration, we show the three dual polygons in red, * |λ ∈ S * , sharing edge λ. Dual to each of these polygons is an edge of the triangle h = λ * ∈ S. Along one of these edges, ⊂ h, we define the second of the triad vectors, e = ê . Finally the normalized vector from the center of edge perpendicular to edge λ, defines our final triad vector, e λ = m λêm .
More specifically, Definition 2.1. The Ricci tensor associated to the circumcentric dual edge λ, Rc λ , is given by In constructing Rc λ we make use of the reduced hybrid tetrahedron, V * λ , as well as reduced hinge volumes, * λ , as shown in Fig. 4. The sums are over the three edges dual to λ. The volume of this tetrahedron, V * λ = (1/6)m λ λ, is the fraction of the set of points interior to the hybrid, V * that are closest to the dual edge, λ. This provides a decomposition of the hybrid cell into reduced hybrid cells, m λ λ Figure 4: The reduced hybrid block illustrated in this diagram is the set of points of the hybrid block - * of Fig. 2 that are closest to edge λ and edge . It is a tetrahedron. This tetrahedron is spanned by three mutually orthogonal line segments, (1) edge , (2) edge λ and (3) the "moment arm" m λ defined as the line segment that reaches from the center of edge to the point on lambda so that it is perpendicular to λ. These reduced hybrid cells are central to the weighted averages we utilize in SRF. The mutual orthogonality of the three line segments also provide a local orthogonal triad basis for the construction of the SRF equations Since the Ricci tensor is naturally constructed at each λ, in order to construct the Rc tensor at an edge we need to do a similar weighted sum over all dual edges λ ∈ * .
Definition 2.2. The Ricci tensor associated to the simplicial edge , Rc , is given by The orthonormal triad as shown in Fig. 4 provides a natural transcription of the continuum in Eq. 1. In three dimensions, our discrete avatar of SRF deforms K by modifying the edge-lengths of K * via the following equation. Definition 2.3. Let K be a piecewise-linear submanifold of Euclidean space and let K * be its dual circumcentric graph. The SRF equation for each dual circumcentric edge λ of K * is given by Using the weighted sum of Eq. 12 one can reformulate this system of equations in terms of codimension-1 facets of K as follows. These equations reproduce (with 2nd-order convergence) the traditional Ricci flow dynamics in the continuum limit. We remark that this is a sparsely-coupled system of non-linear equations for the squared edge-lengths of K. In three dimensions, these equations take on the following relatively simple form since the edges are also the hinges h: The V λ k j represents the hybrid volume restricted to the dual-edge λ k and simplicial-edge j as shown in Fig. 4. These angles, volumes and edge lengths are all defined in [37, Section 3.2] and the references therein. The partial derivative, ∂λ k /∂ j is zero unless j ∈ star (λ k ), i.e. j is one of the 9 edges of the two tetrahedrons that share triangle λ * j . While SRF is one approach to extend combinatorial Ricci flow from two to higher dimensions, there are other active and independent approaches that are being explored. In particular, Yin et al. [50] study discrete curvature flow for hyperbolic 3-manifolds whose boundaries consist of high genus surfaces, where Glickenstein [24,25] studies discrete conformal variations and scalar curvature on piecewise flat two and three-dimensional manifolds and constructs discrete Laplacians on manifolds. Additionally, Ge and Xu [21] define discrete quasi-Einstein metrics (where Einstein metrics are those which are proportional to the Ricci tensor) as critical points for discrete total curvature on simplicial 3-manifolds, where Forman [19] defined a new notion of curvature for cell complexes corresponding to the Ricci curvature for Riemannian manifolds. More recently, there have been combinatorial analyses of curved 3-manifolds by Trout [48] providing, in part, a generalization of the Bonnet-Myers theorem, and Gu and Saucan [45] exploring a combinatorial curvature approach developed by Stone [46].
Our emphasis in this work, however, is on the use of SRF to evolve various geometries on S 2 and S 3 to exhibit singularities. These geometries are used as the theory of formation of singularities in them is well-developed. Then, we apply the methods of persistent homology to a global analysis of these singular models. Previous experience has indicated stiffness in some of the SRF algorithms. To alleviate such stiffness, we exploit the results of [36] which show that the discrete and continuum representations of RF are shown to be equivalent so that one can either begin with the discrete and take the continuum limit or begin with the continuum and discretize. In the following, we numerically evaluate (hence discretize) the continuum RF equations for the considered examples on S 2 and S 3 as it allows for use of a faster algorithm. Then, we form a data structure for analysis via persistent homology.
To proceed, we will discuss the formation of singularities in RF.

Singularities in Ricci Flow
We review the work of [11,29,30]. Consider a solution (M, g(t)) of RF which exists on a maximal time interval [0, T ). Such a solution is maximal if |Rm| -the maximum of the absolute values of the eigenvalues of the Riemann tensor Rm -is unbounded as t → T , T < ∞ or T = ∞. If T < ∞ and |Rm| becomes unbounded as t → T , then a maximal solution develops singularities, and T is called the singularity time [27]. There exist several types of singularities.
) be a solution of RF that exists up to a maximal time T ≤ ∞.
We say a solution to the RF develops a 1. Type-I singularity at a maximal time We investigate singularity formation for the dimpled sphere on S 2 and dumbbells on S 3 in this work. The types of dumbbells include a big-lobed symmetric dumbbell, a small-lobed symmetric dumbbell, a dimpled dumbbell ("genetically modified peanut" [5]), and a degenerate dumbbell [27]. This is because the theory of singularity formation in such examples is well-understood [4,5,11,27].
The dimpled sphere is modeled by a metric whose radial function depends on the polar angle θ and time and is given by where g can = dθ 2 + sin 2 θdφ 2 is the canonical metric on a 2-sphere. The angular measures are usually −π/2 < θ < π/2 (polar) and 0 ≤ φ < 2π (azimuthal). For our code, it is convenient to modify the polar angle so that 0 < θ < π; this is the angle over which we construct the initial radial profile r(θ, 0). By the uniformization theorem, any 2-geometry will evolve under RF to a constant curvature sphere, plane, or hyperboloid [29]. In the case of unnormalized RF, this will lead to the collapse of a sphere to a round point. Regarding singularity formation, the dimpled sphere will Ricci flow to a sphere of constant curvature before collapsing to a round point. The shrinking round sphere is a Type-I singularity.
The dumbbells are constructed by puncturing S 3 at the poles {N, S} and identifying S 3 \ {N, S} with (−c, c) × S 2 where c is a constant usually taken as 1. This identification is for convenience to avoid working in multiple patches [4]. Letting x denote the coordinate on (−c, c) and g can denote the canonical unit sphere metric on S 2 , an arbitrary family g(t) of smooth SO(3)-invariant metrics on S 3 may be written in geodesic polar coordinates as The function ψ 2 (x, t) is a "radial" function for the profile of the dumbbell with one dimension suppressed. This manifests as corseting of the dumbbell. An alternative representation of this metric -the warped product metric -is to introduce a geometric coordinate which normalizes the metric in the variable associated with the interval and leaves only a radial function present [5]. The distance from the "equator" {0} × S n is given by It follows that ∂s/∂x = ϕ(x, t), and this changes the metric to We will work mainly with (15) since we are tasked with formulating an initial-value problem with appropriate boundary conditions. The dumbbells will form two kinds of "neckpinch" singularities: nondegenerate and degenerate. An example of a nondegenerate neckpinch was established in [5]. Here, we review the notation and some necessary definitions [4].
Such a sequence has a corresponding pointed singularity model if the sequence of parabolic dilation metrics g j (x, t) := |Rm(x j , t j )|g(x, t j + |Rm(x j , t j )| −1 t) has a complete smooth limit.
Definition 2.7. Given a RF solution (M, g(t)), a neckpinch singularity develops at a time T if there is some blow-up sequence at T whose corresponding pointed singularity model exists and is given by the self-similarly shrinking Ricci soliton on the cylinder R × S n . Definition 2.8. A neckpinch singularity is nondegenerate if every pointed singularity model of any blow-up sequence corresponding to T is a cylindrical solution. The following are basic assumptions in [4] for such singularity formation in SO(n + 1)-invariant solutions of RF with an initial set of data of the form (17): 1. The sectional curvature L of planes tangent to each sphere {s} × S n is positive.
2. The Ricci curvature Rc = nKds 2 + [K + (n − 1)L]ψ 2 g can (where K is the sectional curvature of a plane orthogonal to {s} × S n ) is positive on each polar cap.
4. The metric has at least one neck and is "sufficiently pinched" in the sense that the value of the radial function ψ at the smallest neck is sufficiently small relative to its value at either adjacent bump.
5. The metric is reflection symmetric, and the smallest neck is at x = 0. Here, Nondegenerate neckpinches are Type-I singularities.
An obvious consequence of the positivity of the tangential sectional curvature L is that |ψ s | ≤ 1. This makes construction of suitable models more challenging and places constraints on any randomized construction.
We will now discuss degenerate neckpinches.
Definition 2.9. A neckpinch singularity is degenerate if there is at least one blow-up sequence at T with a pointed singularity model that is not a cylindrical solution. The basic assumptions in [4] for an open set of initial data are as follows for degenerate neckpinches: 1. The solutions of RF are SO(n + 1)-invariant such that criteria (1) -(3) for nondegenerate neckpinches hold.
2. The initial data may have at least one neck or no necks (hence degenerate dumbbells [27]).
3. It is assumed that a singularity occurs at the "right" pole (x = +1) at some time T < ∞.
As we will demonstrate below, we are able to interpolate between the small-lobed (symmetric) dumbbell and the degenerate dumbbell by adjusting α between 1 and 0, respectively.

Persistent Homology
As mentioned before, we will provide a brief outline of persistent homology and ask the reader to consult one of the many available surveys [7,17,23,40] for further details. Basic information on simplicial homology groups may be found for instance in Hatcher's text [31].
Recall that a filtration consists of a one-parameter family of triangulable topological spaces X p indexed by some real number p ∈ R subject to the constraint that X p is a subspace of X q whenever p ≤ q. At its core, persistent homology associates to each filtration a sequence of intervals [b * , d * ) which are indexed by homology classes across all of the X p 's as p ranges from −∞ to ∞. Here we briefly describe how those intervals are obtained in the special case where each intermediate topological space in the filtration is in fact a finite simplicial complex K p . For the purposes of this section, we do not require an ambient Euclidean space containing all the simplicial complexes -it suffices to assume finiteness and the subcomplex relation K p → K q whenever p ≤ q.
Definition 2.11. Given a simplicial filtration {K p | p ∈ R}, the d-dimensional -persistent homology group of K p is the image of the map H d (K p ) → H d (K p+ ) induced on (ordinary) homology groups by the inclusion of K p into K p+ .
In general, even the space of simplicial filtrations is too large for persistent homology groups to provide a useful invariant and so one adds a local finiteness assumption as follows. A simplicial filtration {K p | p ∈ R} is said to be tame whenever the set of indices p for which the inclusion K p− → K p+ does not induce an isomorphism on homology for all dimensions is finite. Henceforth we will assume that all filtrations in sight are tame.
One may compute homology with coefficients in a field to obtain, for each simplicial filtration {K p } and for each dimension d ≥ 0, a one-parameter family of vector spaces {H d (K p )} which comes equipped with linear maps φ p→q : H d (K p ) → H d (K q ) whenever p ≤ q. These maps are induced on homology by including K p into K q and therefore satisfy the associative property: the composite across any increasing triple p ≤ q ≤ r of positive real numbers. Such a family of vector spaces and linear maps is called a persistence module.
Two key results involving persistence modules make persistent homology an ideal candidate for investigating the topological changes in a filtered space. The first has to do with a finite representation. Associated to any interval 1 [a, b] with b ≥ a is the interval module I [a,b] whose constituent vector spaces are trivial whenever p / ∈ [a, b] and one-dimensional otherwise; the linear maps are identities whenever the source and target vector spaces are both nontrivial. The following proposition was first proved in [51] and has been substantially generalized since [8].
Proposition 2.12. Every persistence module arising from a tame simplicial filtration is canonically isomorphic to a direct sum of interval modules.
In particular, the isomorphism classes of persistence modules are conveniently indexed by a finite family of intervals of the form [a, b]. The d-dimensional persistence diagram of a tame simplicial filtration {K p } is defined to be the collection of intervals which arise from the decomposition of its d-dimensional persistence module.
The second key result confers robustness to persistent homology and is usually called the stability theorem. Rather than stating it in full generality here, we invite the reader to consult [8] directly. The basic idea is that the space of all simplicial filtrations and the space of persistence diagrams may both be assigned natural metric structures so that the process of associating to each filtration its persistence diagrams is 1-Lipschitz.

Data Generation and Preparation Algorithm
A Mathematica code generates the data, in this case the scalar curvature at a collection of points of our object for a sample of discrete times. To obtain the data, we first use NDSolve to solve the Ricci flow. Then, we compute the scalar curvature.
The initial radial profile for the dimpled sphere is obtained by constructing a randomized table of values for a collection of angles θ. To solve RF, we evolve over the angle θ and time. The interval for θ is [0.001, π − 0.001]. (Coming in on both ends of the interval allows for more stable evolution of the code.) To evolve the dumbbells, we have to specify initial and boundary conditions for the functions ψ(x, t) and ϕ(x, t). The initial and boundary conditions for the radial function ψ(x, t) of the dumbbells depend on the model. We use Neumann conditions on the boundaries to satisfy derivative requirements on the poles, which can be expressed in terms of x since (∂x/∂s)| t=0 = 1/ϕ(x, 0) [5]. The boundary conditions at the poles are necessary such that the metric on the dumbbell extends smoothly to a metric on S 3 . In evolving the dumbbells, all of which have topology (−c, c) × S 2 , for the purposes of obtaining a numerical solution, we have to approximate the topology of the dumbbells as [−c Boundary , c Boundary ] × S 2 . The endpoints of the closed interval are c Boundary = c − ε where ε = 100 . c Boundary is, variously, ±100(π/2 − ) (symmetric dumbbells), ±100(π − ) (dimpled dumbbell), or −100(π/2 − ) and another value on the right pole (degenerate dumbbell). For the degenerate dumbbell, the location of the right boundary is determined by the condition that the radial values ψ match on each pole (i.e., the function is the same height above the x-axis). This must be done because, by coming in a fixed amount on each pole, we have a difference in radial values and must make further adjustments.
The first symmetric dumbbell (big-lobed) has initial radial profile The quantity r M in = 1/50 (so that 1 − r M in = 49/50) is chosen to provide adequate pinching. This dumbbell satisfies most of the requirements listed in Definition 2.8, though the scalar curvature is nonpositive near the poles, initially.
To properly address the work of [27] (this will be used in Section 4), we need for the initial profile a function, depending on a parameter α ∈ [0, 1], which satisfies the derivative requirements on the poles for the smooth extension of the metric onto S 3 ; we can impose this on g α (0). Such a function for the initial profile is given by We have a freely-specifiable set of parameters {α, L, k, µ}. The factor k appears in the definition of h(α, k) = −kα + 2k; this function controls the neck thickness. The factor (x/µ − (1 − α)L) 2 + h(α, k) controls the position of the neck. The parameter µ is the scale factor; we select µ = 100. This allows for longer time evolution and delay of stiffness due to incrementation limitations. The constants a and b depend on the choice of these parameters and are determined by imposing the condition of the derivatives on the poles. For the α-class of dumbbells (small-lobed and degenerate), our left pole is fixed at −50π where we come in by 0.1 unit; our right endpoint is a function of α and L, µ π 2 − L α + L , where we come in by 0.1 unit. In general, we come in more on the right pole as the radial profile function will otherwise be at a different height above the x-axis than on the left pole, and we are interested in examining the pinching behavior on the right pole while eliminating artifacts of the construction (i.e., that it's artificially pinching due to closer proximity with the x-axis than the left pole). So, we solve for the value on the x-axis which yields the same radial function value on the right pole as on the left (fixed) pole.
To consider nondegenerate neckpinches satisfying the conditions of [4], we consider a rotationally and reflection symmetric (about the "equator") dumbbell. This is obtained from (21) with α = 1.
The initial radial profile is a symmetric dumbbell obtained by ψ(x) = −12.6948 x 100 + π 2 x 100 The choice of parameters L = 1 and k = 1/25 provides for sufficient pinching. This model satisfies all of the conditions of Definition 2.8.
Interesting topological signatures occur when the filtration parameter or function has many critical points (e.g., the height function [16]). A way to achieve such a function is to consider the dimpled dumbbell or "genetically modified peanut" [5]. This is an object with multiple necks and multiple bumps (i.e., multiple critical points of the radial profile function) and, consequently, an object with multiple critical values of the scalar curvature. For the dimpled dumbbell, we construct an interpolation function over the interval [−100π, 100π]. This interval is larger than for the other dumbbells and is chosen for more stable evolution (though we still come in by 0.1 unit on each pole for the RF computation) and to ease placement of the necks while respecting the derivative conditions for dumbbells.
Motivated by the work of [27], we investigate the profile (21) above with α = 0. This parameter choice generates a degenerate dumbbell, a dumbbell possessing only one bump and no neck which looks figuratively like an ear dropper. This will allow us, per Lemma 2.10, to investigate whether we have evolution to a Type-II singularity or to a round point. Our initial radial profile is given by To prepare the data for the persistent homology analysis using Perseus, we construct a suitable triangulation (Fig. 5). Our triangulation is used as it allows for the RF to be evolved more quickly than an algorithm directly implementing SRF. This approach is justified for the examples we consider here since different choices of triangulations have been shown recently to be equally valid in the equivalence of simplicial Ricci flow (SRF) with Hamilton's Ricci flow (RF) for 2-spheres and for warped product metrics [36]. For each vertex chosen, we obtain the scalar curvature. To compute the scalar curvature along the edges of the triangulation, we take the average of the scalar curvatures of the endpoints. This information is then used to determine scalar curvature values of the edges for each time index, specifically searching for the maximum edge curvature and the minimum edge curvature. Perseus implements this data to compute persistence intervals. It turns out that for all of our models, we can implement the same triangulation. This is due to the redundancy of the Ricci tensor in the dumbbell models. One of the dimensions of S 2 is suppressed; thus, each circular cross-section of the dumbbell represents a sphere. This is due to the symmetry of the dumbbell: in computing the components of the Ricci tensor, the θθ and φφ components are redundant so that, out of three, only the xx and θθ components need be computed to determine the scalar curvature. This redundancy allows suppression of one of the dimensions, thus allowing use of a similar triangulation to the rotational solid. In the triangulation, the modifications from the rotational solid to the dumbbells are θ → x and

Persistence Computations: Filtering by Curvature
The methodology of the preceding section produces, at each time scale, a triangulated 2 or 3-manifold whose vertices or edges have been assigned curvature values. More generally, one may confront a triangulated manifold K of dimension n so that each d-dimensional simplex σ has been assigned a real number ι(σ). From this data, one wishes to construct a filtration {K p | p ≥ 0} compatible with ι in the sense that for each σ in K of dimension d, we have In other words, the smallest value of p for which a given d-simplex σ lies in K p is required to equal ι(σ). There are various ad-hoc ways to extend ι-values to simplices of dimension other than d so that a compatible filtration may be imposed on K. A more principled route is via the star filtration which has been used, for instance, in the persistent homological analysis of image data [44].
Definition 3.1. Given a triangulated n-manifold K and a real-valued function ι defined on the dsimplices for some 0 ≤ d ≤ n, the star filtration along ι is defined via the following containment relation for each p ∈ R and each simplex σ ∈ K: we have σ ∈ K p if and only if one of the following conditions holds: • dim σ < d and there is some d-dimensional co-face τ σ with ι(τ ) ≥ p, or The extreme cases d = 0 and d = n are called upper and lower star filtrations. In practice, one constructs star filtrations via an elementary method: consider a simplex in K. If its dimension equals d, then its filtration index simply equals its ι-value. Otherwise, if its dimension exceeds d then it inherits the largest ι-value encountered among its d-dimensional faces. Finally, if its dimension is smaller than d, then it inherits the smallest ι value encountered among all d-dimensional simplices in K which contain this simplex as a face. Note that in order for such a construction to be well-defined, each simplex of dimension smaller than d must have at least one d-dimensional co-face. But since we work entirely with triangulated manifolds, this condition is automatically satisfied.
Our methodology for analyzing simplicial Ricci flow via persistent homology may now be described via the following pipeline which begins with a triangulated surface K 0 evolving via Ricci flow with the resulting complexes being labeled K t for some discrete indices t ≥ 0: The superscript, associated to discrete time, distinguishes from the filtration, which we denote with a subscript. We then plot the differences against the logarithm (for convenience) of discretized values of t ≥ 0 to get a topological signature of the surface as it undergoes evolution via Ricci flow. As we shall see in the following section, each singularity has its own characteristic signature. We remark that this pipeline is extremely general and dimension-agnostic: given a general triangulated manifold of dimension n, we could simply construct the star filtration associated to the curvature values inherited from the n − 2 dimensional simplices.
All persistence computations were performed using the Perseus software [39] which relies on efficient reduction algorithms [38] based on discrete Morse theory [18].

Results
For all of the models, we plot their initial and final radial profiles from Mathematica. Then, we plot the associated persistence intervals and investigate the outputs from the different data tables generated by Perseus. We refer to these plots and tables in terms of their Betti numbers, so that we have plots and tables associated to β 0 (number of connected components) and β 1 (number of tunnels). The symmetry of our models means that we have only dimensions 0 and 1. Perseus also outputs a total summary table that, at each time index, goes through an ascending collection of values of the filtration parameter and indicates the number of connected components and the number of tunnels present for cells born at or below this value.

Dimpled Sphere
The rotational solid in Fig. 6 has the topology of S 2 and the appearance of a "dimpled" sphere. The dimpling is obtained by using a random number generator to modulate the initial radius of the solid so that r(θ, 0) is an interpolation function built over a table of angles and radii. The solid begins as dimpled, but RF uniformizes the curvature, producing a spherical object that then shrinks to a round point. For the rotational solid, the triangulation is taken over a 50 × 50 grid for 78 time indices. Figure  7 features the length of the persistence intervals (death -birth) associated to the Betti numbers β 0 and β 1 versus the logarithm (base 10) of time.   ; this is the first and only time index in the evolution for which only one such interval is present. There is an infinite persistence interval of only one connected component. Also at this time index, there are no intervals of infinite persistence in the β 1 table and only a single finite interval which is born and dies at the same scalar curvature value. The interpretation is that the solid has uniformized and begun its collapse; this is motivated by the data in the tables for β 0 and β 1 . With only one infinite interval of persistence for the connected component and none for the tunnels, the scalar curvature appears uniform across the object.
The evolution continues after this and by the end, there are 17 intervals of finite persistence and one of infinite persistence in the β 0 table. Our algorithm stiffens after this time. These results, in tandem with the stiffness, indicate that the increase in the number of short -relative to the order of magnitude of the scalar curvature values -intervals of finite persistence is due to the computational process of the generating code and the triangulation: noise becomes prevalent.
In the Betti tables at each time index, we have one connected component and zero holes for the lowest and highest maximum birth scalar curvature with varying numbers of connected components and holes for the scalar curvatures between these values. Also at each time index approaching 30000, the difference between the lowest and highest maximum birth scalar curvatures becomes smaller; this is an indication of the uniformization in scalar curvature of the object. Under RF, it is expected that the dimpling smooths so that the object takes on a uniform curvature, then collapses. Thus, the information regarding appearance and disappearance of topological features for the values of birth curvature between lowest and highest indicates noise. After this time, the difference grows; we interpret this to indicate the increase in curvature as the object collapses.  We examine the different values of scalar curvature arising at the vertices of the triangulation, at each time index, to compute averages for the scalar curvature along edges. The triangulation is over a 50 × 3 grid for 63 time indices. The number of points in the φ-direction is somewhat arbitrary: the persistence diagrams we generate depend on the partition of x. As in the case of the rotational solid, the code encounters numerical stiffness quickly unless we come in a little bit from the ends. Figure 9 features the length of the persistence intervals (death -birth) associated to the Betti numbers β 0 (connected components) and β 1 (tunnels) versus the logarithm (base 10) of time. The tables for the β 0 numbers show a trend of a number of intervals of finite persistence with one interval of infinite persistence at each time index. For the β 1 tables, we see a similar trend of finite persistence but an absence of infinite persistence. In both tables, one of the finite intervals is longer than the other finite intervals. Beginning with time index 60000 (t = 0.6), we see an increase in the number of extremely short finite intervals in the subsequent tables of connected components (β 0 ) and tunnels (β 1 ). While not clear from the tables for earlier time indices, the later tables show an increase in the longest finite interval at each time index. An interpretation of this is that the shortest finite intervals in the tables represent noise, while the longer finite interval and infinite interval (present only in β 0 ; there are no infinite persistences for β 1 ) represent a signal persistent throughout the data. Essentially, a low scalar curvature persists throughout the data, and a larger scalar curvature -associated with neckpinchpersists at each time index though for a finite number of birth scalar curvatures. From the definition of a Type-I singularity, it is consistent that for each time index, we see a somewhat longer finite persistence which grows towards the end of the evolution.

Big-Lobed Dumbbell
In the Perseus tables, initially (t = 10 −5 ), the maximum number of connected components appearing for any maximum birth scalar curvature is 4; only four of these scalar curvatures have any holes present, ranging from 9 to 18. For the smallest and largest maximum birth scalar curvatures at this time index, only one connected component is present. This is the trend across all of the Perseus outputs: more connected components and holes appear for the intermediate maximum birth scalar curvatures, but the smallest and largest curvatures each have one connected component. The curvatures also grow across the time indices, reaching a highest value of the maximum birth scalar curvature between t = 0.7 and t = 0.9 before falling off. This detail is related to the presence of negative scalar curvature near the poles and interpolation errors.  We obtain the following persistence tables for a 50 × 50 triangulation (Fig. 11): For the β 0 tables, we have one interval of finite persistence and one interval of infinite persistence of low scalar curvature for all time indices. The lengths of the finite intervals range from 1.1853352 for a birth curvature of 0.0242748 at t = 10 −6 (initial) to 103.689689 for a birth curvature of 0.0243114 at t = 0.785 (final). Because the lengths of the intervals are a minimum of two and a maximum of four orders of magnitude larger than the listed birth curvature, this indicates a strong presence of topological signal for the connected components in this model. The β 1 tables have multiple intervals of negligible length. This indicates that any tunnels appearing represent topological noise.

Small-Lobed Dumbbell
Notably, different triangulations capture different curvature features. A coarser triangulation does not capture the high-curvature value for the pinching which occurs at and near the origin for α = 1. However, comparing coarser (50 × 50) and finer (100 × 100) triangulations, we see an almost self-similarity between the respective β 0 tables and β 1 tables in the plots of persistence for each of the triangulations. Modulo scale (hence the almost self-similarity), we have the following persistence plots for β 0 and β 1 for the 100 × 100 triangulation (Fig. 12): The importance of the persistence diagrams is that they provide a picture of the evolution of a collection of data across time.
We focus on the 50×50 case. The outputs of Perseus for the initial and final time indices display a trend. For the first time index, we see that for edges whose scalar curvatures are less than or equal to 0.0242748, there is one connected component. For just a slightly higher birth curvature, we have what appear to be two connected components. This trend continues until we encounter edges born with curvatures less than or equal to 1.20961; then, it appears we have one connected component and 49 holes. However, for the remaining birth curvatures higher than this, the indication is that we have only one connected component.
This pattern continues for the various outputs of Perseus, though the values of the birth curvatures for which the filtration is carried out at each time index grow. For the final time index, we see that for birth curvatures less than or equal to 0.0243113, we have only one connected component. Then, for birth curvatures less than 0.0243114, we have two connected components; this continues until we get to the collection where the birth curvatures are less than or equal to 103.714, where we then have one connected component and 49 holes. For higher birth curvatures, we then have just one connected component. The plots in Fig. 11 corroborate these outputs.

Dimpled Dumbbell
The dimpled dumbbell is featured below (Figure 13). The necks are at x = −180, x = −66, x = 0, and x = 130. The greatest pinching occurs at the neck x = 0, whose neck radius is initially at 1.5 and finally at 0.00138269, a decrease of three orders of magnitude. The scalar curvature at this neck grows by seven orders of magnitude. The other necks demonstrate pinching but within one order of magnitude in both neck radius and scalar curvature. Most values of ψ remain constant across the evolution. The triangulation is on a 50 × 10 grid for 70 time indices. The persistence diagrams (Fig. 14) resemble a hybrid between the dimpled sphere and the symmetric dumbbells previously. For the β 0 tables, we have four intervals of finite persistence and one interval of infinite persistence of low scalar curvature for all time indices. Two of the finite intervals show a modest increase in birth value but two orders of magnitude increase in death value from t = 10 −5 to t = 1.1599. An interpretation is that these intervals represent topological signal because of their increasing values of persistence over the evolution of the object. The remaining finite intervals show an order of magnitude increase from birth value to death value. Of these, one interval does not change much over the entire evolution, while the other changes but remains within the same order of magnitude. The β 1 tables have multiple intervals of variously short lengths but no infinite intervals. Though some of the intervals grow to lengths similar to those in the β 0 tables, they are of the same order of magnitude. The growth corresponds to an increase in scalar curvature but not the presence of actual tunnels. This perspective is corroborated in the tables output by Perseus. The smallest and largest maximum birth scalar curvatures each have one connected component for all time indices. An increase in connected components and the appearance of tunnels occurs as we sweep across the birth scalar curvatures, but the final birth scalar curvature always returns to a single connected component. Further, this largest birth value becomes six orders of magnitude larger than the smallest value so that the growth in curvature is detected.

Degenerate Neckpinch
The case where α = 0 gives us an asymmetric dumbbell with a pinch on the right polar cap under RF ( = 0.001 so that we come in by 0.1 unit before adjusting for equal function height to the left pole) in Figure 15. The right pole is initially at 0.103238 and finally at 0.00255471, a decrease of two orders of magnitude. The scalar curvature on this pole increases by three orders of magnitude, from 155.988 to 268,650. The rest of the plot is essentially static with little change in the values of ψ.
We obtain a good result for pinching with a high scalar curvature value on the right pole as the radii of the spheres in that region approach zero with two orders of magnitude difference between the initial (t = 0) and final (t = 0.206) profile at the right pole. This contrasts with the radii on the left pole which are almost unchanged in the same time interval.
After reductions at each time index, we obtain only tables for connected components. In each table, there appears to be only a single interval of infinite persistence of a low scalar curvature value. In the Perseus output, we again see a single connected component across all birth values of scalar curvature, both for the initial and final times. This is not surprising: the scalar curvature (filtration parameter) only has one critical point, so we do not expect a lot of output. Because of the reductions which occur in PH, we are left with only connected components for each birth value.

Conclusions and Future Directions
The application of PH to SRF provides a unique opportunity to analyze topologically a collection of data over which we have substantial command, in this case meaning that we have a ready filtration (scalar curvature). This differs from what is usually encountered in problems of PH, where the assignment of a metric to the data or the choice of a filtration parameter to implement is unclear. We are able to compute scalar curvature at any point on our objects, at any time in the evolution. Given a sample of points, we then can construct a triangulation and define the scalar curvature on edges as an average of scalar curvatures of endpoints sharing the edges. We find from these outputs that persistent homology detects the formation and type of singularity across multiple spatial resolutions.
At each time step, looking at the outputs of Perseus, we find for the different models three classifications of lengths of persistence intervals: negligible finite, finite, and infinite. The tables for connected components display all of these types of intervals; those for the tunnels only have negligibly finite intervals. The first classification consists of extremely short intervals representing features that do not persist over many values of scalar curvature. The second classification consists of longer finite intervals, one or more orders of magnitude longer than the first class of finite intervals. The third classification is the presence of infinite persistence intervals associated to low values of scalar curvature. As intervals are born, we see the birth of connected components. When these die, we see the birth of tunnels. As we filter over higher values of scalar curvature and look at the birth values of new cells, we see the disappearance or reappearance of certain features.
The dimpled 2-sphere uniformizes in curvature before collapsing to a point. The persistence intervals reflect this transition as most of the finite intervals vanish at the time of uniformization, returning only near the end of the evolution. Infinite intervals with increasing birth scalar curvature at each time step appear in the tables of persistence intervals for dimension 0. This corresponds with the formation of a singularity.
At each time index, for the connected components, we have an infinite interval, a longer finite interval, and a number of shorter finite intervals. For the tunnels, we have a number of short finite intervals. This holds until time index 30000, where we have only a brief finite interval and an infinite interval for the connected components, and similarly brief finite intervals for the tunnels. After this time index, until the end, we see an increase in the number of short, finite intervals in both the tables for connected components and tunnels. At the final time index, we see an infinite interval and brief finite intervals for the connected components, as well as brief, finite components for the tunnels. Also, for each time index, we find that the Betti numbers fluctuate but begin and end with 1 connected component and 0 tunnels.
For the big-lobed dumbbell, at each time index for the connected components we have an infinite interval, a longer finite interval, and a number of shorter time intervals. For the tunnels, we have a number of brief finite intervals. When connected components die, this signals the formation of tunnels; these appear and vanish at the same values of scalar curvature. This trend holds uniformly across all time indices, with only the longest finite interval becoming longer.
The small-lobed dumbbell, at each time index for the connected components, has an infinite interval and one finite interval. For the tunnels, we have a number of brief, finite intervals. This is true for the different spatial resolutions, where only the individual birth and death values and the scale of their differences change. The difference between the big-lobed dumbbells and these dumbbells is that for these, the scalar curvature is everywhere positive initially.
The genetically modified peanut has, for the connected components at each time index, an infinite persistence interval, between two and four brief finite intervals, and two or more longer finite intervals. As time progresses, one of these becomes much longer. For the tunnels, there are shorter and longer finite intervals. Unlike the small-lobed dumbbell, we expect that we have more intervals because we have many more local extrema in the radii of the spheres comprising the dumbbell (hence more local extrema in the filtration parameter).
For the degenerate dumbbell, at each time index, there is only an infinite persistence interval in the tables for connected components. No tables for tunnels are generated: for all values of the filtration, there is only a single connected component.
In the Betti number tables for all models but the degenerate dumbbell, multiple connected components and tunnels appear and disappear; we interpret this as indicative of topological noise since with our construction, we expect to have only one connected component. The presence of negligible persistence intervals is associated with this as the appearance of multiple connected components coincides with the lowest of the birth values of finite intervals, while the appearance of tunnels coincides with the death values of the finite intervals for connected components. The source of the noise may be due to the method of determining scalar curvature on edges (as an average of scalar curvatures at points), resulting in cells that are lacking vertices and edges with certain values of scalar curvature. This leads to multiple connected components as we raise the scalar curvature, only to retrieve a single connected component once we reach a suitable threshold (i.e., scalar curvature value for which all vertices and edges are greater than or equal). The argument holds likewise for the tunnels. The longer finite intervals indicate increasing curvature, with multiple order-of-magnitude increases between birth and death values at each time step. The presence of these different classes of lengths of persistence intervals necessitates careful delineation of topological noise (here, due to the method of determining scalar curvature on edges and to numerical errors) from legitimate topological signal.
We conclude that the infinite intervals and the longest of the finite intervals represent a measure of topological signal (i.e., increase of scalar curvature and formation of singularities). The values associated with the longest of the finite intervals grow with each time index, and they grow by orders of magnitude. The brief finite intervals represent topological noise as they coincide with the appearance of multiple connected components or tunnels that are absent at the beginning and end of the scalar curvature values for filtration.
For future work, we would like to use PH to investigate systems with incomplete or sparse data. This includes problems involving the interpretation of data for hard-to-compute higher-dimensional scenarios which may appear in the context of physical problems with more than 4 dimensions or systems with a large number of degrees of freedom (e.g., phase spaces). Another is to take sparse data, such as that obtained for density matrices, and consider different filtrations for filling in the missing data.
We are also interested in the use of PH and RF to explore quantum entanglement. The former would be able to investigate quantum entanglement via filtration over a function such as entropy [3], the tangle [14], or other invariants [49]. RF is understood on continuum manifolds both real and complex, and matrix product states (of which cluster states are) have recently been given a differential geometric representation [28].
Understanding of the global behavior of the data here is useful in approaching new problems that present computational challenges that defy well-developed methods but are amenable to topological techniques.