# Topological Signals of Singularities in Ricci Flow

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries and Background

#### 2.1. Persistent Homology

**Definition**

**1.**

**Proposition**

**1.**

#### 2.2. Ricci Flow

**Definition**

**2.**

- 1.
- Type-I singularity at a maximal time $T<\infty $ if$$\underset{t\in [0,T)}{sup}(T-t)max\{|\mathbf{Rm}(x,t)|:\phantom{\rule{0.222222em}{0ex}}x\in \mathcal{M}\}<\infty ;$$
- 2.
- Type-IIa singularity at a maximal time $T<\infty $ if:$$\underset{t\in [0,T)}{sup}(T-t)max\{|\mathbf{Rm}(x,t)|:\phantom{\rule{0.222222em}{0ex}}x\in \mathcal{M}\}=\infty ;$$
- 3.
- Type-IIb singularity at a maximal time $T<\infty $ if:$$\underset{t\in [0,\infty )}{sup}max\{|\mathbf{Rm}(x,t)|:\phantom{\rule{0.222222em}{0ex}}x\in \mathcal{M}\}=\infty ;$$
- 4.
- Type-III singularity at a maximal time $T<\infty $ if:$$\underset{t\in [0,\infty )}{sup}max\{|\mathbf{Rm}(x,t)|:\phantom{\rule{0.222222em}{0ex}}x\in \mathcal{M}\}<\infty .$$

**Definition**

**3.**

**Definition**

**4.**

## 3. Methodology

#### 3.1. Models

**Definition**

**5.**

- 1.
- The sectional curvature L of planes tangent to each sphere $\{s\}\times {S}^{n}$ is positive.
- 2.
- The Ricci curvature $\mathbf{Rc}=nKd{s}^{2}+[K+(n-1)L]{\psi}^{2}{\mathbf{g}}_{can}$ (where K is the sectional curvature of a plane orthogonal to $\{s\}\times {S}^{n}$) is positive on each polar cap.
- 3.
- The scalar curvature $R=2nK+n(n-1)L$ is positive everywhere.
- 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 $x=0$.

#### 3.2. Persistence Computations

**Definition**

**6.**

- $dim\sigma <d$ and there is some d-dimensional co-face $\tau \succ \sigma $ with $\iota (\tau )\ge p$, or
- $dim\sigma =d$ and $\iota (\sigma )\le p$, or
- $dim\sigma >d$ and there is some d-dimensional face $\tau \prec \sigma $ with $\iota (\tau )\le p$.

- for each ${\mathcal{K}}^{t}$, compute curvature values assigned to all vertices,
- construct the upper star filtration along these values,
- produce the corresponding PDs in dimensions 0 and 1,
- instead of birth-death pairs $(b,d)$, restrict attention to the differences $d-b$, called persistence intervals or lifespans.

#### 3.3. Data Generation and Preparation Algorithm

## 4. Results

#### 4.1. Dimpled Sphere

#### 4.2. Nondegenerate Neckpinch

#### 4.2.1. Symmetric Dumbbell

#### 4.2.2. Dimpled Dumbbell

## 5. Conclusions and Future Directions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

PH | Persistent Homology |

RF | Ricci Flow |

TDA | Topological Data Analysis |

PDE | Partial Differential Equations |

RG | Renormalization Group |

PD | Persistence Diagram |

## References

- Hamilton, R.S. Three manifolds with positive Ricci curvature. J. Differ. Geom.
**1982**, 17, 255–306. [Google Scholar] [CrossRef] - Chow, B.; Knopf, D. The Ricci Flow: An Introduction. In Mathematical Surveys and Monographs; American Mathematical Society: Providence, RI, USA, 2004; Volume 110. [Google Scholar]
- Thurston, W. Three-Dimensional Geometry and Topology; Princeton Mathematical Series 35; Levy, S., Ed.; Princeton University Press: Princeton, NJ, USA, 1997; Volume 1. [Google Scholar]
- Angenent, S.; Isenberg, J.; Knopf, D. Formal matched asymptotics for degenerate Ricci flow neckpinches. Nonlinearity
**2011**, 24, 2265–2280. [Google Scholar] [CrossRef] - Angenent, S.; Isenberg, J.; Knopf, D. Degenerate neckpinches in Ricci flow. J. Reine Angew. Math. Crelle
**2015**, 709, 81–118. [Google Scholar] [CrossRef] - Angenent, S.; Knopf, D. An example of neckpinching for Ricci flow on S
^{n+1}. Math. Res. Lett.**2004**, 11, 493–518. [Google Scholar] [CrossRef] - Gu, H.-L.; Zhu, X.-P. The Existence of Type II Singularities for the Ricci Flow on S
^{n+1}. arXiv, 2007; arXiv:0707.0033. [Google Scholar] - Perelman, G. The entropy formula for the Ricci flow and its geometric applications. arXiv, 2003; arXiv:math.DG/0211159. [Google Scholar]
- Perelman, G. Ricci flow with surgery on three manifolds. arXiv, 2003; arXiv:math.DG/0303109. [Google Scholar]
- Carfora, M. Renormalization group and the Ricci flow. Milan J. Math.
**2010**, 78, 319–353. [Google Scholar] [CrossRef] - Carfora, M. Ricci flow conjugated initial data sets for Einstein equations. Adv. Theor. Math. Phys.
**2011**, 15, 1411–1484. [Google Scholar] [CrossRef] - Carfora, M. The Wasserstein geometry of non-linear sigma models and the Hamilton-Perelman Ricci flow. Rev. Math. Phys.
**2017**, 29, 1–71. [Google Scholar] [CrossRef] - Raamsdonk, M.V. Building up spacetime with quantum entanglemen. arXiv, 2010; arXiv:1005.3035. [Google Scholar]
- Woolgar, E. Some applications of Ricci flow in physics. Can. J. Phys.
**2008**, 86, 645. [Google Scholar] [CrossRef] - Yu, X.; Yin, X.; Han, W.; Gao, J.; Gu, X. Scalable routing in 3D high genus sensor networks using graph embedding. In Proceedings of the INFOCOM 2012, Orlando, FL, USA, 25–30 March 2012; pp. 2681–2685. [Google Scholar]
- Wang, Y.; Shi, J.; Yin, X.; Gu, X.; Chan, T.F.; Yau, S.-T.; Toga, A.W.; Thompson, P.M. Brain surface conformal parameterization with the Ricci flow. IEEE Trans. Med. Imaging
**2012**, 31, 251–264. [Google Scholar] [PubMed] - Miller, W.A.; McDonald, J.R.; Alsing, P.M.; Gu, D.; Yau, S.-T. Simplicial Ricci flow. Commun. Math. Phys.
**2014**, 239, 579–608. [Google Scholar] [CrossRef] - Carlsson, G. Topology and Data. Bull. Am. Math. Soc.
**2009**, 46, 255–308. [Google Scholar] [CrossRef] - Edelsbrunner, H.; Harer, J. Computational Topology; American Mathematical Society: Providence, RI, USA, 2009. [Google Scholar]
- Kaczynski, T.; Mischaikow, K.; Mrozek, M. Computational Homology; Applied Mathematical Sciences 157; Springer: New York, NY, USA, 2004. [Google Scholar]
- Nanda, V.; Sazdanovic, R. Simplicial Models and Topological Inference in Biological Systems. In Discrete and Topological Models in Molecular Biology; Jonoska, N., Saito, M., Eds.; Springer: Heidelberg, Germany, 2014. [Google Scholar]
- Goullet, A.; Kramár, M.; Kondic, L.; Mischaikow, K. Evolution of Force Networks in Dense Particulate Media. Phys. Rev. E
**2014**, 90, 052203. [Google Scholar] - Bhattachayra, S.; Ghrist, R.; Kumar, V. Persistent homology in ℤ
_{2}coefficients for robot path planning in uncertain environments. IEEE Trans. Robot.**2015**, 31, 578–590. [Google Scholar] [CrossRef] - De Silva, V.; Ghrist, R. Coverage in sensor networks via persistent homology. Algebraic Geom. Topol.
**2007**, 7, 339–358. [Google Scholar] [CrossRef] - Bendich, P.; Chin, S.P.; Clark, J.; Desena, J.; Harer, J.; Munch, E.; Newman, A.; Porter, D.; Rouse, D.; Watkins, A.; et al. Topological and Statistical Behavior Classifiers for Tracking Applications. arXiv, 2014; arXiv:1406.0214. [Google Scholar]
- Weygaert, R.V.D.; Vegter, G.; Edelsbrunner, H.; Jones, B.J.T.; Pranav, P.; Park, C.; Hellwing, W.A.; Eldering, B.; Kruithof, N.; Patrick Bos, E.G.; et al. Alpha, Betti, and the Megaparsec Universe: On the Topology of the Cosmic Web. In Transactions on Computational Science XIV; Springer: Berlin, Germany, 2011; pp. 60–101. [Google Scholar]
- Garfinkle, D.; Isenberg, J. Numerical Studies of the Behavior of Ricci Flow. Contemp. Math.
**2005**, 367, 103. [Google Scholar] - Garfinkle, D.; Isenberg, J. The Modelling of Degenerate Neck Pinch Singularities in Ricci Flow by Bryant Solitons. arXiv, 2009; arXiv:0709.0514. [Google Scholar]
- Ghrist, R. Barcodes: The persistent topology of data. Bull. Am. Math. Soc.
**2008**, 45, 61–75. [Google Scholar] [CrossRef] - Miller, W.A.; Alsing, P.M.; Corne, M.; Ray, S. Equivalence of simplicial Ricci flow and Hamilton’s Ricci flow for 3D neckpinch geometries. arXiv, 2014; arXiv:1404.4055. [Google Scholar]
- Zomorodian, A.; Carlsson, G. Computing persistent homology. Discret. Comput. Geom.
**2005**, 33, 249–274. [Google Scholar] [CrossRef] - Chazal, F.; de Silva, V.; Glisse, M.; Oupoint, S. The structure and stability of persistence modules. arXiv, 2012; arXiv:1207.3674. [Google Scholar]
- Bishop, R.L.; Goldberg, S.I. Tensor Analysis on Manifolds; Dover Publications Inc.: New York, NY, USA, 1980. [Google Scholar]
- Nakahara, M. Geometry, Topology, and Physics; Institute of Physics Publishing: Philadelphia, PA, USA, 2003. [Google Scholar]
- Cao, H.-D.; Chow, B.; Chu, S.-C.; Yau, S.-T. (Eds.) Collected Papers on Ricci Flow; Series in Geometry and Topology 37; International Press: Somerville, MA, USA, 2003. [Google Scholar]
- Hamilton, R.S. The Formation of Singularities in the Ricci Flow; Surveys in Differential Geometry II; International Press: Cambridge, MA, USA, 1995. [Google Scholar]
- Chow, B.; Chu, S.-C.; Glickenstein, D.; Guenther, C.; Isenberg, J.; Ivey, T.; Knopf, D.; Lu, P.; Luo, F.; Ni, L. The Ricci Flow: Techniques and Applications, Part 1: Geometric Aspects; Mathematical Surveys and Monographs 135; American Mathematical Society: Providence, RI, USA, 2007. [Google Scholar]
- Robins, V.; Wood, P.J.; Sheppard, A.P. Theory and algorithms for constructing discrete Morse complexes from grayscale digital images. IEEE Trans. Pattern Anal. Mach. Learn.
**2011**, 33, 1646–1658. [Google Scholar] [CrossRef] [PubMed] - Nanda, V. Perseus: The Persistent Homology Software. Available online: http://people.maths.ox.ac.uk/nanda/perseus/index.html (accessed on 28 July 2017).
- Edelsbrunner, H.; Harer, J. Persistent Homology: A Survey; Surveys on Discrete and Computational Geometry; American Mathematical Society: Providence, RI, USA, 2008. [Google Scholar]
- Anand, K.; Bianconi, G. Entropy measures for networks: Toward an information theory of complex topologies. Phys. Rev. E
**2009**, 80, 045102. [Google Scholar] [CrossRef] [PubMed] - Coffman, V.; Kundu, J.; Wootters, W.K. Distributed Entanglement. Phys. Rev. A
**2000**, 61, 052306. [Google Scholar] [CrossRef] - Walck, S.N.; Glasbrenner, J.K.; Lochman, M.H.; Hilbert, S.A. Topology of the three-qubit space of entanglement types. Phys. Rev. A
**2005**, 72, 052324. [Google Scholar] [CrossRef] - Haegeman, J.; Mariën, M.; Osborne, T.J.; Verstraete, F. Geometry of matrix product states: Metric, parallel transport, and curvature. J. Math. Phys.
**2014**, 55, 021902. [Google Scholar] [CrossRef] [Green Version] - Saucan, E.; Jost, J. Network Topology vs. Geometry: From Persistent Homology to Curvature. In Proceedings of the NIPS 2016 Workshop on Learning in High Dimensions with Structure, Barcelona, Spain, 05–10 December 2016; Available online: http://www.cs.utexas.edu/~rofuyu/lhds-nips16/papers/11.pdf (accessed on 24 July 2017).

**Figure 1.**Example triangulation where the index i corresponds with the polar angle $\theta $, and the index j corresponds with the azimuthal angle $\varphi $. Vertices are indexed as $(i,j)$ or as the single digit v, given by $v=j+(i-1)N\varphi $. The numbers of points sampled along the directions are given by the notation $N\theta $ or $N\varphi $. The increments of angles are computed by $\u25b5\theta =\frac{\pi -2\epsilon}{N\theta -1}$ and $\u25b5\varphi =\frac{2\pi}{N\varphi -1}$.

**Figure 2.**This illustration is of the dimpled sphere and the corresponding radial profiles $r(\theta ,t)$ at time (a) $t=0$; (b) $t=0.03$; (c) $t=0.3$; and (d) $t=0.57105$.

**Figure 3.**Lifespans (d–b) for ${\beta}_{0}$ computed for (

**a**) $N\theta =15$ and (

**b**) $N\theta =25$ triangulations.

**Figure 4.**Cardinalities (number of points in PDs) at each time index for ${\beta}_{0}$ computed for (

**a**) $N\theta =15$ and (

**b**) $N\theta =25$.

**Figure 5.**${\beta}_{0}$: interpolation functions of average scalar curvature (denoted by R_avg, solid black line), bottleneck (gray), Wasserstein-1 (dark gray, dashed), and Wasserstein-2 (light gray, dashed) distances for (

**a**) $N\theta =15$ and (

**b**) $N\theta =25$.

**Figure 6.**${\beta}_{0}$: Interpolation functions of ratios of bottleneck distance to Wasserstein-1 distance (black dots) and Wasserstein-2 distance (gray dots) for (

**a**) $N\theta =15$ and (

**b**) $N\theta =25$.

**Figure 7.**This is an illustration of the symmetric dumbbell. The three-dimensional surfaces (above) are the plots of the dumbbell on the interval $[-50\pi ,50\pi ]$ and close-ups of the necks at initial time $t=0$ (

**a**) and final time $t=0.785$ (

**b**). The initial (long dash) and final (short dash) radial profiles are plotted below on the interval $[-50\pi ,50\pi ]$.

**Figure 8.**Lifespans $d-b$ for ${\beta}_{0}$ computed for (

**a**) $Nx=50$ and (

**b**) $Nx=100$ triangulations.

**Figure 9.**${\beta}_{0}$: Interpolation functions of average scalar curvature (denoted by R_avg, solid black line), bottleneck (gray), Wasserstein-1 (dark gray, dashed), and Wasserstein-2 (light gray, dashed) distances for (

**a**) $Nx=50$ and (

**b**) $Nx=100$ triangulations.

**Figure 10.**${\beta}_{0}$: Interpolation functions of ratios of bottleneck distance to Wasserstein-1 distance (black dots) and Wasserstein-2 distance (gray dots) for (

**a**) $Nx=50$ and (

**b**) $Nx=100$ triangulations.

**Figure 11.**This is an illustration of the dimpled dumbbell. The three-dimensional surfaces (above) are the plots of the dumbbell on the interval $[-100\pi +0.1,100\pi -0.1]$ and close-ups of some of the necks, including the most-pinched neck, on the interval $[-130,150]$ at initial time $t=0$ (

**a**) and final time $t=1.1599$ (

**b**). The initial (long dash) and final (short dash) radial profiles are plotted below on the interval $[-130,150]$.

**Figure 12.**Lifespans $d-b$ for ${\beta}_{0}$ computed for (

**a**) $Nx=25$ and (

**b**) $Nx=50$ triangulations.

**Figure 13.**${\beta}_{0}$: Interpolation functions of average scalar curvature (denoted by R_avg, solid black line), bottleneck (gray), Wasserstein-1 (dark gray, dashed), and Wasserstein-2 (light gray, dashed) distances for (

**a**) $Nx=25$ and (

**b**) $Nx=50$ triangulations.

**Figure 14.**${\beta}_{0}$: Interpolation functions of ratios of bottleneck distance to Wasserstein-1 distance (black dots) and Wasserstein-2 distance (gray dots) for (

**a**) $Nx=25$ and (

**b**) $Nx=50$ triangulations.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Alsing, P.M.; Blair, H.A.; Corne, M.; Jones, G.; Miller, W.A.; Mischaikow, K.; Nanda, V.
Topological Signals of Singularities in Ricci Flow. *Axioms* **2017**, *6*, 24.
https://doi.org/10.3390/axioms6030024

**AMA Style**

Alsing PM, Blair HA, Corne M, Jones G, Miller WA, Mischaikow K, Nanda V.
Topological Signals of Singularities in Ricci Flow. *Axioms*. 2017; 6(3):24.
https://doi.org/10.3390/axioms6030024

**Chicago/Turabian Style**

Alsing, Paul M., Howard A. Blair, Matthew Corne, Gordon Jones, Warner A. Miller, Konstantin Mischaikow, and Vidit Nanda.
2017. "Topological Signals of Singularities in Ricci Flow" *Axioms* 6, no. 3: 24.
https://doi.org/10.3390/axioms6030024