Not just theory

Applications

Ricci flow was built to answer a question about pure topology, and Perelman's proof did not come with a product attached. But the mathematics around it reaches a long way. Discrete versions of the flow map brain surfaces and route network traffic. Its curvature measures fragility in markets and bottlenecks in neural networks. Its cousins segment medical images every day. And the topology it completed frames the question of the shape of the universe.

Each application is labelled twice, and each one cites the papers behind it. The first label says how established it is. The second says how directly it descends from Ricci flow.

14applications
4in use
5demonstrated
5early research

How established

  • In use Used in practice: in widely used software, or as a routine method in its field.
  • Demonstrated Shown to work in published studies, but not yet routine practice.
  • Early research Proposed and tested in a few studies, or still an open research programme.

How it connects

  • From Ricci flow itself Uses Ricci flow (often a discrete version of it) or Perelman’s and Thurston’s results directly.
  • Same mathematics Uses the ideas Ricci flow is built from: Ricci curvature, curvature flows, optimal transport, Ricci-flat metrics.
  • From the prerequisites Uses the mathematics you learn on the way to Perelman: topology, Laplacians and heat kernels on manifolds.

About industry groups: there is no industry association, lobby or corporate consortium devoted to Ricci flow, and we found none for the specific applications below either. The organised communities that do exist — professional societies, conference series, research collaborations and open-source projects — are listed with each application. If you know of one we've missed, tell me.

From Ricci flow itself

Uses Ricci flow (often a discrete version of it) or Perelman’s and Thurston’s results directly.

Computer graphics and computer vision

Conformal maps for 3D shapes, faces and surfaces

DemonstratedFrom Ricci flow itself

The uniformization theorem says every surface can be deformed, without distorting angles, to have constant curvature. On a computer that is a powerful tool: it flattens a curved 3D surface onto a plane, sphere or disk while keeping its local shape, so textures can be painted and surfaces compared. Discrete surface Ricci flow, developed by Gu, Luo and collaborators from Chow and Luo's combinatorial flow, computes these maps on triangle meshes by flowing the mesh's metric to one of constant curvature.

It has been used to register and compare 3D shapes. One example is matching 3D face scans using the hyperbolic version of the flow. The underlying discrete theory, including a discrete uniformization theorem, is now proved rigorously. The method is established in research software; most commercial graphics tools use other, simpler parameterization methods.

The papers behind it

  1. 2008
    Discrete Surface Ricci Flow M. Jin, J. Kim, F. Luo, X. Gu · IEEE Transactions on Visualization and Computer Graphics, vol. 14 Jin, Kim, Luo and Gu: discrete surface Ricci flow as a practical algorithm for graphics. DOI 10.1109/TVCG.2008.57
  2. 2010
    Ricci Flow for 3D Shape Analysis Wei Zeng, D. Samaras, D. Gu · IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 32 Zeng, Samaras and Gu: Ricci flow for 3D shape analysis and registration. DOI 10.1109/tpami.2009.201
  3. 2008
    3D face matching and registration based on hyperbolic Ricci flow Wei Zeng, Xiaotian Yin, Yun Zeng et al. · 2008 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops Zeng et al.: matching and registering 3D faces with hyperbolic Ricci flow. DOI 10.1109/cvprw.2008.4563053
  4. 2013
    A discrete uniformization theorem for polyhedral surfaces Xianfeng Gu, Feng Luo, Jian Sun, Tianqi Wu · arXiv:1309.4175 Gu, Luo, Sun (and Wu): a discrete uniformization theorem, the mathematics underneath. PDFarXiv:1309.4175

Who's involved

  • Eurographics The European association for computer graphics. It runs the annual Symposium on Geometry Processing, where conformal and curvature-based mesh methods are presented.
Discrete & graph RicciSurfacesNumerics & visualization

Medical imaging

Brain mapping and medical surface registration

DemonstratedFrom Ricci flow itself

Brain surfaces are folded differently in every person. To compare them, or to follow one brain over time, researchers map each cortical surface onto a common shape without distorting angles. Wang, Gu, Thompson, Yau and colleagues computed these maps with discrete Ricci flow, which handles the surface's topology cleanly. Related conformal-geometry methods from the same school have been used to register colon surfaces between CT scans taken in different patient positions, for virtual colonoscopy.

These are published research methods, tested on real imaging data. They are not standard clinical tools.

The papers behind it

  1. 2012
    Brain Surface Conformal Parameterization With the Ricci Flow Yalin Wang, Jie Shi, Xiaotian Yin et al. · IEEE Transactions on Medical Imaging, vol. 31 Wang et al.: brain surface conformal parameterization with Ricci flow. DOI 10.1109/tmi.2011.2168233
  2. 2010
    Supine and Prone Colon Registration Using Quasi-Conformal Mapping Wei Zeng, J Marino, K Chaitanya Gurijala et al. · IEEE Transactions on Visualization and Computer Graphics, vol. 16 Zeng et al.: registering supine and prone colon scans with quasi-conformal maps. DOI 10.1109/tvcg.2010.200

Who's involved

No industry association, consortium or organised group works on this niche, as far as we could find. The work is happening in university research groups, through the papers above.

Discrete & graph RicciSurfaces

Networking

Routing in wireless sensor networks

DemonstratedFrom Ricci flow itself

In a sensor network with holes (lakes, buildings), simple "send it to the neighbour closest to the destination" routing gets stuck at the edges of holes. Gao, Gu, Luo and collaborators used discrete Ricci flow to re-embed the network into a space where greedy routing always succeeds: a disk with circular holes, or a hyperbolic covering space for resilience. It is a direct engineering use of the uniformization ideas behind Ricci flow on surfaces, demonstrated in simulations and published at the main networking venues.

The papers behind it

  1. 2010
    Resilient Routing for Sensor Networks Using Hyperbolic Embedding of Universal Covering Space Wei Zeng, Rik Sarkar, Feng Luo et al. · 2010 Proceedings IEEE INFOCOM Zeng, Sarkar, Luo, Gu and Gao: resilient routing using a hyperbolic embedding of the network's universal cover. DOI 10.1109/infcom.2010.5461988
  2. 2016
    Discrete Ricci Flow for Geometric Routing Jie Gao, Xianfeng David Gu, Feng Luo · Encyclopedia of Algorithms Gao, Gu and Luo: *Discrete Ricci flow for geometric routing*, a short encyclopedia overview. DOI 10.1007/978-1-4939-2864-4_602

Who's involved

No industry association, consortium or organised group works on this niche, as far as we could find. The work is happening in university research groups, through the papers above.

Discrete & graph RicciSurfaces

Network science

Communities and structure in networks

DemonstratedFrom Ricci flow itself

On a graph, Ollivier's Ricci curvature measures whether the neighbourhoods of two connected nodes overlap (positive curvature, as inside a tight community) or pull apart (negative curvature, like a bridge between communities). Running a discrete Ricci flow on the edge weights stretches the negatively curved bridges until communities separate. Ni, Lin, Luo and Gao showed this detects communities competitively. Earlier, curvature had been used to describe the structure of the Internet's network.

There is an open-source Python library, GraphRicciCurvature, and the method is now one of the standard curvature tools in network science research.

The papers behind it

  1. 2015
    Ricci curvature of the Internet topology Chien-Chun Ni, Yu-Yao Lin, Jie Gao et al. · 2015 IEEE Conference on Computer Communications (INFOCOM) Ni, Lin, Gao, Gu and Saucan: Ricci curvature of the Internet topology. DOI 10.1109/infocom.2015.7218668
  2. 2019
    Community Detection on Networks with Ricci Flow Chien-Chun Ni, Yu-Yao Lin, Feng Luo, Jie Gao · arXiv:1907.03993 Ni, Lin, Luo and Gao: community detection on networks with Ricci flow. PDFarXiv:1907.03993
  3. 2019
    Ollivier-Ricci Curvature-Based Method to Community Detection in Complex Networks Jayson Sia, Edmond Jonckheere, Paul Bogdan · Scientific Reports, vol. 9 Sia, Jonckheere and Bogdan: a curvature-based community detection method. DOI 10.1038/s41598-019-46079-x
  4. 2017
    Characterizing complex networks with Forman-Ricci curvature and associated geometric flows Melanie Weber, Emil Saucan, Jürgen Jost · Journal of Complex Networks, vol. 5 Weber, Saucan and Jost: Forman–Ricci curvature and geometric flows for complex networks. DOI 10.1093/comnet/cnw030

Who's involved

  • GraphRicciCurvature Open-source Python library for Ollivier and Forman Ricci curvature and Ricci flow on graphs, implementing the methods of Ni and colleagues.
  • Network Science Society The professional society for network science, which runs the annual NetSci conference where these methods are presented.
Discrete & graph Ricci

Cosmology

The shape of the universe

Early researchFrom Ricci flow itself

General relativity fixes the local curvature of space but not its global shape: space could be finite and "wrap around". Each possible shape is a 3-manifold, and the geometrization theorem says every closed 3-manifold is built from the eight model geometries. So the catalogue of candidate universes is complete. Most cosmology focuses on the flat and spherical cases, which were classified long before Perelman.

If space wraps around on a scale we can see, the cosmic microwave background should show pairs of matching circles (Cornish, Spergel and Starkman). Searches have found none so far, which rules out small universes of several shapes. Recent work by the COMPACT collaboration argues that many shapes remain allowed and that more refined tests are needed. This is open observational research, and the answer is not known.

The papers behind it

  1. 1996
    Circles in the Sky: Finding Topology with the Microwave Background Radiation Neil J. Cornish, David N. Spergel, Glenn D. Starkman · Class.Quant.Grav. 15 (1998) 2657-2670 Cornish, Spergel and Starkman: circles in the sky, finding topology with the microwave background. PDFarXiv:gr-qc/9602039
  2. 2022
    Cosmic topology. Part I. Limits on orientable Euclidean manifolds from circle searches Pip Petersen, Yashar Akrami, Craig J. Copi et al. · JCAP 01 (2023) 030 Petersen et al. (COMPACT): limits on Euclidean topologies from circle searches, and what remains open. PDFarXiv:2211.02603Journal version

Who's involved

No industry association, consortium or organised group works on this niche, as far as we could find. The work is happening in university research groups, through the papers above.

Geometrization3-manifolds

Same mathematics

Uses the ideas Ricci flow is built from: Ricci curvature, curvature flows, optimal transport, Ricci-flat metrics.

Medical imaging, simulation, visual effects

Image segmentation and moving interfaces

In useSame mathematics

Curve shortening and mean curvature flow, the extrinsic cousins of Ricci flow, have been working engineering tools for decades. Osher and Sethian's level-set method (1988) represents a moving curve or surface as the zero set of a function, so it can change topology freely. It is now a standard technique for simulating moving interfaces, from flames and fluids in films to etching in chip manufacturing. Caselles, Kimmel and Sapiro's geodesic active contours segment an image by flowing a curve by curvature, weighted towards object edges. They are built into standard medical-imaging software such as ITK.

This is the clearest case of the flow mathematics at the core of this site running in everyday software.

The papers behind it

  1. 1988
    Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations Stanley Osher, James A Sethian · Journal of Computational Physics, vol. 79 Osher and Sethian: fronts moving with curvature-dependent speed, the level-set method. DOI 10.1016/0021-9991(88)90002-2
  2. 1997
    Geodesic Active Contours Vicent Caselles, Ron Kimmel, Guillermo Sapiro · International Journal of Computer Vision, vol. 22 Caselles, Kimmel and Sapiro: geodesic active contours for image segmentation. DOI 10.1023/a:1007979827043

Who's involved

  • Insight Software Consortium A non-profit consortium that maintains ITK, the open-source medical image analysis toolkit, which includes level-set and geodesic active contour segmentation.
Curve shorteningMean curvature flowOther curvature flows

Machine learning and statistics

Optimal transport in machine learning

In useSame mathematics

Lott, Villani and Sturm showed that "Ricci curvature is bounded below" can be restated as a fact about optimal transport, the cheapest way to move one distribution of mass onto another. That restatement is the backbone of modern synthetic Ricci curvature. The same optimal transport mathematics is now a mainstream tool in machine learning. Cuturi's Sinkhorn algorithm made transport distances fast to compute. Wasserstein distances are used to train generative models, align datasets and compare distributions. Open-source libraries make it routine.

The connection to Ricci flow is through shared mathematics, not through the flow itself.

The papers behind it

  1. 2004
    Ricci curvature for metric-measure spaces via optimal transport John Lott, Cedric Villani · arXiv:math/0412127 Lott and Villani: Ricci curvature for metric measure spaces via optimal transport. PDFarXiv:math/0412127
  2. 2013
    Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances Marco Cuturi · Advances in Neural Information Processing Systems 26, pages 2292--2300, 2013 Cuturi: Sinkhorn distances, lightspeed computation of optimal transport. PDFarXiv:1306.0895
  3. 2017
    Wasserstein GAN Martin Arjovsky, Soumith Chintala, Léon Bottou · arXiv:1701.07875 Arjovsky, Chintala and Bottou: Wasserstein GAN, transport distances for training generative models. PDFarXiv:1701.07875

Who's involved

Limit spaces & RCD

Machine learning

Graph neural networks: fixing bottlenecks with curvature

DemonstratedSame mathematics

Graph neural networks pass messages between neighbouring nodes. Topping, Di Giovanni, Chamberlain, Dong and Bronstein showed that edges with very negative Ricci-type curvature are bottlenecks. They cause over-squashing, where information from far away is compressed into too little space. Their fix rewires the graph guided by curvature, in a procedure they call stochastic discrete Ricci flow. Curvature-based rewiring has since become an active topic in geometric deep learning, studied mostly on benchmarks rather than in deployed systems.

The papers behind it

  1. 2021
    Understanding over-squashing and bottlenecks on graphs via curvature Jake Topping, Francesco Di Giovanni, Benjamin Paul Chamberlain et al. · arXiv:2111.14522 Topping et al.: over-squashing and bottlenecks in graph neural networks, understood through curvature. PDFarXiv:2111.14522
  2. 2021
    Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges Michael M. Bronstein, Joan Bruna, Taco Cohen, Petar Veličković · arXiv:2104.13478 Bronstein, Bruna, Cohen and Veličković: *Geometric Deep Learning*, the field's framework text. PDFarXiv:2104.13478

Who's involved

  • Learning on Graphs Conference An annual research conference on machine learning on graphs and geometry, where curvature-based methods appear regularly.
Discrete & graph Ricci

Finance

Systemic risk in financial markets

Early researchSame mathematics

Build a network of stocks, linked by how strongly their returns move together, and compute its Ollivier–Ricci curvature. Sandhu, Georgiou and Tannenbaum found that this curvature tends to rise around market crashes. Their reading is that the market becomes more tightly coupled, and so more fragile. They proposed curvature as an indicator of systemic risk. This is a research result on historical data, not a validated trading or regulatory tool. We know of no industry group or standard built around it.

The papers behind it

  1. 2016
    Ricci curvature: An economic indicator for market fragility and systemic risk Romeil S. Sandhu, Tryphon T. Georgiou, Allen R. Tannenbaum · Science Advances, vol. 2 Sandhu, Georgiou and Tannenbaum: Ricci curvature as an economic indicator of market fragility and systemic risk. DOI 10.1126/sciadv.1501495

Who's involved

No industry association, consortium or organised group works on this niche, as far as we could find. The work is happening in university research groups, through the papers above.

Discrete & graph Ricci

Biology and medicine

Cancer and brain networks

Early researchSame mathematics

The same graph curvature has been applied to biological networks. In gene-interaction networks built from tumour data, Sandhu and colleagues found that cancer networks have different (higher) curvature than normal tissue. They link this to robustness, the ability of cancer networks to absorb perturbations. In brain imaging, Farooq and colleagues used the curvature of structural brain networks to assess where the brain's connectivity is robust to damage and where it is fragile. These are research findings that suggest hypotheses. They are not clinical tests.

The papers behind it

  1. 2015
    Graph Curvature for Differentiating Cancer Networks Romeil Sandhu, Tryphon Georgiou, Ed Reznik et al. · Scientific Reports, vol. 5 Sandhu et al.: graph curvature for differentiating cancer networks. DOI 10.1038/srep12323
  2. 2019
    Network curvature as a hallmark of brain structural connectivity Hamza Farooq, Yongxin Chen, Tryphon T. Georgiou et al. · Nature Communications, vol. 10 Farooq et al.: network curvature as a hallmark of brain structural connectivity. DOI 10.1038/s41467-019-12915-x

Who's involved

No industry association, consortium or organised group works on this niche, as far as we could find. The work is happening in university research groups, through the papers above.

Discrete & graph Ricci

Energy and control engineering

Congestion in power grids

Early researchSame mathematics

Jonckheere and collaborators have proposed using Ollivier–Ricci curvature of the grid's graph to find where congestion builds up and to manage it cheaply. The idea is that negatively curved links are bottlenecks, as in the graph neural network setting. It has been studied in conference papers and models only.

The papers behind it

  1. 2019
    Ollivier-Ricci Curvature Approach to Cost-Effective Power Grid Congestion Management Edmond Jonckheere, Eugenio Grippo · 2019 Chinese Control And Decision Conference (CCDC) Jonckheere and Grippo: an Ollivier–Ricci curvature approach to power grid congestion management. DOI 10.1109/ccdc.2019.8832819

Who's involved

No industry association, consortium or organised group works on this niche, as far as we could find. The work is happening in university research groups, through the papers above.

Discrete & graph Ricci

Theoretical physics

Calabi–Yau metrics for string theory

Early researchSame mathematics

String theory needs Ricci-flat metrics on Calabi–Yau manifolds, exactly the fixed points of Ricci flow. Yau proved they exist but gave no formula. Computing them numerically is needed to turn string compactifications into numbers, such as particle masses and couplings. Headrick and Wiseman computed Ricci-flat metrics on K3 surfaces numerically, and used Ricci flow itself as a numerical tool for black-hole metrics. Recent work trains neural networks to approximate Calabi–Yau metrics far faster (Ashmore–He–Ovrut; Douglas et al.).

This is fundamental physics, with no engineering use in sight, and it is still early.

The papers behind it

  1. 2005
    Numerical Ricci-flat metrics on K3 Matthew Headrick, Toby Wiseman · Class.Quant.Grav. 22 (2005) 4931-4960 Headrick and Wiseman: numerical Ricci-flat metrics on K3. PDFarXiv:hep-th/0506129Journal version
  2. 2006
    Ricci flow and black holes Matthew Headrick, Toby Wiseman · Class.Quant.Grav.23:6683-6708,2006 Headrick and Wiseman: Ricci flow and black holes, Ricci flow as a numerical tool. PDFarXiv:hep-th/0606086Journal version
  3. 2019
    Machine learning Calabi-Yau metrics Anthony Ashmore, Yang-Hui He, Burt Ovrut · arXiv:1910.08605 Ashmore, He and Ovrut: machine learning Calabi–Yau metrics. PDFarXiv:1910.08605Journal version
  4. 2020
    Numerical Calabi-Yau metrics from holomorphic networks Michael R. Douglas, Subramanian Lakshminarasimhan, Yidi Qi · arXiv:2012.04797 Douglas, Lakshminarasimhan and Qi: numerical Calabi–Yau metrics from holomorphic networks. PDFarXiv:2012.04797

Who's involved

No industry association, consortium or organised group works on this niche, as far as we could find. The work is happening in university research groups, through the papers above.

Kähler geometryPhysicsNumerics & visualization

From the prerequisites

Uses the mathematics you learn on the way to Perelman: topology, Laplacians and heat kernels on manifolds.

Data science, geometry processing

Manifold learning and shape signatures from heat kernels

In useFrom the prerequisites

The Laplacian and its heat kernel are core tools on the road to Perelman. Perelman and Bamler use the heat kernel of the conjugate heat equation to probe the geometry of a Ricci flow. The same objects are standard tools in data science:

  • Laplacian eigenmaps (Belkin–Niyogi) and diffusion maps (Coifman–Lafon) find the low-dimensional shape hidden in high-dimensional data by approximating the Laplacian of an underlying manifold. Spectral embedding of this kind is built into common machine-learning libraries.
  • Heat kernel signatures (Sun–Ovsjanikov–Guibas) describe each point of a 3D shape by how heat spreads from it. They are used for matching and retrieving shapes.

The papers behind it

  1. 2003
    Laplacian Eigenmaps for Dimensionality Reduction and Data Representation Mikhail Belkin, Partha Niyogi · Neural Computation, vol. 15 Belkin and Niyogi: Laplacian eigenmaps for dimensionality reduction. DOI 10.1162/089976603321780317
  2. 2006
    Diffusion maps Ronald R. Coifman, Stéphane Lafon · Applied and Computational Harmonic Analysis, vol. 21 Coifman and Lafon: diffusion maps. DOI 10.1016/j.acha.2006.04.006
  3. 2009
    A Concise and Provably Informative Multi‐Scale Signature Based on Heat Diffusion Jian Sun, Maks Ovsjanikov, Leonidas Guibas · Computer Graphics Forum, vol. 28 Sun, Ovsjanikov and Guibas: a multi-scale shape signature from heat diffusion. DOI 10.1111/j.1467-8659.2009.01515.x

Who's involved

No industry association, consortium or organised group works on this niche, as far as we could find. The work is happening in university research groups, through the papers above.

Heat kernel

Data science

Topological data analysis

In useFrom the prerequisites

Algebraic topology — homology, the fundamental group, the language in which the Poincaré conjecture is stated — is a prerequisite for understanding Perelman's work. Persistent homology applies it to data. Build a family of shapes from a point cloud at every scale, and track which holes and loops persist across scales. Features that persist are real structure; short-lived ones are noise. The method is used in materials science, neuroscience and biology, with mature open-source software.

The papers behind it

  1. 2002
    Topological Persistence and Simplification Edelsbrunner, Letscher, Zomorodian · Discrete & Computational Geometry, vol. 28 Edelsbrunner, Letscher and Zomorodian: topological persistence and simplification. DOI 10.1007/s00454-002-2885-2
  2. 2009
    Topology and data Gunnar Carlsson · Bulletin of the American Mathematical Society, vol. 46 Carlsson: *Topology and data*, the survey that introduced the field to many mathematicians. DOI 10.1090/s0273-0979-09-01249-x

Who's involved

3-manifolds