Navigating the Geometric Speedup of Innovation: A Topology/Granular to Geometric Assoiciative Memory Approach to Creative and Competitive Spaces

Co-created by divergent thinking of the Catalyzer Think Tank and Gemini Deep Research tool

The analysis of contemporary datasets presents increasing challenges due to their inherent complexity and high dimensionality. Traditional analytical methods often fall short in revealing the intricate underlying structures, particularly within abstract domains such as the creative process and the competitive landscape of unmet needs. In response to these limitations, Topological Data Analysis (TDA) has emerged as a powerful suite of techniques capable of discerning hidden shapes and relationships within data.1 This approach moves beyond conventional statistical methods by focusing on the fundamental topological features of data, such as connectivity, loops, and voids, which are invariant under continuous deformations.1 This inherent robustness allows TDA to effectively handle noisy and high-dimensional datasets, offering a novel lens through which to explore complex phenomena.4

Complementing TDA, hyperbolic geometry, a non-Euclidean space characterized by constant negative curvature, offers unique advantages for representing hierarchical data and intricate relationships.7 Unlike Euclidean space, where volume grows polynomially with radius, hyperbolic space exhibits exponential volume growth, a property analogous to the branching structure of trees and hierarchies.10 This characteristic makes hyperbolic geometry particularly well-suited for embedding knowledge structures, which often possess inherent hierarchical organizations, with lower distortion than traditional Euclidean methods.13

Furthermore, the Thurston Geometrization Conjecture, a foundational concept in the field of 3-manifold topology, posits that all closed 3-manifolds can be decomposed into fundamental geometric pieces, each possessing one of eight distinct geometric structures.15 This conjecture provides a potential framework for understanding the global structure of knowledge by drawing an analogy between the decomposition of 3-manifolds and the organization of diverse knowledge domains.15

This report investigates the potential of integrating TDA methods, particularly those related to hyperbolic 3-manifolds and the concept of lensing, for analyzing the “unknown-unknown creative space” and the “competitive space of unmet and unseen needs.” It also explores how the principles of the Thurston Geometrization Conjecture might be incorporated to represent the global structure of knowledge within these complex landscapes.

Topological Data Analysis (TDA): A New Lens for Data Exploration

Topological Data Analysis (TDA) offers a distinct paradigm for exploring complex datasets by focusing on their underlying shape and connectivity.1 Unlike traditional methods that rely on specific metrics or linear assumptions, TDA employs concepts from topology to extract robust and invariant features from data.1 The core principle of TDA lies in the idea that the inherent structure of data, often referred to as its “shape,” contains meaningful information that can be crucial for understanding complex systems.1

In TDA, data is typically treated as a discrete metric space, where distances between all pairs of data points are defined.1 From this metric space, TDA constructs abstract simplicial complexes, such as the Vietoris-Rips and Cech complexes, to represent the topological structure of the data.1 These complexes are built by connecting data points that are within a certain distance of each other, forming higher-dimensional geometric shapes that capture the relationships between the points.1 This process effectively converts a discrete set of data points into a continuous topological structure, enabling the application of powerful mathematical tools to analyze its shape.1

A key technique within TDA is persistent homology, which computes topological features like connected components, loops, and voids across multiple scales or levels of resolution.2 By analyzing how these features appear (birth) and disappear (death) as the scale changes, persistent homology provides a robust way to distinguish between genuine topological signals and noise within the data.2 The results of this analysis are often visualized as persistence diagrams, where longer bars or points represent more significant and persistent topological features.2

TDA offers several advantages for analyzing complex data. Its ability to handle high-dimensional data effectively makes it suitable for modern datasets with numerous variables.1 Furthermore, TDA’s robustness to noise and outliers ensures that the identified topological features are less likely to be influenced by spurious data points.1 By providing a global perspective on the data structure, TDA can reveal overarching patterns and relationships that might be missed by local or linear methods.1 Importantly, TDA is not a standalone technique; it can be combined with other machine learning and statistical methods to enhance their performance and provide deeper insights.1

The Power of Hyperbolic Geometry in Representing Hierarchical Data

While Euclidean space has long been the standard for data representation, its limitations become apparent when dealing with hierarchical data. The polynomial growth of volume in Euclidean space makes it inefficient for embedding data with exponential branching structures, such as those found in taxonomies, organizational charts, and potentially, knowledge domains.7 To represent these hierarchical relationships effectively in Euclidean space often requires high embedding dimensions, leading to increased computational complexity and potential loss of information.7

Hyperbolic geometry, a non-Euclidean geometry characterized by a constant negative curvature, offers a compelling alternative for embedding hierarchical data.7 A key property of hyperbolic space is its exponential growth of volume with radius, which mirrors the branching nature of hierarchical structures like trees.7 This allows for embedding hierarchical data with significantly lower distortion in fewer dimensions compared to Euclidean space.7 Various models of hyperbolic space exist, including the Poincaré ball and the Lorentz model, each offering different advantages for computation and visualization.7

Embedding knowledge graphs, which often exhibit hierarchical and logical patterns, into hyperbolic space has shown promising results.7 Hyperbolic embeddings can capture both the hierarchical relationships between entities (e.g., a concept and its sub-concepts) and the logical patterns governing their interactions.7 Techniques like hyperbolic rotations and reflections can be used to model complex relational patterns within the hyperbolic embedding space.7 Leveraging hyperbolic geometry for knowledge graph embeddings can lead to more accurate and efficient representations of complex knowledge structures, potentially revealing insights that are obscured in Euclidean embeddings.7

The Thurston Geometrization Conjecture: A Blueprint for Global Knowledge Structure?

The Thurston Geometrization Conjecture, now a theorem proven by Grigori Perelman, provides a profound understanding of the topology of 3-manifolds.15 Analogous to the uniformization theorem for 2-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries (Euclidean, spherical, or hyperbolic), the geometrization conjecture extends this idea to three dimensions.15 It posits that every closed 3-manifold can be decomposed in a canonical way into pieces, each of which has one of eight types of geometric structure.15 These eight Thurston geometries include Euclidean geometry, hyperbolic geometry, spherical geometry, and five others.16

The conjecture involves a two-level decomposition of 3-manifolds. First, every closed 3-manifold has a prime decomposition into an essentially unique collection of prime 3-manifolds.16 Then, irreducible orientable compact 3-manifolds have a canonical Jaco-Shalen-Johannson (JSJ) torus decomposition, where they are cut along a minimal collection of disjointly embedded incompressible tori into pieces that are either atoroidal or Seifert fibered.16 Thurston’s conjecture states that after these decompositions, each resulting piece admits exactly one of the eight geometric structures.16

This conjecture offers a potential analogy for understanding the global structure of knowledge. The eight Thurston geometries could represent fundamental types of knowledge organization or domains. For instance, well-defined and bounded domains of knowledge might be analogous to spherical geometry, while hierarchical knowledge structures could be related to hyperbolic geometry.16 Flat, well-established fields of knowledge might correspond to Euclidean geometry. The decomposition process itself, breaking down complex 3-manifolds into more manageable geometric pieces, could mirror how complex knowledge is often organized into more fundamental components or disciplines.16 The Thurston Geometrization Conjecture, therefore, presents a high-level framework for conceptualizing a “global structure of knowledge,” where different knowledge areas are mapped to these eight geometric structures, and their relationships are governed by the decomposition process outlined in the conjecture.

Integrating TDA, Hyperbolic 3-Manifolds, and Lensing for Innovation Analysis

The “unknown-unknown creative space” represents the realm of ideas and innovations that are yet to be conceived or even imagined. Analyzing this space requires methods capable of exploring vast possibilities and identifying truly novel concepts. Topological Data Analysis, when combined with the representational power of hyperbolic 3-manifolds, offers a potential approach. By representing the space of potential creative ideas as a high-dimensional dataset, where each dimension corresponds to a feature or attribute of an idea, TDA can be employed to identify underlying topological features.1 Connected components in this topological representation might correspond to broad themes or categories of ideas, while loops could indicate potential novel intersections or combinations of existing concepts.1 Voids, or higher-dimensional holes, might represent unexplored areas within the creative space, suggesting entirely new directions for innovation.1

Embedding this high-dimensional space of creative ideas into a hyperbolic 3-manifold could allow for capturing the inherent hierarchical relationships between different concepts.7 More general or foundational ideas could be positioned closer to the origin of the hyperbolic space, with more specific or derivative ideas located further away, reflecting the hierarchical structure of knowledge and creativity.7

Applying “lensing” techniques within this framework could involve using specific filter functions in algorithms like Mapper, a TDA tool, to focus on particular aspects of the data that are relevant to innovation triggers or emerging trends.4 For example, a filter function could be designed to highlight data points associated with identified unmet needs.18 Dimensionality reduction techniques, acting as a “lens,” could project the high-dimensional hyperbolic embedding onto lower dimensions for visualization and focused analysis, while still preserving the essential topological features of interest.20 This combination of TDA in a hyperbolic space with lensing techniques might enable the identification of novel combinations of ideas or entirely new creative directions that are not apparent when using traditional analytical methods in Euclidean space.

Similarly, analyzing the “competitive space of unmet and unseen needs” can benefit from this integrated approach. Market data, customer feedback, and competitor analysis can be represented as a high-dimensional dataset.22 TDA can then be used to find topological features within this data that are indicative of unmet needs.25 For instance, voids in the topological representation might correspond to gaps in the market that are not currently being addressed by existing products or services.25 Clusters of similar complaints or requests in customer feedback could manifest as connected components, highlighting areas of high unmet demand.25

Embedding this competitive space into a hyperbolic 3-manifold could help model the competitive landscape and potential niches within it.7 Different market segments or competitor offerings could be positioned based on their relationships and hierarchical structures within the space. Applying “lensing” in this context could involve focusing on non-customers or analyzing weak signals in the market to identify underserved segments or emerging needs.28 By focusing on these less obvious areas, the combination of TDA, hyperbolic geometry, and lensing could provide a powerful framework for identifying “blue ocean” markets and predicting future needs more accurately than traditional market analysis techniques.28

Capturing Long-Range Dependencies and Hierarchical Structures with TDA

A significant advantage of Topological Data Analysis lies in its ability to capture both long-range dependencies and hierarchical structures within complex datasets. Persistent homology, a core technique in TDA, analyzes data across multiple scales, allowing it to identify features that persist over a wide range of parameters.44 These persistent features can often reveal long-range relationships within the data that might be missed by methods focusing on local patterns or fixed scales.47 For instance, in the context of a knowledge graph representing the “globality of knowledge,” persistent loops in the TDA representation could indicate non-obvious or long-range semantic connections between concepts that are not directly linked.49 Multi-parameter persistent homology, an extension of the standard technique, allows for the analysis of data with multiple filtration parameters simultaneously, potentially capturing even more intricate and complex dependencies within the data.2

Furthermore, the combination of hyperbolic embeddings with TDA provides a powerful approach for analyzing hierarchical structures. Hyperbolic spaces, with their tree-like properties, are naturally suited for embedding hierarchical data with low distortion.7 When TDA techniques, such as persistent homology, are applied to data embedded in a hyperbolic space, they can effectively identify and quantify the hierarchical clustering structures present.55 This synergy allows for a more nuanced understanding of the inherent hierarchies within both creative and needs spaces. For example, in analyzing the “competitive space of unmet needs,” a hierarchical structure might emerge where broad, fundamental needs branch into more specific, granular requirements. TDA in a hyperbolic space could effectively map and analyze these hierarchical relationships, providing valuable insights for innovation and market strategy.

Computational Challenges and Potential Benefits

Applying the advanced topological and geometric methods discussed in this report to the problem of identifying novel ideas and predicting future needs presents both significant computational challenges and substantial potential benefits.

One of the primary challenges lies in the computational intensity of TDA, particularly when dealing with large and high-dimensional datasets.55 The construction of simplicial complexes and the computation of persistent homology can be resource-intensive, requiring significant processing power and memory.56 Similarly, embedding data into hyperbolic space and performing subsequent analytical operations on hyperbolic manifolds can also be computationally demanding.7 Efficient algorithms and access to adequate computational resources would be crucial for the practical application of these methods to real-world innovation and market analysis scenarios.15 Techniques aimed at mitigating the computational costs of TDA, such as focusing on graph characteristics 59, could prove valuable in making these methods more feasible for large-scale datasets.

Despite these computational challenges, the potential benefits of employing TDA and hyperbolic geometry for innovation analysis are considerable. These methods offer the possibility of identifying novel patterns and insights that might be missed by traditional analytical approaches.1 By providing a deeper understanding of the underlying structure of complex, abstract spaces like the “unknown-unknown creative space” and the “competitive space of unmet and unseen needs,” these techniques could unlock new avenues for innovation and strategic decision-making. Furthermore, the ability to capture long-range dependencies and inherent hierarchical structures within data could lead to more accurate predictions of future trends and a more comprehensive understanding of the “globality of knowledge”.58

Alternative Geometric and Topological Methods in Data Analysis

While this report focuses on the potential of TDA, hyperbolic 3-manifolds, and lensing, it is important to acknowledge that other geometric and topological methods in data analysis might also be relevant to the user’s query. The Mapper algorithm, another key technique in TDA, provides a powerful approach for visualizing high-dimensional data by constructing a simplified, low-dimensional representation that preserves the essential topological features.1 Manifold learning techniques, such as Isomap, Locally Linear Embedding (LLE), t-distributed Stochastic Neighbor Embedding (t-SNE), and Uniform Manifold Approximation and Projection (UMAP), offer alternative methods for dimensionality reduction and revealing underlying data structures, potentially acting as a “lens” onto the data.20

Furthermore, the field of complexity science, with its focus on emergent behaviors in complex systems, and network analysis, which studies the relationships and structures within networks, might provide complementary insights into the dynamics of creative and needs spaces.62 These methods could help in understanding how individual ideas or needs interact and give rise to larger, system-level patterns.

Conclusion and Future Directions

The analysis presented in this report suggests that Topological Data Analysis, particularly when integrated with the representational power of hyperbolic geometry and the focused exploration offered by lensing techniques, holds significant potential for analyzing complex, abstract spaces such as the “unknown-unknown creative space” and the “competitive space of unmet and unseen needs.” The ability of TDA to uncover hidden topological structures, combined with hyperbolic geometry’s proficiency in embedding hierarchical data, provides a novel framework for understanding the shape and relationships within these intricate landscapes. The analogy drawn from the Thurston Geometrization Conjecture offers a compelling, high-level perspective on the global organization of knowledge, potentially guiding the structuring and interpretation of knowledge graphs representing these spaces.

Developing a novel approach by combining these concepts with knowledge graph theory appears to be a promising direction for future research. Knowledge graphs, with their capacity to represent entities and relationships, could serve as a natural data structure for encoding the topological and geometric information extracted through TDA and hyperbolic embeddings. The Thurston geometries could potentially inform the schema and organization of such knowledge graphs, providing a foundational structure for the “globality of knowledge.”

Future research should focus on several key areas. Investigating the development of new TDA methods specifically tailored for data embedded in hyperbolic spaces could enhance the analysis of hierarchical knowledge structures. Further exploration of the analogy between the eight Thurston geometries and different domains or organizations of knowledge could lead to new ways of structuring and understanding complex information. Addressing the computational challenges associated with these methods through the development of more efficient algorithms and leveraging high-performance computing resources will be crucial for practical applications. Finally, fostering interdisciplinary collaboration between mathematicians, data scientists, and domain experts in innovation, market analysis, and knowledge management will be essential for realizing the full potential of these advanced topological and geometric approaches.

Works cited

1. Topological Data Analysis – Siobhan K Cronin, accessed March 31, 2025, https://siobhankcronin.com/notes/math/topology/
2. Topological data analysis | Christian Hirsch, accessed March 31, 2025, https://christian-hirsch.netlify.app/project/tda/
3. An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists – Frontiers, accessed March 31, 2025, https://www.frontiersin.org/journals/artificial-intelligence/articles/10.3389/frai.2021.667963/full
4. Machine Learning Explanations with Topological Data Analysis – Nathaniel Saul, accessed March 31, 2025, https://sauln.github.io/blog/tda_explanations.html/
5. Topological data analysis – Wikipedia, accessed March 31, 2025, https://en.wikipedia.org/wiki/Topological_data_analysis
6. Demystifying Complex Patterns with Topological Data Analysis (TDA) | by Janhavi Giri, accessed March 31, 2025, https://medium.com/@janhavi.giri/demystifying-complex-patterns-with-topological-data-analysis-tda-acde223f3ba9
7. pubmed.ncbi.nlm.nih.gov, accessed March 31, 2025, https://pubmed.ncbi.nlm.nih.gov/39386606/#:~:text=Using%20ALBATROSS%20(Stier%2C%20A.%20J.%2C,and%20beyond%20a%20traditional%20Euclidean
8. Navigating Memorability Landscapes: Hyperbolic Geometry Reveals Hierarchical Structures in Object Concept Memory – PubMed, accessed March 31, 2025, https://pubmed.ncbi.nlm.nih.gov/39386606/
9. Navigating Memorability Landscapes: Hyperbolic Geometry Reveals Hierarchical Structures in Object Concept Memory | bioRxiv, accessed March 31, 2025, https://www.biorxiv.org/content/10.1101/2024.09.22.614329v1.full
10. Knowledge Association with Hyperbolic Knowledge Graph Embeddings – ACL Anthology, accessed March 31, 2025, https://aclanthology.org/2020.emnlp-main.460.pdf
11. Knowledge Graph Representation via Hierarchical Hyperbolic Neural Graph Embedding, accessed March 31, 2025, https://par.nsf.gov/servlets/purl/10324491
12. Fully Hyperbolic Rotation for Knowledge Graph Embedding – arXiv, accessed March 31, 2025, https://arxiv.org/html/2411.03622v2
13. Complex Hyperbolic Knowledge Graph Embeddings with Fast Fourier Transform – ACL Anthology, accessed March 31, 2025, https://aclanthology.org/2022.emnlp-main.349.pdf
14. Low-Dimensional Knowledge Graph Embeddings via Hyperbolic Rotations, accessed March 31, 2025, https://grlearning.github.io/papers/101.pdf
15. How does Thurston’s geometrisation conjecture imply Poincaré’s conjecture?, accessed March 31, 2025, https://math.stackexchange.com/questions/324554/how-does-thurstons-geometrisation-conjecture-imply-poincar%C3%A9s-conjecture
16. Thurston’s Geometrization Conjecture — from Wolfram MathWorld, accessed March 31, 2025, https://mathworld.wolfram.com/ThurstonsGeometrizationConjecture.html
17. Geometrization conjecture – Wikipedia, accessed March 31, 2025, https://en.wikipedia.org/wiki/Geometrization_conjecture
18. Identification of innovation potential – 6 important triggers for innovation, accessed March 30, 2025, https://www.lead-innovation.com/en/insights/english-blog/identification-of-innovation-potential-6-important-triggers-for-innovation
19. Innovation Validation – The Complete Guide – Horizon, accessed March 30, 2025, https://www.gethorizon.net/guides/innovation-validation-the-complete-guide
20. 2.2. Manifold learning — scikit-learn 1.6.1 documentation, accessed March 31, 2025, https://scikit-learn.org/stable/modules/manifold.html
21. Visualizing High-Dimensional Data: Advances in the Past Decade – ResearchGate, accessed March 31, 2025, https://www.researchgate.net/publication/307513199_Visualizing_High-Dimensional_Data_Advances_in_the_Past_Decade
22. Market Trend Analysis: A Simple Step-by-Step Guide – Exploding Topics, accessed March 30, 2025, https://explodingtopics.com/blog/market-trend-analysis
23. The Role of Market Research in Predicting Trends – Comparables.ai, accessed March 30, 2025, https://www.comparables.ai/articles/role-of-market-research-in-predicting-trends
24. Ecommerce Trend Forecasting with Secondary Market Research – Mailchimp, accessed March 30, 2025, https://mailchimp.com/resources/secondary-market-research/
25. Disruptive Innovation: A Game-Changing Product Strategy for …, accessed March 30, 2025, https://medium.com/design-bootcamp/disruptive-innovation-a-game-changing-product-strategy-for-modern-businesses-a6766dbfc9d1
26. The Knowns and Unknowns Framework for Design Thinking …, accessed March 30, 2025, https://marvelapp.com/blog/known-unknowns-framework-design-thinking/
27. How to Use UX Research Methods in Design Thinking – Survicate, accessed March 30, 2025, https://survicate.com/blog/design-thinking-ux-research/
28. What is Blue Ocean Strategy, accessed March 30, 2025, https://www.blueoceanstrategy.com/what-is-blue-ocean-strategy/
29. Blue Ocean Strategy for New Product Development – Beyond the Backlog, accessed March 30, 2025, https://beyondthebacklog.com/2024/11/09/blue-ocean-strategy/
30. Searching for Blue Oceans in Beverages | BevSource, accessed March 30, 2025, https://www.bevsource.com/news/searching-for-blue-oceans-in-beverages
31. Blue Ocean Strategy: Examples & How to Apply It, accessed March 30, 2025, https://www.clearpointstrategy.com/blog/blue-ocean-strategy
32. How To Be Creative – 6 Proven Ways To Be More Creative …, accessed March 30, 2025, https://www.blueoceanstrategy.com/blog/how-to-be-creative/
33. Maximize Angel Returns with Blue Ocean Strategy, accessed March 30, 2025, https://businessangelinstitute.org/blog/2024/10/10/maximize-angel-returns-blue-ocean-opportunities/
34. Blue Ocean Strategy: Create Uncontested Market Space – Dotwork, accessed March 30, 2025, https://www.dotwork.com/post/blue-ocean-strategy-create-uncontested-market-space
35. What Is Blue Ocean Strategy And How Can You Create One? – Bernard Marr, accessed March 30, 2025, https://bernardmarr.com/what-is-blue-ocean-strategy-and-how-can-you-create-one/
36. Unveiling the Secrets of Blue Ocean Strategy for Business Growth – LeanScape, accessed March 30, 2025, https://leanscape.io/unveiling-the-secrets-of-blue-ocean-strategy-for-business-growth/
37. Known Unknowns: Unmeasurable Hazards and the Limits of Risk Regulation – University of Oklahoma College of Law Digital Commons, accessed March 30, 2025, https://digitalcommons.law.ou.edu/cgi/viewcontent.cgi?article=2304&context=olr
38. Weak Signals Analyse ⇒ Risiken & Trends erkennen | 4strat, accessed March 30, 2025, https://www.4strat.com/future/weak-signal-analysis/
39. What is Weak Signals? – Really Good Innovation, accessed March 30, 2025, https://www.reallygoodinnovation.com/glossaries/weak-signals
40. Weak signals: don’t miss them during monitoring work | Technology Knowledge Metrix, accessed March 30, 2025, https://tkminnovation.io/en/weak-signal-articles/
41. The Blue Ocean Strategy: How to Create Innovative Markets | Optimum7, accessed March 30, 2025, https://www.optimum7.com/blog/blue-ocean-strategy-creating-uncontested-markets.html
42. Identifying and Meeting Customer Needs – Conductor, accessed March 30, 2025, https://www.conductor.com/academy/customer-needs/
43. Recognizing Unmet Needs And Potential Business Ideas …, accessed March 30, 2025, https://fastercapital.com/topics/recognizing-unmet-needs-and-potential-business-ideas.html
44. Statistical inference for dependence networks in topological data analysis – PMC, accessed March 31, 2025, https://pmc.ncbi.nlm.nih.gov/articles/PMC10752923/
45. Statistical inference for dependence networks in topological data analysis – Frontiers, accessed March 31, 2025, https://www.frontiersin.org/journals/artificial-intelligence/articles/10.3389/frai.2023.1293504/full
46. Topological Data Analysis Approaches to Uncovering the Timing of Ring Structure Onset in Filamentous Networks – PubMed Central, accessed March 31, 2025, https://pmc.ncbi.nlm.nih.gov/articles/PMC7811524/
47. Full article: Topological data analysis and machine learning – Taylor & Francis Online, accessed March 31, 2025, https://www.tandfonline.com/doi/full/10.1080/23746149.2023.2202331
48. Position Paper: Challenges and Opportunities in Topological Deep Learning – arXiv, accessed March 31, 2025, https://arxiv.org/html/2402.08871v1
49. Topological Data Analysis for Multivariate Time Series Data – MDPI, accessed March 31, 2025, https://www.mdpi.com/1099-4300/25/11/1509
50. www.mdpi.com, accessed March 31, 2025, https://www.mdpi.com/1099-4300/24/8/1116#:~:text=The%20field%20of%20topological%20data,%2C%20filtrations)%20on%20the%20data.
51. Multiscale Methods and Machine Learning – KDnuggets, accessed March 31, 2025, https://www.kdnuggets.com/2018/03/multiscale-methods-machine-learning.html
52. Multiscale Methods for Signal Selection in Single-Cell Data – MDPI, accessed March 31, 2025, https://www.mdpi.com/1099-4300/24/8/1116
53. NeuMapper: A scalable computational framework for multiscale exploration of the brain’s dynamical organization – PubMed Central, accessed March 31, 2025, https://pmc.ncbi.nlm.nih.gov/articles/PMC9207992/
54. Why you should use Topological Data Analysis over t-SNE or UMAP? – DataRefiner, accessed March 31, 2025, https://datarefiner.com/feed/why-tda
55. Clustering and Topological Data Analysis: Comparison and Application – ScholarSpace, accessed March 31, 2025, https://scholarspace.manoa.hawaii.edu/bitstreams/0c2a0e74-fa74-4658-abf3-be2ae5a1f3b1/download
56. Computational topology – Wikipedia, accessed March 31, 2025, https://en.wikipedia.org/wiki/Computational_topology
57. Computing Varieties of Representations of Hyperbolic 3-Manifolds into SL(4, R) – UCSB Math, accessed March 31, 2025, https://web.math.ucsb.edu/~cooper/44.pdf
58. The Classification of 3-Manifolds — A Brief Overview, accessed March 31, 2025, https://pi.math.cornell.edu/~hatcher/Papers/3Msurvey.pdf
59. Explaining the Power of Topological Data Analysis in Graph Machine Learning – arXiv, accessed March 31, 2025, https://arxiv.org/html/2401.04250v1
60. Why is the mapping class group of hyperbolic manifolds finite? – MathOverflow, accessed March 31, 2025, https://mathoverflow.net/questions/66459/why-is-the-mapping-class-group-of-hyperbolic-manifolds-finite
61. On Some Mathematics for Visualizing High Dimensional Data – George Mason University, accessed March 31, 2025, http://binf.gmu.edu/jsolka/PAPERS/MathVisRevision.pdf
62. Emergent Behavior in Complex Systems of Interacting Agents – IMSI, accessed March 30, 2025, https://www.imsi.institute/activities/emergent-behavior-in-complex-systems-of-interacting-agents/
63. Can we predict the unpredictable? Big Brains podcast with J. Doyne Farmer, accessed March 30, 2025, https://news.uchicago.edu/big-brains-podcast-can-we-predict-unpredictable-j-doyne-farmer

Can Machines Time Markets? The Virtue of Complexity in Return Prediction, accessed March 30, 2025, https://www.aqr.com/Insights/Research/Alternative-Thinking/Can-Machines-Time-Markets-The-Virtue-of-Complexity-in-Return-Prediction