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Open research questions in Topological and Geometric Data Analysis

81 gap statements mined from Topological and Geometric Data Analysis papers in our 4.5M-paper local library, which holds 463 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.

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What the literature leaves open

  • The study suggests future research on the synthesis of advanced Mycielski-type graph families. The study suggests future research on the discovery of new configurations with specific topological properties.

    Comparing the Reverse Topological Indices of Mycielski Graph of Cycle Graphs · 2026 · DOI

  • The study identifies a need for a comprehensive basis for evaluating the connectivity properties of both chemical structures and technological networks.

    Comparing the Reverse Topological Indices of Mycielski Graph of Cycle Graphs · 2026 · DOI

  • The exact value of the maximum diameter of a 2-dimensional simplicial complex was not known prior to this work. The problem of packing squares of Hamilton cycles in the complete graph is identified as an open problem.

    The Maximum Diameter of 2-Dimensional Simplicial Complexes · 2026 · DOI

  • Further development of the DTL framework is needed. Experimental verification of the paper's findings is required. The application of the DTL framework to biological systems and processes could be explored.

    WORKING DRAFT The Ontological Status of Molecular Shape: A Structural Realist Approach via Discrete Topological Lattices · 2026 · DOI

  • The lack of a unified resolution to the fine-grained dynamic complexity lower bound problem. The lack of a framework to translate the computational limits of cell-probe models into absolute geometric invariants.

    A Five Part Validator Grade Metric Solution to Topological Foundations of Dynamic Fine-Grained Complexity: A Unified Resolution via K-Theoretic Index Closures and 8D Simplicial Regularization · 2026 · DOI

  • The lack of a coherent framework for switching between tree and ring representations. The need for a generalization of the Amoeba Monte Carlo algorithm for generating ensembles of trees with fluctuating sizes.

    Coherent modeling of double-folded ring polymers and their underlying random tree structure · 2026

  • The lack of a unified counts-only protocol for curvature estimation. The need for a rigorous convergence rate for discrete constructions. The absence of a quantitative identification theorem with explicit rate.

    Geometry of Deformed Cellular Spaces · 2026 · DOI

  • Future research will focus on deriving more explicit quantitative bounds for topological complexity in terms of noise parameters and geometric invariants, exploring different notions of stability, and applying these theoretical insights to concrete models in physics, engineering, and biological sciences.

    Topological Complexity and Global Stability of Non-Local Stochastic Flows on Compact Riemannian Manifolds · 2026 · DOI

  • The quest for a quantum theory of gravity remains an open problem in high-energy physics. The paper identifies a gap in the current understanding of the emergence of gravity from topological quantum field theory.

    Emergent Gravity from Topological Quantum Field Theory: Stochastic Gradient Flow Perspective away from the Quantum Gravity Problem · 2026 · DOI

  • Future research could explore the application of the paper's techniques to other problems in topology and geometry. The study of the topology of spaces of tight frames could be continued, with a focus on other topological properties.

    Simply connectedness of spaces of tight frames · 2026 · DOI

  • Macroeconomic models often lack kinematics, leading to violations of accounting identities. There is a need for a topological framework for understanding financial networks.

    The Topology of Conservation: Double-Entry Accounting as a Discrete Gauge Theory of Macroeconomics · 2026 · DOI

  • Future research should apply the pipeline to more complex tasks and datasets. Future research should investigate the use of T-AVO in other domains. Future research should explore the relationship between topological features and neural coding.

    A Falsificationist Pipeline for Neural Representational Analysis: · 2026 · DOI

  • The lack of understanding of phase transitions in higher-order interaction systems. The need to study the effects of higher-order interactions on phase transitions. The gap in understanding the role of group size in phase transitions.

    Phase transitions in the simplicial Ising model on hypergraphs · 2026 · DOI

  • The paper suggests future research directions in psychotherapy process research, quantum consciousness hypotheses, and cosmological models. The paper proposes to test the hypothesis that therapeutic breakthroughs correspond to topological phase transitions in the persistent barcode of the relational index τ.

    Topological Persistence of Identity Across Scales: A Category-Theoretic and Persistent Homology Framework for Relational Dynamics, Quantum Coherence, and Cyclic Cosmology · 2026 · DOI

  • The paper identifies the lack of a formal mathematical language capable of expressing what these transitions share without committing the category error of treating metaphors as physical equivalences. The paper notes that the literature lacks a formal bridge connecting relational identity preservation across psychological, quantum, and cosmological scales.

    Topological Persistence of Identity Across Scales: A Category-Theoretic and Persistent Homology Framework for Relational Dynamics, Quantum Coherence, and Cyclic Cosmology · 2026 · DOI

  • Traditional econometric frameworks and deep learning architectures have limitations in capturing deep non-linearities and global structural changes. There is a need for a novel approach to market dynamics that can decode the macro-behavior of financial datasets.

    Topological Quantization of Bitcoin Market Dynamics: A Phase-Space Analysis (2022–2026) · 2026 · DOI

  • The paper suggests studying the collection of homological quasi-trees from a (delta-)matroidal perspective. The paper suggests relating the period matrix to other objects in discrete geometry and combinatorics.

    Period matrices and homological quasi-trees on discrete Riemann surfaces · 2026 · DOI

  • To explore the implications of the LP law for our understanding of physical identity and transformation.

    The Structural Conditions of Physical Identity - How Physics Instantiates Without Deriving the Foundations of Persistent Transformation · 2026 · DOI

  • The current understanding of physical identity is implicit and not grounded. The information paradox and the quantum gravity problem are long-standing issues in physics.

    The Structural Conditions of Physical Identity - How Physics Instantiates Without Deriving the Foundations of Persistent Transformation · 2026 · DOI

  • We would like to emphasize that the robustness of the approach is not limited to the specific control parameter values studied in this study.

    Leveraging topological noise for dynamic state change detection using persistent homology · 2026 · DOI

  • Implementing the algorithm for generating randomly double-folded rings based on sampling wrapping codes. Extending the scheme to interacting systems with more complex interactions.

    Coherent modeling of double-folded ring polymers and their underlying random tree structure · 2026

  • Extending the construction to higher dimensions, - Investigating the application of the unified counts-only protocol to other discrete-curvature…

    Geometry of Deformed Cellular Spaces · 2026 · DOI

  • Future work could explore the exact quantitative relationship between jump parameters…

    Topological Complexity and Global Stability of Non-Local Stochastic Flows on Compact Riemannian Manifolds · 2026 · DOI

  • Further study of the stochastic gradient flow perspective on the emergence of gravity from topological quantum field theory. Development of new approaches to quantum gravity using the paper's framework. Investigation of the implications of the paper's results for our understanding of the nature of gravity and the behavior of matter at very small distances.

    Emergent Gravity from Topological Quantum Field Theory: Stochastic Gradient Flow Perspective away from the Quantum Gravity Problem · 2026 · DOI

  • Applying the topological approach to other financial datasets or markets. Investigating the relationship between the deterministic periodicity and investment decisions. Developing a real-time tracking system for the periodicity.

    Topological Quantization of Bitcoin Market Dynamics: A Phase-Space Analysis (2022–2026) · 2026 · DOI

Most-cited papers in Topological and Geometric Data Analysis

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Open questions raised by papers listed on this page include: (1) The lack of a unified resolution to the fine-grained dynamic complexity lower bound problem (“A Five Part Validator Grade Metric Solution to Topological Foundations of…”, 2026); (2) The lack of a unified counts-only protocol for curvature estimation (“Geometry of Deformed Cellular Spaces”, 2026); (3) Future research could explore the application of the paper's techniques to other problems in topology and geometry (“Simply connectedness of spaces of tight frames”, 2026); (4) There is a need for a topological framework for understanding financial networks (“The Topology of Conservation: Double-Entry Accounting as a Discrete Gauge…”, 2026); (5) Future research should apply the pipeline to more complex tasks and datasets (“A Falsificationist Pipeline for Neural Representational Analysis:”, 2026). 81 gap statements have been mined from Topological and Geometric Data Analysis papers in our 4.5M-paper local library, which holds 463 papers on the topic; the gaps come from whichever of those papers state one. They are mostly the research gaps the authors state and the papers' abstracts, plus future-work, limitations and challenges passages. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one, so you can read the original claim in context.

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