Open research questions in Topological and Geometric Data Analysis

What the literature leaves open

• The need for extensive human intervention in persistent homology. The difficulty in distinguishing between true features and noise in the data. The challenge of applying the framework to real-time data analysis.

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

• The methodology is based on simulated data. The study focuses on zebrafish skin patterns, and it is unclear how the results will generalize to other biological systems.

  Quantifying Topological Features and Irregularities in Zebrafish Patterns Using the Sweeping-Plane Filtration · 2026 · DOI

• To apply the methodology to other biological systems, such as cancer histology and microscopy images. To explore the use of other filtrations, such as the Vietoris-Rips filtration, in agent-based modeling studies. To develop new methods for quantifying features and irregularities in complex patterns.

  Quantifying Topological Features and Irregularities in Zebrafish Patterns Using the Sweeping-Plane Filtration · 2026 · DOI

• The ambiguity in the choice of scale in topological data analysis. The need for a methodology that can distinguish meaningful topology from small-scale fluctuations.

  Persistent Homology in Topological Data Analysis: Theory and Applications · 2026 · DOI

• The current understanding of the fundamental nature of reality is incomplete, with issues such as the fine-tuning problem and dark matter stability. The OFN addresses this gap by providing a unified geometric framework for understanding the fundamental nature of reality.

  MANIFESTO OF THE ONTOLOGY OF THE FUNDAMENTAL NETWORK (OFN) · 2026 · DOI

• Multi-parameter persistence. Differentiable persistence modules. Scalable algorithms for large data.

  Persistent Homology in Topological Data Analysis: Theory and Applications · 2026 · DOI

• Although prior research focuses on geometric accuracy and semantic completeness of HD maps, topology remains underexplored.

  Topological Analysis of OpenDRIVE Models for Advanced Autonomous Vehicle Simulations · 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 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

• The paper identifies a gap in the understanding of discrete period matrices and their relation to combinatorics. The paper addresses a question posed by Richard Kenyon.

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

• Further investigation of the interplay between topological complexity and global stability for non-local stochastic flows on compact Riemannian manifolds. The development of more accurate and robust mathematical models for stochastic systems. The application of the results to modeling complex phenomena across diverse fields.

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

• The lack of understanding of non-local stochastic flows on compact Riemannian manifolds. The need to develop a framework to analyze global stability in probability for these systems. The necessity to characterize the topological complexity of the flow's phase space.

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

• the lack of a comparative structural framework for identifying recurring persistence topology across mathematically distinct systems, - the need for a framework that can analyse recurring transformational structure across different domains

  Persistence Fingerprint Analysis (PFA): A Cross-Scale Mathematical Morphology Framework · 2026 · DOI

• To further develop the LP law and its applications. 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

• 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

• 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

• The application of the framework to other types of dynamical systems. The development of new metrics and approaches to improve the accuracy of the framework.

  Leveraging topological noise for dynamic state change detection using persistent homology · 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 failure to discriminate between competing theories of neural population coding is partly methodological. There is a need for a falsificationist pipeline for neural representational analysis. There is a need for a corrected implementation of persistent homology.

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

• The present work is numerical and exploratory, with no claim made that the observed structures correspond directly to any known physical system. The local motif analysis should be regarded as heuristic, with further investigation required to interpret the results. The topological analysis is limited to the parameter regime explored in the study.

  GQR88: Preliminary Investigation of Three-Dimensional Survivor Ecologies Geometry, Branching and Topological Suppression in Non-Hermitian Memory Fields · 2026 · DOI

Most-cited papers in Topological and Geometric Data Analysis

• Topological Signal Processing Over Simplicial Complexes · IEEE Transactions on Signal Processing · 2020 · 188 citations
• Chemical applicability of Sombor indices · Journal of the Serbian Chemical Society · 2021 · 137 citations
• Exploring Spatial Process Dynamics Using Irregular Cellular Automaton Models · Geographical Analysis · 2001 · 64 citations
• The promises of persistent homology, machine learning, and deep neural networks in topological data analysis of democracy survival · Quality & Quantity · 2023 · 10 citations
• Representing 3D Topological Adjacencies between Volumes Using a 36-Intersection Model · Geomatics and Environmental Engineering · 2022 · 8 citations
• An Invitation to the Euler Characteristic Transform · American Mathematical Monthly · 2024 · 7 citations
• On Relationships of Eigenvalue–Based Topological Molecular Descriptors · Acta chimica slovenica · 2020 · 5 citations
• Beyond distance: quantifying point cloud dynamics with persistent homology and dynamic optimal transport · Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences · 2026 · 1 citations
• Generalizing the Gaussian Network Model: Spanning‐Tree Thermodynamics Shows Entropy‐Driven <scp>KRAS</scp> Activation · Proteins Structure Function and Bioinformatics · 2026 · 1 citations
• Hidden Topology of Spin Chains · Physics · 2025 · 0 citations

Most recent work

• Beyond distance: quantifying point cloud dynamics with persistent homology and dynamic optimal transport · Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences · 2026
• Generalizing the Gaussian Network Model: Spanning‐Tree Thermodynamics Shows Entropy‐Driven <scp>KRAS</scp> Activation · Proteins Structure Function and Bioinformatics · 2026
• Topological Study of β-Sparsified d-Uniform Hypergraph-Based Simplicial Complexes · Mathematics · 2026
• Comparing the Reverse Topological Indices of Mycielski Graph of Cycle Graphs · Journal of Modern Technology and Engineering · 2026
• The Maximum Diameter of 2-Dimensional Simplicial Complexes · Discrete & Computational Geometry · 2026
• WORKING DRAFT The Ontological Status of Molecular Shape: A Structural Realist Approach via Discrete Topological Lattices · Zenodo (CERN European Organization for Nuclear Research) · 2026
• Topological feature engineering and persistent homology hybrid deep learning model for time-series prediction · Complex & Intelligent Systems · 2026
• Tree-Skeleton Collaborative Learning Promotes Topological Completeness in Tree-like Structure Segmentation · International Journal of Computational Intelligence Systems · 2026
• The Toroidal Chip: Converging Biological Substrate, Topological Storage, and Language Models · Zenodo (CERN European Organization for Nuclear Research) · 2026
• Multi-Scale Symmetry Analysis in Molecular Structures · match Communications in Mathematical and in Computer Chemistry · 2026

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