Conflict-Driven Theory Formation: Synthesis of Contradictory Persistent Homology Results via Meta-Analysis

While Bobrowski & Skraba (2024) demonstrate universality in the limiting distributions of persistence diagrams, other works (e.g., Duan et al., 2022; Wang et al., 2023) highlight context-dependent or unstable behavior in PH-derived features. This research direction proposes a meta-analysis of persistent homology literature and datasets, aiming to map out and categorize the types of conflicts—such as differing stability results, or contradictory topological features extracted from similar data modalities. The project would then develop new theoretical foundations (possibly involving multiparameter or stochastic homology frameworks) to reconcile or explain these conflicts, perhaps identifying "regimes" where universality holds and others where it breaks down. Such synthesis could lead to new diagnostic tools for persistent homology practitioners: for example, automatic alerts when a dataset falls into a "conflict-prone" regime, or meta-invariants that predict the reliability of PH features. This not only advances theory but also increases practitioner confidence in topological data analysis.

References:

1. Universality in Random Persistent Homology and Scale-Invariant Functionals. O. Bobrowski, P. Skraba (2024).
2. Corrosion characterization of steel wires based on persistent homology theory for magnetostrictive guided wave testing signal. Shuyu Duan, Xinjun Wu, Yiqing Zou, Lijun Jiang (2022). Structural Health Monitoring.
3. Stability for Inference with Persistent Homology Rank Functions. Qiquan Wang, In'es Garc'ia-Redondo, Pierre Faugere, Anthea Monod, Gregory Henselman-Petrusek (2023). Computer graphics forum (Print).