Bredon’s Trick for Distributed Persistent Homology: Local-to-Global Algorithms for Massive, Heterogeneous Data

Angel (2025) showcases Bredon’s trick as a unifying principle for local-to-global arguments in topology. This research idea adapts that principle to computational persistent homology: develop distributed algorithms that process data in local chunks (possibly on different machines or data sources), compute local persistent homology, and then use Bredon-inspired "gluing" techniques to assemble a coherent global topological summary. Key challenges include ensuring stability, managing overlaps/redundancies, and defining rigorous conditions under which local computations yield accurate global results. While existing PH algorithms (Otter et al., 2015) focus on centralized computation, this approach could revolutionize PH for big data, IoT, or federated learning contexts where central aggregation is infeasible. If successful, it would enable real-time, scalable topological analysis of streaming data—from sensor networks to distributed biological datasets—while retaining rigorous mathematical guarantees.

References:

1. A roadmap for the computation of persistent homology. N. Otter, M. Porter, U. Tillmann, P. Grindrod, H. Harrington (2015). EPJ Data Science.
2. Extensions and Applications of Bredon's Trick in Geometric and Topological Contexts. Mauricio Angel (2025).