Persistent Homology: A Complete Guide to Topological Data Analysis
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Open Problems in Topological Machine Learning
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Topological Machine Learning is a young and rapidly growing field. Open problems include: multiparameter persistence (no complete discrete invariant yet), scalability to million-point clouds, theoretical understanding of what PH features encode in learned representations, unification with sheaf theory, and the design of truly end-to-end differentiable TDA architectures.
TDA for Neuroscience: Topology of Brain Connectivity
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The brain is a network with rich topological structure. Clique topology (Reimann et al., 2017) used persistent homology to show that neocortical microcircuits form high-dimensional simplicial structures, far beyond random. PH-based connectivity fingerprints distinguish subjects, tasks, and disease states from fMRI. This post covers the key findings and open questions in topological neuroscience.
TDA for Biology and Genomics: Shape in the Life Sciences
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Biology is full of shape: protein folding, cell morphology, gene expression manifolds. TDA has found major applications in single-cell RNA-seq analysis (Mapper for trajectory inference), protein structure comparison (PH-based structural fingerprints), and cancer genomics (topological signatures of tumour heterogeneity). This post surveys the key results and tools.
TDA for Graphs: Persistent Homology Meets GNNs
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Persistent homology can be applied directly to graphs by defining filtrations on nodes or edges (e.g., by WL colours, degree, or learned scalars). The resulting persistence diagrams encode global graph topology — connectivity, cycles, cliques — beyond what standard 1-WL GNNs can detect. This post covers WL-filtrations, extended persistence on graphs, and hybrid GNN+PH architectures.