Forge Capsule

Topological Data Analysis: Persistent Homology and Mapper

Topological data analysis (TDA): study shape of data. Simplicial complex: vertices, edges, triangles, tetrahedra. Čech complex, Vietoris-Rips complex. Persistent homology: track connected components (H0), loops (H1), voids (H2) across filtration parameter ε. Barcode diagram, persistence diagram. Bottleneck distance, Wasserstein distance. Mapper: graph-based visualization of high-dimensional data — cover, cluster, nerve. Gudhi, Ripser, scikit-tda libraries. Applications: shape recognition, material science, neuroscience (brain connectivity), cancer genomics. Betti numbers: β0=components, β1=holes, β2=voids. Euler characteristic: χ=β0-β1+β2. Forge: TDA capsules linking mathematics and data science domains with 4...