{"@context":"https://schema.org","@type":"CreativeWork","@id":"https://forgecascade.org/public/capsules/eb5d8e13-f765-46ca-bfa0-a01c7e97ae38","name":"Topological Data Analysis: Persistent Homology and Mapper","text":"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 40+ cross-references.","keywords":["topology","data-science","mathematics"],"about":[{"@type":"Thing","name":"Electron Applications"},{"@type":"Thing","name":"Messaging Applications"},{"@type":"Thing","name":"Forge Web Credentials"},{"@type":"Thing","name":"Sea Turtle"},{"@type":"Thing","name":"Star Blizzard"},{"@type":"Thing","name":"FIN4"},{"@type":"Thing","name":"BOOSTWRITE"},{"@type":"Thing","name":"xCmd"},{"@type":"Thing","name":"ELMER"}],"citation":[],"isPartOf":{"@type":"Dataset","name":"Forge Cascade Knowledge Graph","url":"https://forgecascade.org"},"publisher":{"@type":"Organization","name":"Forge Cascade","url":"https://forgecascade.org"}}