Topological Data Analysis Home

These notes are based on a course taught at UVA by Slava Krushkal in Spring 2025. The course was cross-listed in the Department of Mathematics and the School of Data Science.

Unit 01

• TDA in pictures
• Topological spaces: basic notions
  • 01 Theory - Spaces, subspaces, connectedness
  • 02 Illustration
  • 03 Theory - Metric, open balls, continuity
  • 04 Illustration
  • 05 Theory - Compactness, homeomorphism, continuity, homotopy
• Topological spaces: further notions
  • 06 Theory - Quotients, gluing
  • 07 Illustration
  • 08 Theory - Embedding, limit points, closure, boundary
• Simplicial complexes
  • 09 Theory - Simplices and complexes
  • 10 Illustration
  • 11 Theory - Subdivision and triangulation
• Simplicial homology
  • 12 Theory - Boundaries, chain complexes
  • 13 Illustration

Unit 02

• Delta, Nerve, Čech complexes
  • 01 Theory - Delta complexes
  • 02 Theory - Nerve complex
  • 03 Theory - Čech complex
  • 04 Illustration
• Homotopy type
  • 05 Theory - Homotopy type and homology
  • 06 Illustration
• Rank of homology
  • 07 Theory - Betti numbers, Euler Characteristic
  • 08 Illustration
• Maps and homology
  • 09 Theory - Simplicial maps, chain maps
  • 10 Theory - Induced maps on homology
  • 11 Illustration
• Homology variations
  • 12 Theory - Relative homology
  • 13 Illustration
  • 14 Theory - Cochain complex
  • 15 Theory - Cohomology
  • 16 Illustration
  • 17 Theory - Coefficients
• Manifolds
  • 18 Theory - Topological and smooth manifolds
  • 19 Illustration

Unit 03

• Manifolds cont’d
  • 01 Theory - Local coordinates
  • 02 Illustration
• Morse theory
  • 03 Theory - Gradient, critical points; Hessian, non-degenerate points
  • 04 Illustration
  • 05 Theory - Morse functions on manifolds
  • 06 Illustration
  • 07 Theory - Morse functions and topology
  • 08 Illustration
• Structures from point-cloud data
  • 09 Theory - Gromov-Hausdorff distance
  • 10 Theory - Dendrogram
  • 11 Theory - Filtrations
  • 12 Illustration
  • 13 Theory - Vietoris-Rips complex
  • 14 Theory - Persistent homology, barcodes
  • 15 Illustration
  • 16 Theory - Barcode basis

Unit 04

• Stability
  • 01 Theory - Bottleneck distance
  • 02 Theory - Wasserstein metrics
• Multi-dimensional persistent homology
  • 03 Theory - Several parameters
  • 04 Illustration
  • 05 Theory - Zig-zag persistent homology
  • 06 Illustration
• Mapper algorithm
  • 07 Theory - Clustering and Reeb graph
  • 08 - Mapper algorithm
  • 09 Illustration
• Dimensionality reduction and manifold learning
  • 10 Theory - PCA
  • 11 Theory - MDS
  • 12 Theory - Isomap manifold learning
  • 13 Theory - Local Linear Embedding LLE

Unit 05

• Laplacian eigenmaps
  • 01 Theory - Laplace-Beltrami operator
  • 02 Theory - Laplacian for graphs
• UMAP
  • 03 Theory - UMAP process
  • 04 Theory - Gradient descent
  • 05 Illustration
  • 06 Illustration