BD2K module: TDA

Course materials:

Suggested reading:

Books:

  • Ghrist’s Elementary Applied Topology
    • Section 4.1 discusses homology
    • Section 5.3 discusses persistence/persistent homology
  • Edelsbrunner and Harer’s Computational topology
    • Section III.1 and III.4 discuss simplicial complexes, and then alpha complexes
    • Section IV.1 discusses homology
    • Section VII.1 discusses persistence
    • Section VIII.2 discusses stability
  • Kaczynski, Mischaikow, and Mrozek’s Computational homology

    • Section 2 discusses cubical complexes and homology computations

Surveys/notes:

  • Ghrist’s Barcodes: the persistent topology of data
    • A very readable introduction to what TDA is all about, with (relatively) minimal math pre-requisites. Builds up to an example of TDA in natural image processing.
  • Giusti, Ghrist, and Bassett’s Two’s company, three (or more) is a simplex
    • Another very readable introduction to TDA, with a focus on neuroscience applications. Less formal math shows up in the exposition, but they include formal definitions/statements in side boxes.
  • Ghrist’s Homological algebra and Data
    • Notes from Ghrist’s lectures on the topic at the PCMI “The Mathematics of Data” summer program. Written with more mathematical maturity in mind.
  • Carlsson’s Topology and Data
    • An introduction to TDA with non-specialist mathematicians in mind; maybe a “step-up” from Ghrist’s Barcodes survey.
  • Lum et.al.’s Extracting insights from the shape of complex data using topology
    • An interesting overview of the various ways topological methods can be used to visualize (and get insights from) various kinds of complex data.

Other:

An algebraic topology/neuroscience bibliography by Chad Giusti

Optimal cycles by Wu et. al.