TDA Foundations

Carlsson's survey in the Bulletin of the AMS is the most widely cited introduction to TDA. It covers simplicial complexes, persistent homology, and the mapper algorithm in under 55 pages, making it accessible to anyone with basic algebraic topology. An ideal starting point before the full Edelsbrunner–Harer textbook.

Ghrist's short paper popularised the "barcode" metaphor for persistence diagrams. It is one of the most readable introductions to why topological features persist across scales and what that tells us about the shape of data. Strongly recommended as a first encounter with the key ideas.

A highly accessible demonstration of the mapper algorithm applied to real scientific datasets, including human genetics and basketball statistics. This paper is often used as an entry point for non-mathematicians who want to see what TDA can do before engaging with the formal theory.

A practical introduction to computing homology groups that bridges abstract algebra and implementation. Less focused on persistence than Edelsbrunner–Harer, but an invaluable companion for understanding the algebraic machinery (chain complexes, boundary matrices) that underpins TDA algorithms.

A statistician's introduction to TDA, covering persistent homology and the mapper algorithm from the perspective of statistical inference. Wasserman is unusually clear on confidence sets for persistence diagrams and bootstrapping procedures — essential reading for anyone thinking about uncertainty quantification in TDA.

The standard graduate-level introduction to computational topology. Edelsbrunner and Harer develop persistent homology with rigorous mathematical precision while keeping geometric intuition central. Chapter 7 on persistence is essential reading for anyone working with TDA on real data.

Oudot approaches persistence from the perspective of quiver representations in algebra, providing a more algebraically rigorous foundation than Edelsbrunner–Harer. Essential for readers who want a deep understanding of the theoretical structure underlying the stability and uniqueness theorems that make TDA work.

Zomorodian and Carlsson's 2005 paper established the algebraic foundation of persistent homology by connecting persistence modules to graded modules over a polynomial ring. Their Matrix Reduction algorithm is still the basis for most practical TDA software. A technically demanding but essential paper.

The 2002 paper that introduced topological persistence and the first computationally practical algorithm for computing it. Edelsbrunner, Letscher, and Zomorodian's reduction algorithm transformed an abstract mathematical idea into an implementable procedure. Reading this alongside the Zomorodian–Carlsson paper shows how the field progressed.