Textbook

Textbook#

Main textbook#

  • Tamal K. Dey and Yusu Wang, Computational Topology for Data Analysis, Cambridge University Press, 2022. The authors’ pre-publication version is free to download. Readings marked DW on the schedule refer to this book.

Second reference#

  • Herbert Edelsbrunner and John Harer, Computational Topology: An Introduction, American Mathematical Society, 2010. Marked EH on the schedule. We use it for matrix reduction (IV.2), persistent homology (VII.1), and stability (VIII.2). Available here.

Background and further reading#

Both free online. Dip into them when a definition in class goes by too fast.

  • Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002. Free online. The full story behind homology.

  • Robert Ghrist, Elementary Applied Topology, 2014. Free online. A short, picture-heavy tour of applied topology.

Papers on the schedule#

Listed by the lecture in which they appear. Everything here is either open access or on arXiv.

Lecture 8 — Stability and Persistence Modules

  • David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Stability of persistence diagrams. Discrete & Computational Geometry 37 (2007), 103–120. doi:10.1007/s00454-006-1276-5

  • David Cohen-Steiner, Herbert Edelsbrunner, and Dmitriy Morozov. Vines and vineyards by updating persistence in linear time. SoCG 2006. pdf

Lecture 10 — Reeb Graphs

  • Brian Bollen, Erin Chambers, Joshua A. Levine, and Elizabeth Munch. Reeb graph metrics from the ground up. arXiv:2110.05631 (2021). A survey of the distances covered in class.

Lecture 11 — Mapper

  • Gurjeet Singh, Facundo Mémoli, and Gunnar Carlsson. Topological methods for the analysis of high dimensional data sets and 3D object recognition. Eurographics Symposium on Point-Based Graphics (2007). doi:10.2312/SPBG/SPBG07/091-100 The original Mapper paper.

  • Mathieu Carrière, Bertrand Michel, and Steve Oudot. Statistical analysis and parameter selection for Mapper. Journal of Machine Learning Research 19(12) (2018), 1–39. pdf

Lecture 12 — Directional Transforms

  • Elizabeth Munch. An invitation to the Euler characteristic transform. The American Mathematical Monthly (2025). doi:10.1080/00029890.2024.2409616, arXiv:2310.10395. Start here.

  • Katharine Turner, Sayan Mukherjee, and Doug M. Boyer. Persistent homology transform for modeling shapes and surfaces. Information and Inference 3(4) (2014), 310–344. doi:10.1093/imaiai/iau011

  • Justin Curry, Sayan Mukherjee, and Katharine Turner. How many directions determine a shape and other sufficiency results for two topological transforms. Transactions of the AMS, Series B 9 (2022), 1006–1043. doi:10.1090/btran/122

  • Abigail Hickok. Persistence diagram bundles: a multidimensional generalization of vineyards. arXiv:2210.05124 (2022).

  • Shreya Arya, Barbara Giunti, Abigail Hickok, Lida Kanari, Sarah McGuire, and Katharine Turner. Decomposing the persistent homology transform of star-shaped objects. arXiv:2408.14995 (2024). Monodromy in the PHT.

  • Jessi Cisewski-Kehe, Brittany Terese Fasy, Alexander McCleary, and Eli Quist. Tensor computation of Euler characteristic functions and transforms. SoCG 2026. arXiv:2511.03909. Computing the ECT on a GPU; code in pyECT.

Lecture 13 — Discrete Morse Theory and Multiparameter Persistence

  • Robin Forman. A user’s guide to discrete Morse theory. Séminaire Lotharingien de Combinatoire 48 (2002), B48c.

  • Magnus Bakke Botnan and Michael Lesnick. An introduction to multiparameter persistence. arXiv:2203.14289 (2022).

  • Chad M. Topaz, Lori Ziegelmeier, and Tom Halverson. Topological data analysis of biological aggregation models. PLoS ONE 10(5) (2015), e0126383. doi:10.1371/journal.pone.0126383 Introduces CROCKER plots.

  • RIVET, software for visualizing 2-parameter persistence.

Classic papers and surveys#

Good starting points for the final project, and for the history of the field.

  • Herbert Edelsbrunner, David Letscher, and Afra Zomorodian. Topological persistence and simplification. Discrete & Computational Geometry 28 (2002), 511–533. doi:10.1007/s00454-002-2885-2 Where persistence starts.

  • Afra Zomorodian and Gunnar Carlsson. Computing persistent homology. Discrete & Computational Geometry 33 (2005), 249–274. doi:10.1007/s00454-004-1146-y

  • Robert Ghrist. Barcodes: the persistent topology of data. Bulletin of the AMS 45(1) (2008), 61–75. doi:10.1090/S0273-0979-07-01191-3

  • Gunnar Carlsson. Topology and data. Bulletin of the AMS 46(2) (2009), 255–308. doi:10.1090/S0273-0979-09-01249-X

  • Frédéric Chazal and Bertrand Michel. An introduction to topological data analysis: fundamental and practical aspects for data scientists. Frontiers in Artificial Intelligence 4 (2021). arXiv:1710.04019

  • Nina Otter, Mason A. Porter, Ulrike Tillmann, Peter Grindrod, and Heather A. Harrington. A roadmap for the computation of persistent homology. EPJ Data Science 6, 17 (2017). arXiv:1506.08903 Compares the software below.

  • Ulrich Bauer. Ripser: efficient computation of Vietoris–Rips persistence barcodes. Journal of Applied and Computational Topology 5 (2021), 391–423. doi:10.1007/s41468-021-00071-5

  • Peter Bubenik. Statistical topological data analysis using persistence landscapes. Journal of Machine Learning Research 16 (2015), 77–102. pdf

  • Henry Adams et al. Persistence images: a stable vector representation of persistent homology. Journal of Machine Learning Research 18(8) (2017), 1–35. pdf

  • Felix Hensel, Michael Moor, and Bastian Rieck. A survey of topological machine learning methods. Frontiers in Artificial Intelligence 4 (2021). doi:10.3389/frai.2021.681108

A longer list of suggested project papers, sorted by topic, will be posted on 10/7.

Software#

All of the in-class notebooks use Python. Install numpy, matplotlib, and jupyter, then add the TDA libraries as we reach them.

  • GUDHI — the most complete library: simplicial complexes, persistence, alpha complexes, Mapper, vectorizations. The representations tutorial accompanies Lecture 9.

  • Ripser — fastest Vietoris–Rips persistence; use it from Python through ripser.py in scikit-tda, which also bundles persim (diagram distances and images) and kepler-mapper.

  • giotto-tda — TDA as scikit-learn transformers, convenient for ML pipelines.

  • teaspoon — time series and signal processing with TDA.

  • Topology ToolKit (TTK) — Reeb graphs, merge trees, and Morse–Smale complexes for scientific visualization (ParaView plugin, Python bindings).

  • Mapper Interactive — Mapper in the browser.

  • RIVET — 2-parameter persistence.

  • pyECT — Euler characteristic transforms on the GPU.

More resources#

  • DONUT — a searchable database of applications of TDA; browse it when looking for a project topic.

  • AATRN — recorded talks from the Applied Algebraic Topology Research Network.