On this page I have a number of items to get the interested reader started with persistent homology and topological data analysis (TDA). If you know linear algebra you are ready to start! If you you’ve never heard of linear algebra, you can still learn what TDA is about with this article on TDA and Pokemon

• Connecting the Dots: Discovering the “Shape” of Data, by Michelle Feng, Abigail Hickok, Yacoub H. Kureh, Mason A. Porter, and Chad M. Topaz.

Introduction to Topological Data Analysis and Persistent Homology

• Introduction to Persistent Homology, a great YouTube video, by Matthew Wright
• Studying the Shape of Data Using Topology, a brief non-technical introduction by Michael Lesnick
• A User’s Guide to Topological Data Analysis, by Elizabeth Munch.
• To help learn the basics, complete my worksheet.
• Persistent homology: an introduction via interactive examples, by Federico Iuricich.
• Introductory Topological Data Analysis, by Dayten Sheffar.

Simplicial Homology

The main technical tool for persistent homology is simplicial homology. For persistent homology, we use coefficients in a field. So simplicial k-chains are vectors and the set of simplicial k-chains is a vector space. Furthermore, the boundary map is a linear transformation. For finite simplices, it is represented by a matrix.

Topological Data Analysis with R

If you want to get started doing topological data analysis.

• My Introduction to TDA with R script:  intro_tda.R. Copy and paste or download the file and rename it intro_tda.R.
• My older TDA with R workshop:  instructions and R files tda_functions.R, tda_workshop_script.R, and persistence_script.R. Rename the files from *.txt to *.R.
• Use Jose Bouza’s tda-tools R package.

Topological Data Analysis and Persistent Homology

Here are some recent introductory articles. If you want to learn more about the subject I would recommend starting here. The first three are mathematical, the fourth emphasizes connections to data science, and the fifth is more statistical.

• A Brief History of Persistence, by Jose Perea
• Persistent Homology and Applied Homotopy Theory, by Gunnar Carlsson
• Homological Algebra and Data, by Robert Ghrist
• An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists, by Frédéric Chazal and Bertrand Michel
• Topological Data Analysis, by Larry Wasserman

There is a Wikipedia page.

• Topological Data Analysis

The following slightly older introductory articles provide background, some mathematical details and a few applications.

• Persistent Homology – Theory and Practice, by Herbert Edelsbrunner and Dmitriy Morozov
• Barcodes: The persistent topology of data, by Robert Ghrist
• Topology and data, by Gunnar Carlsson

• Topological pattern recognition for point cloud data, by Gunnar Carlsson

The following are more technical summaries of some of the main results in the field.

• Persistent Homology, by Herbert Edelsbrunner and Dmitriy Morozov
• High Dimensional Topological Data Analysis, by Frederic Chazal

For a serious introduction, I highly recommend the following new book. It is an excellent resource for mathematics graduate students wanting to learn the subject.

•  Persistence Theory: From Quiver Representations to Data Analysis, by Steve Oudot.

Topological Data Analysis and Deep Learning

• PLLay: Efficient Topological Layer based on Persistence Landscapes, by Kwangho Kim, Jisu Kim, Manzil Zaheer, Joon Sik Kim, Frederic Chazal, Larry Wasserman.