On this page I have a number of items to get the interested reader started with persistent homology and topological data analysis. If you know linear algebra you are ready to start!
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 nontechnical introduction by Michael Lesnick
 To help learn the basics, complete my worksheet.
Topological Data Analysis with R
If you want to get started doing topological data analysis.
 Here are the instructions for my TDA with R workshop.
 These are my R files for a workshop: tda_functions.R, tda_workshop_script.R, and persistence_script.R. Rename the files from *.txt to *.R.
Topological Data Analysis and Persistent Homology
Here are three recent introductory articles. If you want to learn more about the subject I would recommend starting here. The first two are mathematical, the third emphasizes connections to data science, and the fourth is more statistical.
 A Brief History of Persistence, by Jose Perea
 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.
The following slightly older introductory articles provide background, some mathematical details and a few applications.
 Persistent Homology – a Survey, by Herbert Edelsbrunner and John Harer
 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.