MATH 4570

Welcome to the homepage for MATH 4570: Introduction to Applied Algebraic Topology. This is an undergraduate course on the mathematics behind recent approaches to data science using tools from algebraic topology. It was taught in Spring 2018 by Crichton Ogle and Tom Needham and in Spring 2019 by Tom Needham. A course description follows below:

In recent years, topology has contributed key ideas to a new discipline sitting at the crossroads of mathematics, computer science, and statistics. These fields interact to create new methods that can be applied to interpret data coming from the life sciences, chemistry, engineering, etc. A key idea of Applied Algebraic Topology is that topology can be used to provide robust descriptions of the geometry of high-dimensional and complex data. Topological data analysis has grown into an active research field over the last decade and still offers many opportunities for research in pure and applied mathematics. Some of the ideas of topological data analysis are introduced in the video available here: https://www.youtube.com/watch?v=XfWibrh6stw

This course will serve as an introduction to applied algebraic topology. The first part of the course will cover background material on topology, starting with metric topology and moving on to some more sophisticated topics in algebraic topology. We will then move toward studying persistent homology of point clouds for applications to data analysis. We will rigorously treat the mathematics behind this active area of current research. Real-world applications to data analysis will be provided.

Course Materials:

• Lecture Notes. These lecture notes were written for the course by Tom Needham. The notes are a work-in-progress and are updated frequently. Check back for updates!
• Syllabus. The syllabus for Spring 2019.