Algebraic topology and its applications, AIMS Senegal, Spring 2023

Instructor: Joseph Maher
Webpage: http://www.math.csi.cuny.edu/~maher/teaching/2023/spring/tda/
Email: joseph.maher@csi.cuny.edu

Outline:

The purpose of this course is to cover the foundational mathematics needed for various techniques in topological data analysis, such as persistent homology. We will illustrate the methods with some basic examples using R.

Topics:

  • Topological spaces, simplicial complexes, metric spaces, manifolds

  • Topological invariants, path connected, simply connected, fundamental group, higher homotopy groups

  • Homology, simplicial homology, homological algebra

  • Persistent homology, diagrams, barcodes, landscapes, stability

Examples:

An example using R

Another example using R

An example with noise in R

Assignments:

Week 1

Week 2

Week 3

References:

There is no specific set text for this course, but the standard reference for a geometric approach to algebraic topology is:

Allen Hatcher, Algebraic Topology, Cambridge University Press, ISBN 0-521-79540-0.

The following two books both give introductions to topological data analysis:

Gunnar Carlsson and Mikael Vejdemo-Johansson, Topological Data Analysis with Applications

Raul Rabadan and Andrew J. Blumberg, Topological Data Analysis for Genomics and Evolution

Here are some websites and papers for further reading and more examples in R:

Peter Bubenik’s website

Topology of Deep Neural Networks, Naitzat, Zhitnikov and Lim, Journal of Machine Learning Research 21 (2020) 1-40

These are the notes I make for class, they are probably not of much use to anyone else. Here are the smartboard pics.