Persistent Homology

Ximena Fernandez

EUTOPIA 2022

This repository contains all the lectures and code related to the course Persistent Homology, as part of the EUTOPIA Summer School 2022.

Lecture 1.1: Topological Data Analysis

This lecture is about a general exposition of Persistent Homology and some applications. The slides of the lecture are available here. The demo of the software Ripser can be found at this link. The synthetic point clouds to test the software are available at the folder data: cicle, sphere and torus.

Lecture 1.2: Hands on: computational topology in action

This lecture is a live-coding/hands-on exposition of the use of the software Ripser to compute Persistent Homology, and some real applications with concrete data.

All the datasets are available at the folder data.

To run the notebooks, it is required to have installed jupyter, as well as the following:

Python (>= 3.6)
pip
NumPy (>= 1.19.1)
SciPy (>= 1.5.0)
scikit-learn (>= 0.23.1)
matplotlib
seaborn
pandas

The specific libraries for TDA we use are:

Fermat
ripser
tadasets
persim
biopython

The notebook Intro_Persistent_Homology.ipynb provides a complete description and implementation of tools related to the computation of persistent homology with the software Ripser. It also contains some simulations in synthetic point clouds to describe properties of persistent homology. The notebook Beyond_Persistent_Homology.ipynb describes how to infer other topological features from a sample (such as orientability, singularities, local dimension and low domensional representation). Some exercises to familiarize yourself with the techniques can be found at the notebook Exercises_Persistent_Homology.ipynb

Lecture 2:

This lecture is about concrete applications of the study of topological descriptors in real data. The slides of the lecture are available here.

• We will apply all the topological methods already learned to study the topology of one of the following real datasets:

2.1. Cyclo Octane

2.2 Grid Cells

• We will apply all we have learned in a concrete problem of classification of the secondary structure of proteins, using persistent homology to construct topological features to feed a ML algorithm. The guided implementation can be found at the notebook Proteins_Structure_Classification.ipynb