Mikael Vejdemo-Johansson: Mini course on applied algebraic topology

This is a 4x60m lecture course on persistent homology and applied algebraic topology. The course will focus on inferring topological invariants from point clouds, assumed to be noisy samples from some underlying shape. The course will be accessible to anyone with experience of linear algebra, simplicial complexes and the simplicial homology functor; recommended background is Topologi fdk or Homologisk Algebra och Algebraisk Topologi.

Course literature will be Computational Topology by Edelsbrunner and Harer. Available from Bokus (see link on the right-hand sidebar). Note the almost week long delivery time, which means that ordering the book should be done as soon as possible to have access to it for those who want to own a copy.

The book is the most current reference work for best practices implementing topological algorithms from a large range, and can be recommended in addition to this lecture course for anyone with an interest in computational or applied topology in general.

I.Lecture Sept 2, 11:00-12:00, room 3733
Introduction

Review of the AG Seminar talk from Sep 1.
A pipeline for computing topological invariants from point clouds.
Detailed worked example.
Viewing persistence as modules over k[t] for a field k.

Edelsbrunner-Harer: VII.1

II. Lecture Sept 3, 11:00-12:00, room 3733
Computational Efficiency

A toolbox of simplicial complexes: Cech, Vietoris-Rips, alpha-shapes, Witness complexes
A toolbox of topological simplifications: collapsing schemes, restricting to skeleta, smoothing and denoising.

Edelsbrunner-Harer: III, VII.2

III. Lecture Sept 5, 11:00-12:00, room 3733
Tools & Applications

Persistence software: Plex, jPlex, javaPlex, Dionysus, CULT, ...
Case study: the Klein bottle of natural image patches.
Dataset: http://www.kyb.mpg.de/bethge/vanhateren/index.php
Setup for topological analysis.
Setup for tool usage.
Primary circle. Secondary circles.
Lab description - try it yourself.

IV. Lecture Sept 6, 11:00-12:00, room 3733
Enriched Information

Instead of just a barcode, get representative cycles.
Instead of homology, compute cohomology.
Using cohomology, compute circular coordinates.
Relative homology.
Extended persistence.
Zig-Zag persistence.
Levelsets, superlevelsets, sublevelsets, combining samples.

Edelsbrunner-Harer: IV.3, V.1, VII.3