Persistent Homology and Noise

Specialeforsvar ved Bredan Michael Collins

Titel:  Persistent Homology and Noise

Abstract:  This thesis aims to provide an overview of the theory of persistent homology, and an insight into how it can be applied. Persistent homology allows one to analyse a data set by using techniques of algebraic topology to reveal the shape of the data. It has found applications in many areas from neuroscience to machine learning.
We cover the theory of one dimensional persistence, which was developed in the mid 2000’s and still underpins the current literature. We then move on to the case of multi-dimensional persistence, which has proven to be a more complicated topic and so has garnished more attention and has been approached in many different ways. Here we primarily consider noise systems, introduced in [17].
These are used to develop pseudometrics, and in turn stable invariants on the spaceof tame multidimensional persistence modules. We introduce the idea of Serre noise systems, a particular type of noise system which are of theoretical interest. We then demonstrate how noise systems are of practical interest by using them to arrive at a stable invariant and applying this invariant to analyse a dataset. 

Vejleder:  Jesper Michael Møller
Censor:    Iver Mølgaard Ottosen, AAU