Properties of generalised persistence modules

The stability of persistent homology is rightly considered to be one of its most important properties, but persistence is still sensitive to choices of metrics, indexing sets, and methods of filtering. This thesis will expand upon previous discussions around stability, considering sources of invariance and symmetry, as well as potential sources of instability. While homology is a large-scale feature which is invariant under homotopy, transferring to the persistent setting does not preserve all of these properties. In this thesis, we show that there exists an excision property for persistent homology. This is a new result even for one-dimensional persistence, but we’ll show that this result holds for persistence modules indexed by any partially ordered set. We will then expand on the theory of generalised persistence modules, building on work by Lesnick and Bubenik, de Silva and Scott. In this thesis we have adapted slightly the definition of a generalised interleaving as seen so that the we can recover the usual definition of an interleaving of one-parameter persistence modules defined as a special case. We will give a new proof of the stability of this generalised distance, and describe how geometric symmetries in data result in interleavings between Vietoris-Rips filtrations. Finally, we will consider reparameterisations of persistence modules, which can be used to rescale persistence. We’ll investigate the effect that reparameterisations have on the interleaving distance. We show that a reparameterisation by a Lipschitz order isomorphism is also a Lipschitz map, and we present several stability results for the operation of rescaling persistence