Topics in multidimensional persistence

Thesis

A multiparameter persistence module is a representation of the lattice quiver Nd, where maps along all squares commute. When d = 1, Gabriel’s theorem applies and these modules admit interval decompositions, allowing us to classify one dimen- sional persistence modules through their associated barcode, a combinatorial invariant first introduced in by Carlsson and Zomorodian. When d > 1, no such classification is possible. In this thesis we study these higher dimensional persistence modules, seeking to over- come their lack of simple classification by defining discrete invariants with as much discriminative power as possible. The thesis is composed of three parts.

First, we give an in depth analysis of barcode bases. These are bases of one-dimensional persistence modules that realise the interval decom- position given by Gabriel’s theorem. We present a novel algorithm that computes these barcode bases, and give theoretical results that characterise the set of barcode bases of a given persistence module. This allows for a decomposition results of certain types of ladder persistence modules. We generalise all these results to zigzag persistence.

Second, we consider Harder-Narasimhan filtrations for quiver representations and define the skyscraper invariant, a novel discrete invariant for multidimensional persistence which is finer than the rank invariant. We further show the skyscraper invariant can be refined to create a complete invariant on certain families of ladder persistence modules.

Finally, we discuss computation methods for the skyscraper invariant. We exhibit an algorithm that computes the skyscraper invariant for ladder persistence modules. This is done by leveraging the decomposition result for ladder persistence modules from the first chapter. In doing so, we introduce the ladder invariant, which is computable and more discriminative than the rank invariant. It coincides with the skyscraper invariant on ladder persistence modules and is non-comparable to the skyscraper invariant in general.

Algorithms from the first chapter are given as pseudo-code. These were later implemented as a python package and we give an overview of this package in the appendix.

Authors: Jacquard, E

• DOI: 10.5287/ora-r1pabmrp8
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: quiver representations; topological data analysis
• Subjects: Topological Data Analysis