Non-commutative Persistent Homology

Abstract

This thesis explores the possibility of defining `non-commutative persistent homology'. Persistent homology is an algorithm in topological data analysis that assigns to each point cloud a `barcode' which contains information about the shape of this point cloud. An important property of the persistent homology map is that it is stable: it is Lipschitz-continuous with respect to the Gromov-Hausdorff distance on point clouds and the 'bottleneck distance' on barcodes. In `non-commutative persistent homology' we aim to formulate persistent homology using non-commutative geometry. We introduce non-commutative metrics, non-commutative Gromov-Hausdorff distances and we formulate versions of non-commutative vertex sets based on the algebraic generalization of points in a topological space. Using this framework we produce four candidate non-commutative persistent homology maps defined on non-commutative metric spaces. The first candidate is demonstrated to be ineligible, as it does satisfy the stability property. The second candidate is shown to be ill-defined. For the remaining candidates the well-definedness (a fortiori the eligibility) remains open.