Representation theory in persistence and quantum symmetries

The first half of this thesis focuses on the application of quivers and their representations in topological data analysis (TDA), particularly persistent homology. After outlining the TDA pipeline, we analyze the stability of the $p$-presentation distance introduced in \cite{BjerkLes}. In doing so, we contribute a bound for the $p$-presentation distance between two persistence modules. We also give criterion on $p$ and on the persistence modules that ensure this distance is finite.