Harder-Narasimhan filtrations of persistence modules

Thesis

Multiparameter persistence modules are central objects in Topological Data Analysis. Unlike ordinary persistence modules, they do not admit a complete discrete invariant such as the barcode. This thesis explores the use of Harder-Narasimhan theory as a way to devise discrete invariants of multiparameter persistence modules that are discriminating, computable, stable and interpretable.

Harder-Narasimhan types are a family of discrete invariants of persistence modules over finite posets. We first study their discriminating power in several settings arising in Topological Data Analysis. We then use Harder-Narasimhan types to define the skyscraper invariant, a novel discrete invariant of multiparameter persistence modules. We show that this invariant is strictly more discriminating than the rank invariant and is stable with respect to the interleaving distance.

Authors: Fersztand, M

• DOI: 10.5287/ora-ovgr8b0qq
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Harder-Narasimhan filtrations; multiparameter persistence; quiver representations; topological data analysis; persistent homology
• Subjects: Topological Data Analysis