Persistence of Common Topological Structures by a Commutative Triple Ladder Quiver

This is a survey paper of our recent results [6]. We present a novel method to detect robust and common topological structures of two geometric objects. The idea is to extend the notion of persistent homology [5, 12] to representations on a commutative triple ladder quiver. Our contributions of this paper are given as follows: (i) We prove that the commutative triple ladder quiver is representation finite. It implies that the persistence modules of this type can be classified by complete discrete invariants. (ii) The Auslander-Reiten quiver of the commutative triple ladder, which lists up all the isomorphism classes of indecomposable persistence modules and irreducible morphisms among them, is explicitly derived. In addition, the notion of persistence diagrams is generalized to graphs on the Auslander-Reiten quiver. (iii) An algorithm for computing indecomposable decompositions by using the AuslanderReiten quiver is presented. (iv) A numerical example to detect robust common topological features is shown.