分野別セミナー

Persistent homology is a tool in topological data analysis for studying the robust topological features of data. In this talk, we extend its applicability to a wider variety of settings by using bound quivers and their representations. Motivated by applications, we focus on the so-called commutative ladder quivers.

We show that the commutative ladder quivers with length n ≤ 4 are representation-finite by computing their Auslander-Reiten quivers. Moreover, using the Auslander-Reiten quivers, we generalize the definition of persistence diagrams, which is a compact way to summarize the persistent homology. In the representation-finite commutative ladder case, we show how to visualize and interpret our generalized persistence diagrams in the context of input data.

We also extend the use of discrete Morse theory to our setting of quiver complexes. Given a quiver complex X and an acyclic matching for X, we show that there is an associated Morse quiver complex A with the property that X and A have isomorphic persistent homology. The Morse quiver complex A tends to be smaller in size, so that computing the persistent homology from A instead of X tends to be less costly.