Title: Generalized Persistence Diagrams for Persistence Modules over Posets
Speaker: Woojin Kim (Ohio State University)
Abstract: When a category $C$ satisfies certain conditions, we define the notion of rank invariant for arbitrary poset-indexed functors $F:P \rightarrow C$ from a category theory perspective. This generalizes the standard notion of rank invariant as well as Patel's recent extension. Specifically, the barcode of any interval decomposable persistence modules $F:P \rightarrow vec$ of finite dimensional vector spaces can be extracted from the rank invariant by the principle of inclusion-exclusion. Generalizing this idea allows freedom of choosing the indexing poset $P$ of $F:P \rightarrow C$ in defining Patel's generalized persistence diagram of $F$. By specializing our idea to zigzag persistence modules, we also show that the zig-zag barcode of a Reeb graph can be obtained in a purely set-theoretic setting without passing to the category of vector spaces. This leads to a promotion of Patel's semicontinuity theorem about type-A persistence diagram to Lipschitz continuity theorem for the category of sets.
Seminar URL: https://tgda.osu.edu/