Topology, Geometry and Data Seminar - Woojin Kim

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Woojin Kim
September 3, 2019
4:10PM - 5:10PM
Location
Cockins Hall 240

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Add to Calendar 2019-09-03 16:10:00 2019-09-03 17:10:00 Topology, Geometry and Data Seminar - Woojin Kim 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/ Cockins Hall 240 Department of Mathematics math@osu.edu America/New_York public
Description

Title: Generalized Persistence Diagrams for Persistence Modules over Posets

SpeakerWoojin 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 URLhttps://tgda.osu.edu/

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