Representation Theory and (Barcoding) Invariants for Persistence

Abstract: Persistent homology is one of the main tools of topological data analysis, which has seen rapid growth recently. In the first part of this talk, I discuss some of the ways representation theory is being used for persistent homology, focusing on "invariants". In particular, the persistence barcode, which can be obtained from an indecomposable decomposition of a persistence module into intervals, plays a central role. For multi-parameter persistent homology, where persistence modules are no longer always interval-decomposable, many alternative invariants have been proposed. Naturally, identifying the relationships among invariants, or ordering them by their discriminating power, is a fundamental question. The second part of this talk, based on arXiv:2412.04995, addresses this. I discuss our formalization of the notion of "barcoding invariants", which generalizes the persistence barcode, and results concerning the comparison of their discriminating powers.

combinatoricsquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

OIST representation theory seminar

Series comments: Timings of this seminar may vary from week to week.