Stability of homological invariants of multiparameter persistence modules

Abstract: Unlike one-parameter persistence modules, for which we have the barcode, persistence modules with two or more parameters do not admit a complete discrete invariant, and thus incomplete invariants must be used to study the structure of such modules in practice. The Hilbert function and the multigraded Betti numbers are among the simplest such incomplete invariants. Although these two invariants are already being used in applications, it is a priori unclear whether they satisfy a stability result analogous to the stability of the one-parameter barcode. Stability results are essential for the interpretability and consistency of practical methods. I will present joint work with Steve Oudot in which we prove stability results for multigraded Betti numbers and for the Hilbert function. I will also discuss ongoing work in which we prove the stability of finer invariants coming from different exact structures on a category of multiparameter persistence modules.

combinatoricscategory theoryrepresentation theory

Audience: researchers in the topic

( paper | slides )

The TRAC Seminar - Théorie de Représentations et ses Applications et Connections