Oliver Vipond - Local Equivalence of Metrics for Multiparameter Persistence Modules

Explore a comprehensive lecture on the local equivalence of metrics for multiparameter persistence modules. Delve into the analysis of the fibered bar code, a stable and computable invariant of multiparameter persistence modules, and its discriminative power. Discover how the fibered bar code, equivalent to the rank invariant, encodes bar codes of 1-parameter submodules. Examine the local completeness of the fibered bar code for finitely presented modules through a demonstration of local metric equivalence between the interleaving distance and the matching distance on fibered bar codes. Learn about the bi-Lipschitz inequalities that exist within a neighborhood of a finitely-presented multiparameter module M. Gain insights into the implications of this local equivalence, including the uniqueness of fibered bar codes within the defined neighborhood. Follow the lecture's structure, covering topics such as motivation, preliminaries, 1-parameter indomitable, interleaving distance, matching distance, comparison of multiparameter metrics, main results, cautionary examples, proof sketches, paths between persistence modules, and concluding remarks.

Intro
Motivation
Preliminaries
1-parameter Indomitable
Interleaving Distance
Matching Distance
Comparing Multiparameter Metrics
Main Result
Cautionary Example
Sketch Proof
Paths Between Persistence Modules
Conclusion