Stabilizing decomposition of multiparameter modules

How multiparameter persistence modules decompose as direct sums of indecomposable modules is well-understood from the perspective of pure representation theory. In topological data analysis, however, this understanding is unsatisfying, as decomposition is wildly unstable with respect to noise.

I will explain why naive attempts to build a theory of decomposition of multiparameter modules that is robust to noise tend to fail, and then present recent work showing that a decomposition stability theorem can be proved despite these difficulties. I will also talk about a conjecture that would strengthen this result in some cases that are relevant for other areas of multipersistence, and how a version of this conjecture can be phrased in an elegant way using just basic graph theory and linear algebra.

The talk is based on the arXiv preprint https://arxiv.org/abs/2305.15550