Applied Topology Seminar

Stability is a key property of topological invariants used in data analysis and motivates the fundamental role of metrics in persistence theory. This talk reviews noise systems, a framework for constructing and analysing metrics on persistence modules, and shows how a rich family of metrics enables the definition of metric-dependent stable invariants. Focusing on one-parameter persistence, we discuss algebraic Wasserstein distances and the associated Wasserstein stable ranks, invariants that can be computed and compared efficiently. These invariants depend on interpretable parameters that can be optimised within machine-learning pipelines. We illustrate the use of Wasserstein stable ranks through experiments on synthetic and real datasets, showing how different metric choices highlight specific structural features of the data.