Interaction between geometric scale and algebraic precision in two-parameter persistence

Investigate and characterize the interaction between geometric scale parameter τ and algebraic precision parameter k in two-parameter persistence for network sheaf cohomology, determining how features at different geometric scales manifest distinct precision signatures and defining appropriate invariants to capture this interplay.

Background

The paper proposes viewing precision as a second axis of persistence, alongside geometric filtrations that vary network topology or sheaf data by scale. Vertical (precision) stability is established, while horizontal (geometric) stability follows classical results.

A detailed understanding of the joint behavior across both axes—how geometric features at scale τ correspond to precision thresholds k—would enable richer multiparameter invariants and stability results tailored to applications with both hierarchical geometry and quantized precision.