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In topological data analysis, cohomological ideas have recently been incorporated to enhance computations of persistence diagrams in software like Ripser. The cup product endows cohomology with a graded ring structure that captures richer topological information than the vector space structure alone. The cup-length, defined as the maximum number of cocycles with nontrivial cup product, serves as a useful invariant of the cohomology ring.

In this talk, we lift cup-length to a persistent invariant that tracks the evolution of the cohomology ring structure across a filtration. We provide a polynomial-time algorithm for its computation from representative cocycles and prove its stability under interleaving distances. To further organize the information encoded by cup products, we introduce the persistent cup module, a two-parameter persistence module indexed by filtration and cup-length, and show its Gromov-Hausdorff stability. In addition, we compare the distinguishing power of these invariants with persistent homology and with each other, highlighting cases where they better distinguish between metric spaces.