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Explore advanced concepts in persistent homology theory through this 58-minute conference talk that examines interval modules and their fundamental role in topological data analysis. Learn how intervals encode the lifetimes of topological features emerging from data and discover an explicit formula for computing interval multiplicities in persistence modules indexed by arbitrary finite posets, which generalizes the classical one-parameter persistence formula relating multiplicities of birth-death pairs to persistent Betti numbers. Understand how this result provides immediate corollaries for computing compression multiplicity and interval rank invariants of persistence modules. Discover the essential-cover technique, an innovative method that enables efficient computation of interval multiplicities by transforming persistence modules into restricted modules indexed by simpler, algorithmically tractable posets such as zigzag posets where fast algorithms can be applied. Gain insights into how this technique has the potential to compute interval multiplicities, compression multiplicity invariants, and interval rank invariants directly from filtrations of topological spaces, advancing the computational aspects of applied algebraic topology.
