Speaker
Description
We study the structure of persistent modules over arbitrary coefficient fields and related algebraic systems, with an emphasis on explicit basis constructions and algorithmic realizations rooted in linear algebra. Persistent homology assigns to a filtered topological space a persistence module, traditionally decomposed via the structure theorem over a field. However, extending such decompositions to more general coefficient systems and enhancing computational tractability remains an open challenge.
In this work, we extend and unify basis construction techniques originally motivated by the algorithms in Ezra Miller’s work on multigraded modules and combinatorial commutative algebra, as well as computational approaches in the works of Vidit Nanda and Gunnar Carlsson. We develop a generalized algorithm that constructs a canonical basis for persistence modules by leveraging matrix reductions and pivot strategies analogous to those in classical linear algebra (e.g., Gaussian elimination and Smith normal form), while respecting the graded structure inherent to persistent settings.
Our contributions include (1) a rigorous formulation of persistence basis extension over fields and select non-field coefficient rings, (2) an efficient reduction algorithm that generalizes standard persistence matrix reductions, providing structural insights into birth–death pairs and algebraic invariants, and (3) explicit connections between algorithmic choices and linear algebraic concepts such as rank, invariant factors, and module presentations. Through theoretical analysis and experimental validation, we demonstrate that our methods improve both conceptual clarity and computational performance in persistent module decomposition.
This framework bridges combinatorial topology, computational algebra, and classical linear algebra, offering new tools for both theoretical investigations and practical implementations in topological data analysis.