Learning from Two-Parameter Persistent Homology Using Graphcodes

Explore a novel mathematical approach to two-parameter persistent homology through the introduction of graphcodes in this 55-minute conference talk. Discover how graphcodes provide an informative and interpretable summary of two-parameter persistent homology that can be computed as efficiently as one-parameter summaries. Learn about the compressed version of graphcodes that offers even greater computational efficiency while maintaining a size comparable to minimal module presentations. Understand the elegant translation between combinatorics and algebra, where connected components of the graphcode correspond to module summands and isolated paths represent intervals. Examine practical applications including accelerated decomposition of modules into indecomposable summands and seamless integration into machine learning pipelines using graph neural networks. Gain insights into how the graphcode perspective enables a straightforward algorithm for determining whether a persistence module is interval-decomposable, making this advanced topological data analysis technique more accessible and computationally tractable.