Real multiparameter persistent homology

Ezra Miller (May 22, 2018)

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Abstract

Persistent homology with multiple continuous parameters presents fundamental challenges different from those arising with one real or multiple discrete parameters: no existing algebraic theory applies (even poorly or inadequately). In part that is because the relevant modules are wildly infinitely generated. This talk explains how and why real multiparameter persistence should nonetheless be practical for data science applications. The key is a finiteness condition that encodes topological tameness -- which occurs in all modules arising from data -- robustly, in equivalent combinatorial and homological algebraic ways. Out of the tameness condition surprisingly falls much of ordinary (that is, noetherian) commutative algebra, crucially including finite minimal primary decomposition and a concept of minimal generator. The geometry and relevance of these algebraic notions will be explained from scratch, assuming no prior experience with commutative algebra, in the context of two genuine motivating applications: summarizing probability distributions and topology of fruit fly wing veins.