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This lecture explores the concept of monodromy in the context of bi-parameter persistence modules in topological data analysis. Dive into how monodromy tracks objects circling around singularities, particularly in biparameter filtrations obtained from sublevel sets of continuous functions. Learn about the fundamental group of admissible open subspaces of lines defining linear one-parameter reductions, and understand how non-trivial monodromy occurs when this group acts on the persistence space. Discover the formalization of monodromy behavior in algebraic terms under tameness assumptions, and explore how this translates to persistence module presentations as bigraded modules. Examine the proof that non-trivial monodromy involves generators within the same summand in direct sum decompositions, leading to the conclusion that interval-decomposable persistence modules must have trivial monodromy groups. This ongoing research is a collaboration between Sara Scaramuccia and Octave Mortain from École Normale Superieure, Paris.
