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Explore persistent homotopy groups of metric spaces in this one-hour conference talk from the Applied Algebraic Topology Network. Begin with an overview of discrete homotopy groups and their intersection with persistence concepts. Delve into the study of persistent homotopy groups for compact metric spaces, examining their stability properties in the Gromov-Hausdorff sense. Discover how the classical fundamental group exhibits a tree-like structure and associated ultrametric under certain conditions. Compare filtrations that are indistinguishable by persistent homology but differentiated by persistent homotopy groups. Investigate persistent rational homotopy groups as a more manageable alternative that still provides additional information beyond persistent homology. Conclude with a characterization of persistent rational homotopy groups of the circle, combining insights from Adams and Adamaszek's work on Vietoris-Rips complexes with Serre's classical findings on rational homotopy groups of spheres.
