Persistent Homotopy Groups of Metric Spaces

This course covers the study of persistent homotopy groups of compact metric spaces and their stability properties in the Gromov-Hausdorff sense. By blending homotopy groups with persistence, the course aims to capture both geometric and topological features of datasets. The learning outcomes include understanding the underlying tree-like structure and associated ultrametric of the classical fundamental group, as well as distinguishing pairs of filtrations based on their persistent homotopy groups. The course also introduces the notion of persistent rational homotopy groups, providing additional information compared to persistent homology. The teaching method involves a theoretical overview of previous works in the field and the presentation of key concepts and results. This course is intended for individuals interested in algebraic topology, applied mathematics, or data analysis, with a basic understanding of homotopy theory and topology.