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This course aims to teach the geometric interpretation of persistence in a metric space through filtration by complexes and persistent homology. The learning outcomes include understanding how complexes reconstruct the homotopy type of a space at small scales and how persistent homology reveals information about the size of holes at increasing scales. The course covers topics such as the classification of one-dimensional persistent homology, approximation by finite samples, and the detection of contractible geodesics using persistent homology. The teaching method involves presenting results, examples, and technical details to support the viewpoint of multi-scale representation. This course is intended for individuals interested in algebraic topology, geometric interpretation of data, and topological data analysis.
