Barcodes for the Topological Analysis of Gradient-Like Vector Fields

Explore a comprehensive method for topological analysis of fields in this hour-long lecture. Delve into a pipeline that transforms weighted and based chain complexes into tame epimorphic parametrized chain complexes, ultimately encoding them as barcodes of tagged intervals. Learn how to apply this technique to gradient-like Morse-Smale vector fields on compact Riemannian manifolds in both smooth and discrete settings. Discover the isometry between tame epimorphic parametrized chain complexes and barcodes of tagged intervals, and understand the continuity of the map from generic gradient-like vector fields to their barcodes. Gain insights into approximating vector field barcodes using combinatorial versions with arbitrary precision. This talk presents joint work with Clemens Bannwart from the University of Modena and Reggio Emila, with additional details available in their arXiv paper.