"Topological Data Analysis meets Geometric Group Theory: Stratifying the space of barcodes using Coxeter complexes"
Garin, AdélieAt the intersection of data science and algebraic topology, topological data analysis (TDA) is a recent field of study, which provides robust mathematical, statistical and algorithmic methods to analyze the topological and geometric structures underlying complex data. TDA has proved its utility in many applications, including biology, material science and climate science, and it is still rapidly evolving. Barcodes are frequently used invariants in TDA. They provide topological summaries of the persistent homology of a filtered space. Understanding the structure and geometry of the space of barcodes is hence crucial for applications. In this talk, we use Coxeter complexes to define new coordinates on the space of barcodes.These coordinates define a stratification of the space of barcodes with n bars where the highest dimensional strata are indexed by the symmetric group. This creates a bridge between the fields of TDA, geometric group theory and permutation statistics, which could be exploited by researchers from each field. This presentation is based on joint work with B. Brück. No prerequisite on TDA or Coxeter complexes are required.