Por Vadim Lebovici (Univ. Paris-Sud & E.N.S.).
Topological data analysis extracts topological information from data using persistent homology, that is, by applying homology to a 1-parameter filtration of topological space built on the data at hand. This results in a linear quiver with relations called a persistence module. I will explain the remarkable simplicity of the representation theory of these quivers: they are completely described by multisets of intervals in R called persistence barcodes.
However, when dealing with multi-parameter filtrations, I will show how the representation theory of the obtained quivers is significantly more complex and lacks a combinatorial description like the persistence barcode. This motivates the search for subclasses of such modules that have a barcode-like structure. I will expose results in collaboration with Magnus Bakke Botnan and Steve Oudot in the two-parameter case, proving that membership in the class of rectangle-decomposable modules can be algorithmically tested using local observations, and that it is, in some precise sense, a maximal class with this property.
