Anna-Laura Sattelberger: Algebraic Aspects of Multiparameter-TDA

Abstract

Topological data analysis uses persistent homology as a main tool to investigate data. In the one-parameter case, persistence modules naturally are graded modules over the univariate polynomial ring and hence perfectly understood from an algebraic point of view. One associates the so-called barcode, from which one reads topological features of the data.

Generalizing persistent homology to a multivariate setting allows for the extraction of finer information of the data. Its algebraic properties are more subtle. In this talk, I give insights into an ongoing project with René Corbet and Wojciech Chachólski in which we study multipersistence modules. We introduce the shift-dimension. This is a new, stable invariant of multipersistence modules obtained as the hierarchical stabilization of a classical invariant.