Algebraic Tools for Topological Data Analysis in a Multiparameter Setting

Topological data analysis investigates data by topological methods. The main tool is persistent homology. 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. By a classical structure theorem, 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 from data, but its algebraic properties are more subtle. In this talk, I introduce and discuss the shift-dimension. This is a stable invariant of multipersistence modules obtained as the hierarchical stabilization of a classical invariant.

This talk is based on recent work with Wojciech Chachólski and René Corbet.