New invariants for multiparameter persistent homology

Persistent homology (PH) is one of the most successful methods in topological data analysis, and has been used in a variety of applications from different fields, including robotics, material science, biology, and finance. PH allows to study qualitative features of data across different values of a parameter, which one can think of as scales of resolution, and provides a summary of how long individual features persist across the different scales of resolution. In many applications, data depend not only on one, but several parameters, and to apply PH to such data one therefore needs to study the evolution of qualitative features across several parameters. While the theory of 1-parameter persistent homology is well understood, the theory of multiparameter PH is hard, and it presents one of the biggest challenges of topological data analysis.

In this talk I will introduce persistent homology, give a brief overview of the complexity of the theory in the multiparameter case, and then discuss how tools from commutative algebra give invariants suitable for the study of data.

This is based on joint work with Heather Harrington, Henry Schenck, and Ulrike Tillmann.