Rank decompositions and associated exact categories for multi-parameter persistence modules

The motivation for the work I am going to speak about comes from a recent field of application of representation theory: the study of persistent homology in topological data analysis.

I will try to explain how and why one might turn data into a quiver representation. Most classically this will be a representation of a linearly ordered quiver of type A. Such a representation can be depicted as a collection of line segments, corresponding to the supports of the indecomposable summands. This depiction is known as a “bar code”. One interprets the results by considering the longest bars most significant.

In many applications, it would be natural to consider multiple parameters, equivalently representation of tensor products of multiple type A quivers. These algebras are wild in almost all cases, and indecomposables are not determined by their support as in the one parameter case.

The original part of my talk will be based on joint work with Magnus Botnan and Steve Oudot. We introduce a candidate for a bar code of a 2-parameter persistence module, and observe that it is closely related to an exact structure on the representation category.