New Invariants for Multiparameter Persistence Modules

Topological Data Analysis (TDA) is the use of abstract tools from algebraic topology for the concrete task of analysing large and complex datasets. One of the more common data sets TDA applies well to do are finite point clouds in metric spaces. At first glance, as discrete spaces, these finite point clouds carry little topological information. In order to study them across various scales, one assigns filtered simplicial complexes, that is a finite increasing sequence of simplicial complexes, whose vertices coincide with the given points. This is generally achieved using the Cech or Vietoris-Rips complex, as explained in [1, 3.2]. Using this family of simplicial complexes, we may compute the persistent homology of the filtered complex, giving us topological insight into our given data set across an increasing family of scales. As is shown in [2], computing the persistent homology of a filtered complex over a certain field F is equivalent to the classification of finitely generated graded modules over the graded ring F[t], called persistence modules. This classification is a well-known algebraic fact and can therefore be used to completely parametrise these persistence modules using barcodes, as seen in [2]. The power of barcodes is that they form a complete invariant of the filtered homology, in the sense that two persistence modules are isomorphic if and only if they have the same barcodes.

Multiparameter persistence is the study of persistence modules arising from multi-filtrations, that is we no longer have a total order on the space of simplicial complexes built from our finite point cloud. This is explored in [3], and the main difference between this setting and that of single parameter persistent homology is that we are now required to study finitely generated graded modules over the multivariate polynomial ring F[t1,...,tn]. The classification of such objects is substantially harder, and the assignment of complete invariants to multiparameter persistence modules is an area of ongoing research. This subject is explored in [3], where it is proved that no complete discrete invariant exists for multidimensional persistence, unlike barcodes for the single parameter case. There are however some continuous invariants, but these are in practice less useful as they are incomputable. The rank invariant is defined in [3] and is showed to be a useful and computable discrete invariant, although we know it to be far from complete.

The goal of this project is to discover other, richer discrete invariants to help classify multiparameter persistence modules. We require any such invariant to be (a) computable, (b) discriminative and (c) stable to perturbation. Constructing such invariants will require tools from representation theory, algebraic topology, category theory and algebraic geometry.

This project falls within the EPSRC 'Application driven Topological Data Analysis' research area.