Conference Agenda

Multiparameter persistent homology is an area of applied algebraic topology that studies topological spaces, often arising from complex data, simultaneously indexed by multiple parameters. In the usual setting, persistent homology studies a single-parameter filtration associated with a topological space. The homology of such a filtration is a persistence module, which can be conveniently described by its barcode decomposition. In many applications, however, a single-parameter filtration is not adequate to encode the structures of interest in complex data; two or more filtrations may be required. Multiparameter persistence studies the homology of spaces equipped with multiple filtrations. The homological invariants of these spaces are far more complicated than in the single-parameter setting, requiring new algebraic, computational, and statistical techniques. This work has deep connections to representation theory and commutative algebra, with compelling applications to data analysis.

Recent years have seen considerable advances in multiparameter persistent homology, including algorithms for working with large multiparameter persistence modules, software for computing and visualizing invariants, statistical techniques, and applications. This minisymposium will highlight recent work in multiparameter persistence. Talks will include including theoretical results, algorithmic advances, and applications to data analysis. As many important questions remain to be answered in order to advance the theory and to increase the applicability of multiparameter persistence, this minisymposium seeks to cultivate discussion and collaboration that will lead to new results in the practical use of multiparameter persistent homology.

One of the main ideas in Topological Data Analysis is to convert application data into an algebraic object called a persistence module and to calculate distances between such modules. I will introduce these constructions and describe the main examples of such distances, called Wasserstein distances. The weakest of these distances, called the bottleneck distance, has previously been described algebraically (called interleaving distance). This has been useful in theory and in applications. I will give an algebraic description of all of the Wasserstein distances and discuss their generalizations to multiparameter persistence.

This is joint work with Jonathan Scott and Don Stanley.

An important problem in the field of Topological Data Analysis is defining topological summaries which can be combined with traditional data analytic tools. For single parameter persistence modules Bubenik introduced the persistence landscape, a stable representation of persistence diagrams amenable to statistical analysis and machine learning tools. In this talk we generalise the persistence landscape to multiparameter persistence modules providing a stable representation of the rank invariant. We show that multiparameter persistence landscapes are stable with respect to the interleaving distance and persistence weighted Wasserstein distance. Moreover the multiparameter landscapes enjoy many more desirable properties: the collection of multiparameter landscapes associated to a module are interpretable, computable, amenable to statistical analysis, and faithfully represent the rank invariant. We shall provide example calculations to demonstrate potential applications and how one can interpret the multiparameter landscapes associated to a multiparameter module.

Using ideas inspired from geometric and differential topology, we introduce a version of multiparameter persistence, which combines sub-level and zig-zag persistence. Our construction arises from one-parameter families of smooth functions on compact manifolds. We show how to analyse this version of multiparameter persistence in geometric terms with several examples. Furthermore, we focus on practical aspects of this theory, with an emphasis on visualization and potential algorithm development. This is joint work with Peter Bubenik.

I will describe a sequence of fairly naive experiments and small observations, towards a characterization of the persistent homology of noise. This should be viewed as an attempt to quantify what it means for a bar to be "short", vs. "long" or "interesting".