Multi-parameter Persistence

Standard persistent homology tracks topological features across a single scale parameter. This is typically visualized by growning the radii of balls around datapoints, or increasing the height of a sub-level set, and tracking topological features as a function of such. This one-parameter theory is well understood from a variety of mathematical viewpoints including representation theory, graph theory, and from computational perspectives. The output in this simple case is a barcode, a collection of intervals that is provably a complete invariant of the filtration.

We often want to understand real data from a collection of perspectives, meaning there are often multiple natural parameters we’d like to vary simultaneously. A function on a manifold may have both a height and a density filtration; or a dataset of images might be filtered by both intensity and gradient; or we might want to filter the same manifold by multiple directions simultaneously. Multiparameter persistence generalizes the theory of persistence to these settings, but th algebra becomes significantly harder. Unlike the one-parameter case, there is no complete discrete invariant analogous to the barcode. From a representation theoretic perspective, the structure of multiparameter persistence modules is wild in general.

This project develops geometric models to study multiparameter persistence using tools from Morse theory and geometric topology. One classical source of one-parameter persistence is the sublevel-set filtration of a single Morse function on a manifold. The natural generalization is to consider generic families of Morse functions parameterized by a manifold. Thus, as you move through the parameter space, the Morse function changes, and the associated persistence changes with it. Together with Peter Bubenik, we use this setup in (Bubenik & Catanzaro, 2021) to construct and partially decompose multiparameter persistence modules, obtaining structure analogous to a barcode in several cases.