Metrics in Multiparameter Persistence

This workshop has been rescheduled to 19 – 23 July 2021

Persistent homology is a central topic in the burgeoning field of topological data analysis. The key idea is to study topological spaces constructed from data and infer global properties of the data from topological invariants, typically homology vector spaces. The term “persistent” refers to the fact that the construction of these spaces usually depends on one or more parameters, and in order to obtain information about the data in a stable and robust way, it is crucial to consider the family of resulting invariants obtained not just in isolation, but together with morphisms obtained from maps between the individual spaces.

While the bulk of work on persistent homology to data has focused on one-parameter constructions, the need for methods that are robust with respect to noise and outliers naturally leads to methods based on multiple parameters. Although there has been considerable recent progress in the theoretical and computational aspects of multi-parameter persistent homology, applications are still in their infancy. 

Metrics on invariants of data play a central role in both theory and applications of one parameter persistent homology, and it is expected that such metrics should play a similarly important role in the multi-parameter setting. However, the invariants of data provided by this setting are much more complex, and this presents obstacles to naively extending the definitions from the one-parameter case. This raises the question of which metrics are most suitable for practical use.

This workshop will bring together specialists and junior researchers in applied algebraic topology to study the problem of metrizing the space of multi-parameter persistence modules. In particular, the workshop will address the problem of developing computationally tractable metrics suitable for use in applications. The focus will be on both the theoretical foundations and practical computational aspects of such metrics. Considerable time will be devoted to discussion sessions and hands-on work with software implementations and data.