In topological data analysis, one often studies data by associating to the data a filtered topological space whose structure can then be examined using persistent homology. However, in many settings, a single-filtered space is not a rich enough invariant to encode the interesting structure of the data. This motivates the study of multiparameter persistence, which associates to the data a topological space simultaneously equipped with two or more filtrations. The homological invariants of these “multifiltered spaces,” while much richer than their 1-D counterparts, are also far more complicated. As such, adapting the usual 1-D persistent homology methodology for data analysis to the multi-D setting requires new ideas. Recent years have seen significant progress in the development of such ideas. At the same time, many interesting and important questions remain open, particularly in the computational and statistical aspects of multiparameter persistence.
This tutorial, aimed at graduate students, postdocs, and other early-career researchers, will both acquaint participants with recent research developments in multiparameter persistence and also prepare participants to use multiparameter persistence as a tool in their own research. The tutorial will feature in-depth introductions to one-parameter and multiparameter persistence, with an emphasis on computational aspects. Talks will explore how the state-of-the-art in the one-parameter persistence computation can be leveraged in the multi-parameter setting. Statistical foundations and applications of multiparameter persistent homology will also be discussed. There will be ample time for discussion, collaboration, and exploration of software tools.
List of topics
- Introduction to persistent homology
- Persistence computation
- Morse Theory in persistence computation
- Multiparameter persistence
- The RIVET software for multiparameter persistence computation
- Computing bigraded Betti numbers for multiparameter persistence
- Computation of indecomposables
- Statistical foundations of persistence
- Applications of (multiparameter) persistence