Mini-course: Multidimensional Persistent Homology: Theory and Computation
Abstract: In topological data analysis, we often study data by associating to the data a filtered topological space, whose structure we can then examine using persistent homology. However, in many settings, a single filtered space is not a rich enough invariant to encode the interesting structure our data. This motivates the study of multidimensional 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 standard 1-D persistent homology methodology for data analysis to the multi-D setting requires some some new ideas.
We will introduce multidimensional persistent homology and describe our recent work on the development of a computational tool for its visualization.