Presentation description

In topological data analysis, the persistent homology pipeline allows us to understand topological information about a data set in the form of a barcode - a summary of persistent homological features ("holes") across a one-dimensional filtration of simplicial complexes on our data. Some data warrant investigation across multiple dimensions, requiring the use of filtrations indexed by more than one parameter. Key information about these multidimensional filtrations cannot be completely described by one barcode; instead, summarizing the data requires a collection of barcodes, where each barcode corresponds to a one-dimensional filtration of our data. Critical points, points in the filtration where the simplicial complex undergoes a change in homology, partition such one-dimensional filtrations into equivalence classes by barcode. Our research aims to enumerate and describe these equivalence classes. Equivalently, in two dimensions, we want to understand how many ways a line with positive slope can partition a set of points in the plane.

We present preliminary findings, which partially classify when pairs of points cannot be separated by a line of positive slope, as well as precise definitions of the words "left" and "right."