Anna Schenfisch: Representing Simplicial Complexes via Directional Topological Descriptors

Abstract.

In this talk, I explore several ideas related to one of my core research interests, directional transforms. Given a geometric simplicial complex K, suppose we record how the topological properties of sublevel sets of K change as we filter K in a fixed direction. A single such summary does not uniquely correspond to K. That is, many other simplicial complexes produce the same summary, and we say the single summary is not “faithful.” However, for nice enough K, it is known that taking summaries in all (uncountably many) directions does result in a faithful set. The questions I am interested in tend to lie somewhere between these two extremes – can we find a finite set of directions for which the set of summaries is still faithful? How many directions do we actually need? How can we choose them efficiently? And what happens if we change our summary slightly? This talk is intended to be accessible to a wide audience, and the nature of the problem lends itself to many visual explanations.