Organizes: Henry Adams and Ziqing Feng
Bestvina-Brady discrete Morse theory is a topological tool that has historically been most useful in geometric group theory. In this talk I will discuss a version of Bestvina-Brady Morse theory that is particularly conducive to understanding topological properties of Vietoris-Rips complexes of metric spaces, and has applications not only to geometric group theory, but also to applied topology and topological data analysis. In particular I will discuss a recent short proof of a result of Virk, that says the metric space $\mathbb{Z}^n$ with the usual $L^1$ metric has contractible Vietoris-Rips complexes.
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Topological data analysis is naturally suited to “data with shape”. In this talk, I will use a recent joint project with Jakini Auset Kauba as a demonstration of how TDA can uncover shape in geospatial data. In our project, we looked at persistence diagrams given by the demographics of 100 U.S. cities, and used them to perform various investigations and comparisons. Towards the end of the talk, I will highlight some of the pitfalls of using persistent homology on this kind of data, and pitch some challenges for those interested in TDA and geospatial data.
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