**abstract:**
Vietoris-Rips complex, VR(X,r), at a scale r>0 of a sample of points X of a metric space M is a standard tool to approximate the
homotopy type of M. Understanding Vietoris-Rips complexes of spheres contributes to the foundational understanding of patterns
occurring when applying persistent homology techniques to data analysis. Vietoris-Rips metric thickening VR^{m}(X,r) is a metric
space version of VR(X,r) that, in addition to being homotopy equivalent to M (under some conditions), possesses a natural choice
of metric (a property that VR(X,r) lacks). In 1, the authors describe the homotopy type of the Vietoris-Rips metric thickenings of
a circle, VR^{m}(S^{1,} r), without considering the circle group's natural action. The space VR^{m}(S^{1,} r) inherits the circle group
action from the action of the circle on itself. In this talk, we describe the equivariant homotopy type of VR^{m}(S^{1,} r) and pose
some related questions.
This is a joint work with Henry Adams, Michael Moy, Nikola Sadovek, and Aditya De Saha.
1 Michael Moy. Vietorisâ€“Rips metric thickenings of the circle. Journal of Applied and Computational Topology, 7(4):831â€“877,
Jul 2023

Thu 3 Oct, 16:00 - 16:35, Aula Dini

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