Combinatorial Algebraic Topology and Applications II

Equivariant homotopy type of Vietoris-Rips metric thickenings of a circle

speaker: Evgeniya (CONTRIBUTED TALK) Lagoda

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. Vietoris–Rips metric thickenings of the circle. Journal of Applied and Computational Topology, 7(4):831–877, Jul 2023

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