Metric Spaces in Applied Topology Joshua Mirth, Colorado State - PowerPoint PPT Presentation

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Metric Spaces in Applied Topology Joshua Mirth, Colorado State University QG&T at OSU April 27, 2019 Two principles: Datasets are metric spaces. Datasets have shapes. A dataset in R 2 and its VietorisRips complex. 2

Metric Spaces in Applied Topology Joshua Mirth, Colorado State University QG&T at OSU – April 27, 2019
Two principles: • Datasets are metric spaces. • Datasets have shapes. A dataset in R 2 and its Vietoris–Rips complex. 2
Synthesis: make simplicial complexes metric spaces! a d ( 1 3 a + 1 3 b + 1 3 c ) 4 c + 3 ( 1 4 d ) b c • Metrizes the simplicial complex topology when finite. • More natural when the vertex set is a (possibly infinite) metric space. Qestions: • If M is a manifold, for what r is VR m ( M ; r ) ≃ M ? • When VR m ( M ; r ) is not homotopy equivalent to M , what is its homotopy type? 3

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