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

▶

Dec 20, 2023 95 likes •140 views

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

SLIDE 1

Metric Spaces in Applied Topology

Joshua Mirth, Colorado State University QG&T at OSU – April 27, 2019

SLIDE 2

Two principles:

Datasets are metric spaces.
Datasets have shapes.

A dataset in R2 and its Vietoris–Rips complex.

2

SLIDE 3

Synthesis: make simplicial complexes metric spaces! ( 1

3a + 1 3b + 1 3c)

a b c d ( 1

4c + 3 4d)

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 VRm(M; r) ≃ M?
When VRm(M; r) is not homotopy equivalent to M, what is its