Topology, Geometry and Data Seminar - Osman Okutan

September 16, 2018
Tuesday, September 25, 2018 - 4:10pm to 5:10pm
Cockins Hall 240
Osman Okutan

Title: Metric Graph Approximations of Geodesic Spaces

SpeakerOsman Okutan (Ohio State University)

Abstract: A standard result in metric geometry is that every compact geodesic metric space can be approximated arbitrarily well by finite metric graphs in the Gromov-Hausdorff sense. It is well known that the first Betti number of the approximating graphs may blow up as the approximation gets finer.

In our work, given a compact geodesic metric space $X$, we define a sequence $(\delta^X_n)_{n \geq 0}$ of non-negative real numbers by $$\delta^X_n:=\inf \{d_{GH}(X,G): G \text{ a finite metric graph, } \beta_1(G)\leq n \} .$$

By construction, and the above result, this is a non-increasing sequence with limit $0$. We study this sequence and its rates of decay with $n$. We also identify a precise relationship between the sequence and the first Vietoris-Rips persistence barcode of $X$. Furthermore, we specifically analyze $\delta_0^X$ and find upper and lower bounds based on hyperbolicity and other metric invariants. As a consequence of the tools we develop, our work also provides a Gromov-Hausdorff stability result for the Reeb construction on geodesic metric spaces with respect to the specific function given by distance to a reference point.

Seminar URLhttps://tgda.osu.edu/activities/tdga-seminar/

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