Topology, Geometry and Data Seminar - Osman Okutan

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Osman Okutan
September 25, 2018
4:10PM - 5:10PM
Location
Cockins Hall 240

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Add to Calendar 2018-09-25 16:10:00 2018-09-25 17:10:00 Topology, Geometry and Data Seminar - Osman Okutan Title: Metric Graph Approximations of Geodesic Spaces Speaker: Osman 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 URL: https://tgda.osu.edu/activities/tdga-seminar/ Cockins Hall 240 Department of Mathematics math@osu.edu America/New_York public
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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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