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Metric Graph Approximations of Geodesic Spaces
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 theExpand
The Distortion of the Reeb Quotient Map on Riemannian Manifolds
Given a metric space $X$ and a function $f: X \to \mathbb{R}$, the Reeb construction gives metric a space $X_f$ together with a quotient map $X \to X_f$. Under suitable conditions $X_f$ becomes aExpand
Vietoris-Rips Persistent Homology, Injective Metric Spaces, and The Filling Radius
In the applied algebraic topology community, the persistent homology induced by the Vietoris-Rips simplicial filtration is a standard method for capturing topological information from metric spaces which arises by first embedding a given metric space into a larger space and then considering thickenings of the original space. Expand
Quantitative Simplification of Filtered Simplicial Complexes
We introduce a new invariant defined on the vertices of a given filtered simplicial complex, called codensity , which controls the impact of removing vertices on the persistent homology of this filtered complex. Expand
Reeb posets and tree approximations
This work was partially supported by NSF grants IIS-1422400 and CCF-1526513.A well known result in the analysis of finite metric spaces due to Gromov says that given any $(X,d_X)$ there exists a \emph{tree metric} $t_X$ on $X$, a quantity that measures the treeness of $4$-tuples of points in $X$. Expand
Free actions on products of spheres at high dimensions
Ankara : The Department of Mathematics and the Graduate School of Engineering and Science of Bilkent University, 2012.
Decorated Merge Trees for Persistent Topology
This paper introduces decorated merge trees (DMTs) as a novel invariant for persistent spaces. DMTs combine both π0 and Hn information into a single data structure that distinguishes filtrations thatExpand