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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
TLDR
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
TLDR
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
TLDR
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