Johnathan Bush
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The connectivity of Vietoris-Rips complexes of spheres Open
We survey what is known and unknown about Vietoris-Rips complexes and thickenings of spheres. Afterwards, we show how to control the homotopy connectivity of Vietoris-Rips complexes of spheres in terms of coverings of spheres and projectiv…
Topological feature selection for time series data Open
We use tools from applied topology for feature selection on time series data. We develop a method for scoring the variables in a multivariate time series that reflects their contributions to the topological features of the corresponding po…
Toroidal Coordinates: Decorrelating Circular Coordinates with Lattice Reduction Open
The circular coordinates algorithm of de Silva, Morozov, and Vejdemo-Johansson takes as input a dataset together with a cohomology class representing a 1-dimensional hole in the data; the output is a map from the data into the circle that …
View article: Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes
Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes Open
We explore emerging relationships between the Gromov--Hausdorff distance, Borsuk--Ulam theorems, and Vietoris--Rips simplicial complexes. The Gromov--Hausdorff distance between two metric spaces $X$ and~$Y$ can be lower bounded by the dist…
View article: The topology of projective codes and the distribution of zeros of odd maps
The topology of projective codes and the distribution of zeros of odd maps Open
We show that the size of codes in projective space controls structural results for zeros of odd maps from spheres to Euclidean space. In fact, this relation is given through the topology of the space of probability measures on the sphere w…
Representations of energy landscapes by sublevelset persistent homology: An example with <i>n</i>-alkanes Open
Encoding the complex features of an energy landscape is a challenging task, and often, chemists pursue the most salient features (minima and barriers) along a highly reduced space, i.e., two- or three-dimensions. Even though disconnectivit…
Operations on Metric Thickenings Open
Many simplicial complexes arising in practice have an associated metric space\nstructure on the vertex set but not on the complex, e.g. the Vietoris-Rips\ncomplex in applied topology. We formalize a remedy by introducing a category of\nsim…