## Johnathan Bush

Colorado State University

I am interested in computational, applied, and algebraic topology and geometry.

# Papers

**"On the Nonlinear Statistics of Optical Flow"**

With Henry Adams, Brittany Carr, Lara Kassab, and Joshua Mirth.

Proceedings of Computational Topology in Image Context, LNCS volume 11382 (2019)

[Abstract, arXiv:1812.00875]

In *A naturalistic open source movie for optical flow evaluation*, Butler et al. create a database of ground-truth optical flow from the computer-generated video short *Sintel*.
We study the high-contrast \(3\times 3\) patches from this video, and provide evidence that this dataset is well-modeled by a *torus* (a nonlinear 2-dimensional manifold).
Our main tools are persistent homology and zigzag persistence, which are popular techniques from the field of computational topology.
We show that the optical flow torus model is naturally equipped with the structure of a fiber bundle, which is furthermore related to the statistics of range images.

**"Vietoris–Rips Thickening of the Circle and Centrally–Symmetric Orbitopes"**

With Henry Adams.

Submitted in partial fulfillment of the requirements for the degree of Master of Science at Colorado State University (2018).

[Abstract, Master's Thesis]

Given a metric space \(X\) and a scale parameter \(r>0\), the associated Vietoris–Rips simplicial complex, denoted \(\text{VR}(X;r)\), has as its simplices all finite subsets of \(X\) of diameter at most \(r\).
In the case that \(X\) is a Riemannian manifold, a result of Jean–Claude Hausmann states that the homotopy type of \(X\) is achieved by \(\text{VR}(X;r)\) for sufficiently small \(r\).
However, this approach does not recover metric information about \(X\), and this deficiency motivates the consideration of a related construction, called the Vietoris–Rips thickening of \(X\), defined via the theory of optimal transport.
This construction, which does preserve metric information about \(X\), additionally satisfies an analogue of Hausmann's theorem for sufficiently small \(r\).
On the other hand, one often encounters such thickenings given instead by increasingly large values of \(r\) in applications of persistent homology, and much less is known about the topological behavior of these constructions.
A recently established result due to Adams and Adamaszek provides the homotopy type of the Vietoris–Rips complex of the circle for arbitrarily large values of \(r\).
Presently, we determine the homotopy type of the Vietoris–Rips thickening of the circle for a range of values of \(r\).
Our primary tools will be an embedding of the metric thickening into Euclidean space via a symmetric moment curve, and the facial structure of the related Barvinok–Novik orbitope.