My research is in applied topology. In particular, I am interested in the structure of Vietoris–Rips simplicial complexes, discrete Morse theory, and persistent homology.

I am co-organizer of the Greenslopes seminar at Colorado State University and of the Category Theory lab, as well as an attendee of the Pattern Analysis Lab and the Topology Seminar.

# Papers

"On the nonlinear statistics of optical flow."

With Henry Adams, Johnathan Bush, Brittany Carr, and Lara Kassab.

To appear in Pattern Recognition Letters, special issue *Topological Analysis and Recognition* (2018+).

[Publisher link, arXiv:1812.00875, Slides, Abstract]

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.

"A fractal dimension for measures via persistent homology"

With Henry Adams, Manuchehr Aminian, Elin Farnell, Michael Kirby, Rachel Neville, Chris Peterson, and Clayton Shonkwiler.

Submitted September 2018.

[arXiv:1808.01079, Abstract]

We use persistent homology in order to define a family of fractal dimensions, denoted \(\dim_{\text{PH}}^i(\mu)\) for each homological dimension \(i\ge 0\), assigned to a probability measure \(\mu\) on a metric space.
The case of \(0\)-dimensional homology (\(i=0\)) relates to work by Michael J Steele (1988) studying the total length of a minimal spanning tree on a random sampling of points.
Indeed, if \(\mu\) is supported on a compact subset of Euclidean space \(\mathbb{R}^m\) for \(m\ge2\), then Steele's work implies that \(\dim_{\text{PH}}^0(\mu)=m\) if the absolutely continuous part of \(\mu\) has positive mass, and otherwise \(\dim_{\text{PH}}^0(\mu) \lt m\). Experiments suggest that similar results may be true for higher-dimensional homology \(0 \lt i \lt m\), though this is an open question.
Our fractal dimension is defined by considering a limit, as the number of points \(n\) goes to infinity, of the total sum of the \(i\)-dimensional persistent homology interval lengths for \(n\) random points selected from \(\mu\) in an i.i.d. fashion.
To some measures \(\mu,\) we are able to assign a finer invariant, a curve measuring the limiting distribution of persistent homology interval lengths as the number of points goes to infinity.
We prove this limiting curve exists in the case of \(0\)-dimensional homology when \(\mu\) is the uniform distribution over the unit interval, and conjecture that it exists when \(\mu\) is the rescaled probability measure for a compact set in Euclidean space with positive Lebesgue measure.

"Metric Thickenings of Euclidean Submanifolds"

With Henry Adams.

Submitted December 2017

[Publisher link, arXiv:1709.02492, Abstract, Master's Thesis]

Given a sample \(X\) from an unknown manifold \(M\) embedded in Euclidean space, it is possible to recover the homology groups of \(M\) by building a Vietoris-Rips or Čech simplicial complex on top of the vertex set \(X\).
However, these simplicial complexes need not inherit the metric structure of the manifold, in particular when \(X\) is infinite.
Indeed, a simplicial complex is not even metrizable if it is not locally finite.
We instead consider metric thickenings of \(X\), called the *Vietoris-Rips* and *Čech thickenings*, which are equipped with the 1-Wasserstein metric in place of the simplicial complex topology.
We show that for Euclidean subsets \(M\) with positive reach, the thickenings satisfy metric analogues of Hausmann's theorem and the nerve lemma (the metric Vietoris-Rips and Čech thickenings of \(M\) are homotopy equivalent to \(M\) for scale parameters less than the reach).
In contrast to Hausmann's original result, our homotopy equivalence is a deformation retraction, is realized by canonical maps in both directions, and furthermore can be proven to be a homotopy equivalence via simple linear homotopies from the map compositions to the corresponding identity maps.