Johnathan Bush

Broadly, I am interested in applied, computational, and algebraic topology. Most of my research is motivated by topological data analysis.

Currently, I am working to understand the topological behavior of parametrized simplicial complexes (e.g. Vietoris–Rips or Čech complexes) defined on manifolds at large scales. This has led to interesting connections between topics in applied topology, algebraic topology, and convex geometry. For example, the following theorem follows from knowledge of the homotopy type of the Vietoris–Rips complexes defined on the circle, $$\text{VR}(S^1;r)$$, at all scales.

Theorem. Equip $$S^1$$ with the geodesic metric of total circumference $$2\pi$$. Fix $$k\geq1$$ and let $$f\colon S^1\to\mathbb{R}^k$$ be a continuous map such that $$f(-x)=-f(x)$$ for all $$x\in S^1$$. Then, there exists a subset $$X\subseteq S^1$$ such that $$\text{diam}(X)\leq \frac{2\pi k}{2k+1}$$ and $$\vec{0}\in\text{conv}(f(X))$$, and this diameter bound is optimal.

This can be seen as a generalization of the Borsuk–Ulam theorem in which the dimension of the co-domain of the function may be arbitrarily large with respect to the dimension of the domain. For more details and a proof of this theorem, see the paper "Metric thickenings, Borsuk–Ulam theorems, and orbitopes" below, co-authored by Henry Adams and Florian Frick.

I am working to extend these results to make quantitative statements about arbitrary continuous maps $$S^n\to\mathbb{R}^k$$ in order to gain a better understanding of certain spaces that natually arise in topological data analysis and persistent homology.

Research papers

"Operations on metric thickenings"
With Henry Adams and Joshua Mirth.
Proceedings of Applied Category Theory Conference, Electronic Proceedings in Theoretical Computer Science 328:1-15 (2020).
[Abstract] [Paper accepted to EPTCS]

"Metric thickenings, Borsuk–Ulam theorems, and orbitopes"
With Henry Adams and Florian Frick.
Mathematika 66:79-102 (2020).
[Abstract] [Publisher link, arXiv:1907.06276, Poster, Slides]

"A torus model for optical flow"
With Henry Adams, Brittany Carr, Lara Kassab, and Joshua Mirth.
Journal version published in Pattern Recognition Letters 129:304-310 (2020).
Conference version "On the nonlinear statistics of optical flow" published in Proceedings of Computational Topology in Image Context, LNCS 11382:151–165 (2019).