## Research

My research interests are in computational topology and geometry, combinatorial topology, and applied topology. In particular, I like to study Vietoris-Rips and Čech simplicial complexes, Vietoris-Rips and Čech metric thickenings, applications of topology to data analysis, and applications of topology to sensor networks.I am the co-director of the Applied Algebraic Topology Research Network, which hosts a weekly Online Seminar. Recordings of our seminar are available at our YouTube Channel, which has over 300 subscribers and currently averages over two hours watched per day.

At Colorado State I am a member of the Pattern Analysis Lab, a co-organizer of the Topology Seminar, and an attendee of the Algebraic Combinatorics Seminar.

Here is my PhD thesis, a related research highlight, and research statements from 2019, 2018, 2017, 2016. See section 4 of this column for some open (and some "closed") problems I am interested in.

### Research papers

Metric thickenings, Borsuk-Ulam theorems, and orbitopes.
With Johnathan Bush and Florian Frick.
(2019+).
[arXiv:1907.06276, Abstract, Slides1, Slides2, Poster] |

Multidimensional scaling on metric measure spaces.
With Mark Blumstein and Lara Kassab.
(2019+).
[arXiv:1907.01379, Abstract, Poster, Slides] |

On Vietoris-Rips complexes of planar curves.
With Ethan Coldren and Sean Willmot.
(2019+).
[arXiv:1812.03374, Abstract] |

Vietoris-Rips complexes of regular polygons.
With Samir Chowdhury, Adam Jaffe, and Bonginkosi Sibanda.
(2019+).
[arXiv:1807.10971, Abstract, Slides, Webpage] |

A fractal dimension for measures via persistent homology.
With Manuchehr Aminian, Elin Farnell, Michael Kirby, Chris Peterson, Joshua Mirth, Rachel Neville, Patrick Shipman, and Clayton Shonkwiler.
Accepted to appear in Abel Symposia (2019).
[arXiv:1808.01079, Abstract, Software] |

On the nonlinear statistics of optical flow.
With Johnathan Bush, Brittany Carr, Lara Kassab, and Joshua Mirth.
Proceedings of Computational Topology in Image Context, LNCS volume 11382 (2019), 151-165.
[Publisher link, arXiv:1812.00875, Abstract, Slides, Software] |

Metric thickenings of Euclidean submanifolds.
With Joshua Mirth.
Topology and its Applications 254 (2019), 69-84.
[Publisher link, arXiv:1709.02492, Abstract, Joshua's masters thesis, Slides, Poster] |

Metric reconstruction via optimal transport.
With Michał Adamaszek and Florian Frick.
SIAM Journal on Applied Algebra and Geometry 2 (2018), 597-619.
[PDF, Publisher link, arXiv:1706.04876, Abstract, Poster, Talk Notes, Slides, Video] |

Sweeping costs of planar domains.
With Brooks Adams and Colin Roberts.
In Erin W Chambers, Brittany T Fasy, and Lori Ziegelmeier, eds., Research in Computational Topology, pages 71-92, AWM Springer series, volume 13 (2018).
[Publisher Link, arXiv:1612.03540, Abstract] |

Vietoris-Rips and Čech complexes of metric gluings.
With Michał Adamaszek, Ellen Gasparovic, Maria Gommel, Emilie Purvine, Radmila Sazdanovic, Bei Wang, Yusu Wang, and Lori Ziegelmeier.
Proceedings of the 34th Symposium on Computational Geometry (2018), 3:1-3:15.
[Publisher Link, arXiv:1712.06224, Abstract, Slides] |

Persistence images: A stable vector representation of persistent homology.
With Sofya Chepushtanova, Tegan Emerson, Eric Hanson, Michael Kirby, Francis Motta, Rachel Neville, Chris Peterson, Patrick Shipman, and Lori Ziegelmeier.
Journal of Machine Learning Research 18 (2017), Number 8, 1-35.
[PDF of Paper, Publisher Link, arXiv:1507.06217, Abstract, Software] |

On Vietoris-Rips complexes of ellipses.
With Michał Adamaszek and Samadwara Reddy.
Journal of Topology and Analysis (2017), 1-30.
[Publisher Link, arXiv:1704.04956, Oberwolfach Preprint, Abstract, Slides] |

The Vietoris-Rips complexes of a circle.
With Michał Adamaszek.
Pacific Journal of Mathematics 290 (2017), 1-40.
[Publisher Link, arXiv:1503.03669, Abstract, Slides, Poster 1, Poster 2, Talk Video] |

Random cyclic dynamical systems.
With Michał Adamaszek and Francis Motta.
Advances in Applied Mathematics 83 (2017), 1-23.
[Publisher Link, arXiv:1511.07832, Abstract] |

Nerve complexes of circular arcs.
With Michał Adamaszek, Florian Frick, Chris Peterson, and Corrine Previte-Johnson.
Discrete & Computational Geometry 56 (2016), 251-273.
[PDF of Paper, Publisher Link, arXiv:1410.4336, Abstract] |

Evasion paths in mobile sensor networks.
With Gunnar Carlsson.
International Journal of Robotics Research 34 (2015), 90-104.
[Publisher Link, PDF of Paper, Abstract, Slides, Poster, Talk Video, Multimedia] |

Nudged elastic band in topological data analysis.
With Atanas Atanasov and Gunnar Carlsson.
Topological Methods in Nonlinear Analysis 45 (2015), 247-272.
[Publisher Link, arXiv:1112.1993, Abstract, Slides, Software, Javadoc, Data] |

Javaplex: A research software package for persistent (co)homology.
With Andrew Tausz and Mikael Vejdemo-Johansson.
Proceedings of ICMS 2014, Han Hong and Chee Yap (Eds), LNCS 8592 (2014), 129-136.
Software available at http://appliedtopology.github.io/javaplex/. [Publisher Link, Abstract] |

On the nonlinear statistics of range image patches.
With Gunnar Carlsson.
SIAM Journal on Imaging Sciences 2 (2009), 110-117.
[Publisher Link, PDF of Paper, Abstract, Data] |

### Software tutorials

- Andrew Tausz and I maintain a PDF tutorial and a wiki tutorial for the Javaplex Software for persistent homology.
- I wrote a Matlab tutorial and a BeanShell tutorial for the JPlex Software for persistent homology.

### Mathematica demonstrations

The following interactive Mathematica demonstrations allow one to visualize geometric simplicial complexes at various choices of scale. The user can click on the points to move them around, add new points, delete points, and vary the scale parameter via a slider. Thanks to Jan Segert for help in creating these demonstrations!- Mathematica demo on Čech and Vietoris-Rips complexes
- Mathematica demo on Vietoris-Rips and witness complexes
- Mathematica demo on witness complexes

## What is topology? What are simplicial complexes?

Topology is the study of shapes and surfaces in higher dimensions. One way to build a higher-dimensional space is by gluing simple building blocks together. For example, you can start with a set of vertices (0-dimensional building blocks), glue on a set of edges (1-dimensional building blocks), glue on a set of triangles (2-dimensional building blocks), and then glue on a set of tetrahedra (3-dimensional building blocks). But the fun doesn't stop there - you can continue by gluing on higher-dimensional pieces, to form what's called a*simplicial complex*. I think of this like building a LEGO set. When creating a LEGO set, you follow a list of instructions describing how to attach simple building blocks together. Once done, you can simply look at the LEGO set you built to see and understand its shape. In topology, you are sometimes given a list of instructions for how to glue higher-dimensional building block together. Once done, we can't understand the higher-dimensional shape by simply looking at it, but we can use mathematics to help us visualize the shape we built!