## Research

My research interests are in topology, geometry, data analysis, machine learning, and sensor networks. More specific subfields of interest include applied topology, computational topology and geometry, metric geometry, and combinatorial 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 machine learning, and applications of topology to sensor networks.I am a co-director of the Applied Algebraic Topology Research Network, which hosts a weekly Online Seminar. Recordings of our seminar are available at the AATRN YouTube Channel, which has over 2,000 YouTube subscribers, and averages over 18 hours watched per day. I also maintain a personal YouTube Channel.

At Colorado State I co-organize the Topology Seminar, am a member of the Pattern Analysis Lab, attend the Data Science Seminar, and am an affiliate faculty at the Data Science Research Institute.

I am a member of the Descriptors of Energy Landscapes Using Topological Data Analysis (DELTA) leadership team.

Here is my PhD thesis, a related research highlight, and research statements from 2020, 2019, 2018, 2017, 2016. See section 4 of this column for some open (and some "closed") problems I am interested in.

### I. Press and mathematical writings

How do I ... develop an online research seminar?
Notices of the American Mathematical Society, Volume 67, Number 8, September 2020. [Print Version] |

Anne Manning at CSU wrote a nice press release on my work with the Pattern Analysis Lab at CSU, August 11, 2020.
[Press Release] |

Topological data analysis of collective motion.
With Maria-Veronica Ciocanel, Chad M. Topaz, and Lori Ziegelmeier.
SIAM News, January/February Issue, 2020. [Publisher Link, Print Version] |

### II. Book chapter

Topological data analysis.
With Johnathan Bush and Joshua Mirth.
Book chapter in Data Science for Mathematicians, editor Nathan Carter, Chapman & Hall/CRC, New York (2020), DOI 10.1201/9780429398292.
[Publisher Link, Software, Software tutorial] |

### III. Preprints

Geometric approaches on persistent homology.
With Baris Coskunuzer.
Submitted (2021). [arXiv:2103.06408, Abstract] |

On Vietoris-Rips complexes of hypercube graphs.
With Michał Adamaszek
Submitted (2021). [arXiv:2103.01040, Abstract] |

Efficient evader detection in mobile sensor networks.
With Deepjyoti Ghosh, Clark Mask, William Ott, and Kyle Williams.
Submitted (2021). [arXiv:2101.09813, Abstract] |

Lions and contamination, triangular grids, and Cheeger constants.
With Leah Gibson and Jack Pfaffinger.
Submitted (2021). [arXiv:2012.06702, Abstract] |

Capturing dynamics of time-varying data via topology.
With Lu Xian, Chad Topaz, and Lori Ziegelmeier.
Submitted (2020). [arXiv:2010.05780, Abstract] |

Support vector machines and Radon's theorem.
With Brittany Carr and Elin Farnell.
Submitted (2020). [arXiv:2011.00617, Abstract] |

The persistent homology of cyclic graphs.
With Ethan Coldren and Sean Willmot.
Submitted (2020). [arXiv:1812.03374, Abstract] |

Vietoris-Rips complexes of regular polygons.
With Samir Chowdhury, Adam Jaffe, and Bonginkosi Sibanda (2020).
[arXiv:1807.10971, Abstract, Slides, Webpage] |

### IV. Research papers

Topology applied to machine learning: From global to local.
With Michael Moy.
Accepted to appear in Frontiers in Artificial Intelligence; Machine Learning and Artificial Intelligence (2021). [arXiv:2103.05796, Abstract] |

Representations of energy landscapes by sublevelset persistent homology: An example with n-alkanes.
With Joshua Mirth, Yanqin Zhai, Johnathan Bush, Enrique G Alvarado, Howie Jordan, Mark Heim, Bala Krishnamoorthy, Markus Pflaum, Aurora Clark, and Y Z.
Journal of Chemical Physics 154 (2021), 114114. [Publisher Link, arXiv:2011.00918, Abstract, Software] |

An adaptation for iterative structured matrix completion.
With Lara Kassab and Deanna Needell.
Accepted to appear in the 54th Asilomar Conference on Signals, Systems and Computers (2020). [arXiv:2002.02041, Abstract] |

Metric thickenings and group actions.
With Mark Heim and Chris Peterson.
Journal of Topology and Analysis (2020), DOI 10.1142/S1793525320500569 [Publisher Link, arXiv:1911.00732, Abstract] |

Operations on metric thickenings.
With Johnathan Bush and Joshua Mirth.
In: Spivak, D., Vicary, J. (eds), Applied Category Theory, Electronic Proceedings in Theoretical Computer Science 333:261-275 (2021). [Publisher Link, arXiv:2101.10489, Abstract] |

Metric thickenings, Borsuk-Ulam theorems, and orbitopes.
With Johnathan Bush and Florian Frick.
Mathematika 66 (2020), 79-102. [Publisher Link, arXiv:1907.06276, Abstract, Johnathan's Masters Thesis, Slides1, Slides2, Poster] |

A fractal dimension for measures via persistent homology.
With Manuchehr Aminian, Elin Farnell, Michael Kirby, Chris Peterson, Joshua Mirth, Rachel Neville, and Clayton Shonkwiler.
In: Baas N., Carlsson G., Quick G., Szymik M., Thaule M. (eds), Topological Data Analysis. Abel Symposia, Springer vol 15 (2020), 1-32. [Publisher Link, arXiv:1808.01079, Abstract, Software] |

On homotopy types of Vietoris-Rips complexes of metric gluings.
With Michał Adamaszek, Ellen Gasparovic, Maria Gommel, Emilie Purvine, Radmila Sazdanovic, Bei Wang, Yusu Wang, and Lori Ziegelmeier.
Journal version: Journal of Applied and Computational Topology 4 (2020), 425-454.
Conference version: Vietoris-Rips and Čech complexes of metric gluings in Proceedings of the 34th Symposium on Computational Geometry (2018), 3:1-3:15.
[Journal Version Link, Conference Version Link, arXiv:1712.06224, Abstract, Slides] |

Multidimensional scaling on metric measure spaces.
With Mark Blumstein and Lara Kassab.
Rocky Mountain Journal of Mathematics 50 (2020), 397-413. [Publisher Link, arXiv:1907.01379, Abstract, Lara's Masters Thesis, Poster, Slides] |

A torus model for optical flow.
With Johnathan Bush, Brittany Carr, Lara Kassab, and Joshua Mirth.
Journal version: Pattern Recognition Letters 129 (2020), 304-310.
Conference version: On the nonlinear statistics of optical flow published in Proceedings of Computational Topology in Image Context, LNCS volume 11382 (2019), 151-165.
[Journal Version Link, Conference Version 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] |

On Vietoris-Rips complexes of ellipses.
With Michał Adamaszek and Samadwara Reddy.
Journal of Topology and Analysis 11 (2019), 661-690. [Publisher Link, arXiv:1704.04956, Oberwolfach Preprint, Abstract, Slides] |

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] |

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] |

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] |

### V. Software tutorials

- I co-wrote a software tutorial for point cloud persistent homology with Ripser.
- 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.

### VI. 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!