## Research

My research interests are in topology, geometry, and data analysis. More specific subfields of interest include applied topology, computational topology and geometry, combinatorial topology, metric geometry, machine learning, and sensor networks. I advance the study of Vietoris-Rips and Čech simplicial complexes, and I apply topology to data analysis, machine learning, and sensor networks. My research forms bridges between applied topology and nearby areas of mathematics, including quantitative topology, geometric group theory, Riemannian geometry, metric geometry, optimal transport, equivariant topology, and combinatorics, and I have experience working with datasets arising in interdisciplinary research areas such as computational chemistry, computer vision, collective motion in biological systems, sensor networks, and knowledge-guided machine learning.
**I am excited to be starting a faculty position in the Department of Mathematics at the University of Florida in Fall 2023.**

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 500 videos, over 5,000 YouTube subscribers, and averages about 24 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 an Associate Editor for the journal Foundations of Data Science.

Over 2019-2022 I was 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 2022, 2021, 2020, 2019, 2018, 2017, 2016. See Section 4 of the ACM SIGACT News, Open Problems Column, Vol. 28, No. 3, 2017 for some open (and some "closed") problems I am interested in.

### I. Press and mathematical writings

How to Tutorial-a-thon.
With Hana Dal Poz Kouřimská, Teresa Heiss, Sarah Percival, and Lori Ziegelmeier.
Notices of the American Mathematical Society, Volume 68, Number 9, October 2021.
[Print Version] |

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. Books

Counting Rocks! An Introduction to Combinatorics.
With Kelly Emmrich, Maria Gillespie, Shannon
Golden, and Rachel Pries (2023).
[Book Webpage, Book PDF, Abstract, Videos] |

### III. 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] |

### IV. Preprints

Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes.
With Johnathan Bush, Nate Clause, Florian Frick, Mario Gómez, Michael Harrison, R. Amzi Jeffs, Evgeniya Lagoda, Sunhyuk Lim, Facundo Mémoli, Michael Moy, Nikola Sadovek, Matt Superdock, Daniel Vargas, Qingsong Wang, Ling Zhou.
(2022).
[arXiv:2301.00246, Abstract] |

Additive energy functions have predictable landscape topologies.
With Brittany Story, Biswajit Sadhu, and Aurora Clark.
Submitted (2022), DOI 10.26434/chemrxiv-2022-1669f.
[ChemRxiv link, Abstract] |

Čech complexes of hypercube graphs.
With Samir Shukla and Anurag Singh.
Submitted (2022).
[arXiv:2212.05871, Abstract] |

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

Vietoris-Rips complexes of regular polygons.
With Samir Chowdhury, Adam Jaffe, and Bonginkosi Sibanda (2021).
[arXiv:1807.10971, Abstract, Slides, Webpage] |

### V. Research papers

A Primer on Topological Data Analysis to Support Image Analysis Tasks in Environmental Science.
With Lander Ver Hoef, Emily King, and Imme Ebert-Uphoff.
Artificial Intelligence for the Earth Systems 2 (2023), e220039.
[Publisher Link, arXiv:2207.10552, Abstract] |

Geometric approaches to persistent homology.
With Baris Coskunuzer.
Accepted to appear in SIAM Journal on Applied Algebra and Geometry (2022).
[arXiv:2103.06408, Abstract] |

The persistent topology of optimal transport based metric thickenings.
With Facundo Mémoli, Michael Moy, and Qingsong Wang.
Accepted to appear in Algebraic & Geometric Topology (2022).
[arXiv:2109.15061, Abstract] |

The topology of projective codes and the distribution of zeros of odd maps.
With Johnathan Bush and Florian Frick.
Accepted to appear in Michigan Mathematical Journal (2022).
[arXiv:2106.14677, Abstract] |

Metric thickenings and group actions.
With Mark Heim and Chris Peterson.
Journal of Topology and Analysis 14 (2022), 587-613.
[Publisher Link, arXiv:1911.00732, Abstract] |

Vietoris thickenings and complexes have isomorphic homotopy groups.
With Florian Frick and Žiga Virk.
Journal of Applied and Computational Topology (2022), DOI 10.1007/s41468-022-00106-5.
[Publisher Link, arXiv:2206.08812, Abstract] |

On Vietoris-Rips complexes of hypercube graphs.
With Michał Adamaszek.
Journal of Applied and Computational Topology 6 (2022), 177-192.
[Publisher Link, arXiv:2103.01040, Abstract] |

Support vector machines and Radon's theorem.
With Elin Farnell and Brittany Story.
Foundations of Data Science 4 (2022), 467-494, DOI 10.3934/fods.2022017.
[Publisher Link, arXiv:2011.00617, Abstract] |

Capturing dynamics of time-varying data via topology.
With Lu Xian, Chad Topaz, and Lori Ziegelmeier.
Foundations of Data Science 4 (2022), 1-36.
[Publisher Link, arXiv:2010.05780, Abstract] |

The persistent homology of cyclic graphs.
With Sophia Coldren and Sean Willmot.
International Journal of Computational Geometry and Applications 32 (2022), 1-37.
[Publisher Link, arXiv:1812.03374, Abstract] |

The middle science: Traversing scale in complex many-body systems.
With Aurora E Clark, Rigoberto Hernandez, Anna I Krylov, Anders Niklasson, Sapna Sarupria, Yusu Wang, Stefan M Wild, and Qian Yang.
ACS Central Science 7 (2021), 1271-1287.
[Publisher Link, Abstract, Supplemental Cover Photo] |

Lions and contamination, triangular grids, and Cheeger constants.
With Leah Gibson and Jack Pfaffinger.
Journal version: Accepted to appear in Ellen Gasparovic, Vanessa Robins, and Kate Turner, eds., Research in Computational Topology 2, AWM Springer series (2021).
Conference version: In 37th European Workshop on Computational Geometry (2021), 7:1-7:6.
[arXiv:2012.06702, Conference Version Link, Abstract] |

Topology applied to machine learning: From global to local.
With Michael Moy.
Frontiers in Artificial Intelligence; Machine Learning and Artificial Intelligence 4 (2021), 668302.
[Publisher Link, 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.
Journal version: Foundations of Data Science 3 (2021), 769-791.
Conference version: 54th Asilomar Conference on Signals, Systems, and Computers (2021), 1451-1456.
[Journal Version Link, Conference Version Link, arXiv:2002.02041, Abstract, Software] |

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] |

### VI. 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.

### VII. 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 Vietoris-Rips and Čech 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!