I work in the area of applied topology. This area operates at the interface of math, statistics, computer science, and engineering. My research program
- Develops foundations for topological data analysis (TDA) by blending algebra and topology together with sheaf theory;
- Studies analytic, geometric and combinatorial questions motivated by TDA;
- Prompts interdisciplinary work and computational projects suitable for (under)graduate students.
Click here to view my CV. Updated November 5, 2023.
My PhD thesis developed the theory of sheaves and cosheaves with an eye towards applications. It is meant to be both a pedagogical introduction to the basics of sheaf theory as well as a source of new mathematics, both pure and applied. It has been cited over 182 times and has been the subject of many study groups around the world. You can read it here:
- Algebraic and Geometric Models for Space Networking by William Bernardoni, Robert Cardona, Jacob Cleveland, Justin Curry, Robert Green, Brian Heller, Alan Hylton, Tung Lam, Robert Kassouf-Short. 43 pages, 18 figures. Last update: April 3, 2023.
- A Relative Theory of Interleavings with Magnus Bakke Botnan and Elizabeth Munch. 50 pages. Last Update: April 29, 2020.
- Functors on Posets Left Kan Extend to Cosheaves: an Erratum. 10 pages. Last Update: July 22, 2019.
Conference Papers and Abstracts
- Topologically Attributed Graphs for Shape Discrimination by Justin Curry, Washington Mio, Tom Needham, Osman Okutan, and Florian Russold. 14 pages = 9 pages + references and supplemental section, 4 figures. Accepted June 18, 2023 to the Workshop on Topology, Algebra, and Geometry in Machine Learning (TAGML) as part of the 40th International Conference on Machine Learning (ICML).
- Convergence of Leray Cosheaves for Decorated Mapper Graphs by Justin Curry, Washington Mio, Tom Needham, Osman Okutan, and Florian Russold. 6 pages, 2 figures. Accepted April 18, 2023 to Young Researchers Forum (YRF) of the 39th International Symposium on Computational Geometry (SoCG 2023).
- Towards Time Synchronization in the Solar System Internet by Alan Hylton, Natalie Tsuei, Mark Ronnenberg, Jihuan Hwang, Brendan Mallery, Jonathan Quartin, Colin Levaunt, Jeremy Quail, Justin Curry. 19 pages, 10 figures. Accepted to AeroConf 2023 on November 15, 2022.
- Contact Multigraph Routing: Overview and Implementation by Michael Moy, Robert Short, Nadia Kortas, Jacob Cleveland, Dominic Conricode, Yael Kirkpatrick, Robert Cardona, Brian Heller and Justin Curry. 8 pages, 4 figures. Accepted to AeroConf 2023 on November 15, 2022.
- A Lattice-Theoretic Perspective on the Persistence Map by Brendan Mallery, Adélie Garin, and Justin Curry. 4 pages, 2 figures. Accepted March 26, 2022 to the Young Researchers Forum (YRF) of the 38th International Symposium on Computational Geometry (SoCG 2022).
- The Universal ℓ^p-Metric on Merge Trees with Robert Cardona, Tung Lam and Michael Lesnick. 20 pages, 10 figures. Accepted February 9, 2022 to the 38th International Symposium on Computational Geometry (SoCG 2022).
- Sheaf Theoretic Models for Routing in Delay Tolerant Networks by Robert Short, Alan Hylton, Jacob Cleveland, Michael Moy, Robert Cardona, Robert Green, Justin Curry, Brendan Mallery, Gabriel Bainbridge, Zara Memon. 19 pages, 2 figures. Accepted November 16, 2021 to AeroConf 2022. Published March 9, 2022.
- Introducing Tropical Geometric Approaches to Delay Tolerant Networking Optimization by Jacob Cleveland, Alan Hylton, Robert Short, Brendan Mallery, Robert Green, Justin Curry, Devavrat Vivek Dabke, Olivia Freides. 11 pages, 10 figures. Accepted November 16, 2021 to AeroConf 2022. Published March 9, 2022.
- A Survey of Mathematical Structures for Lunar Networks by Alan Hylton, Robert Short, Jacob Cleveland, Olivia Freides, Zara Memon, Robert Cardona, Robert Green, Justin Curry, Sriram Gopalakrishnan, Devavrat Vivek Dabke, Brittany Story, Michael Moy, Brendan Mallery. 17 pages, 6 figures. Accepted November 16, 2021 to AeroConf 2022. Published March 9, 2022.
- A Sheaf-Theoretic Construction of Shape Space with Shreya Arya and Sayan Mukherjee. 45 pages, 11 figures. Accepted on September 4, 2023 to Foundations of Computational Mathematics (FoCM).
- From Trees to Barcodes and Back Again II: Combinatorial and Probabilistic Aspects of a Topological Inverse Problem with Jordan DeSha, Adélie Garin, Kathryn Hess, Lida Kanari, and Brendan Mallery. 39 pages, 17 figures. Accepted on June 9, 2023 to Computational Geometry: Theory and Applications (CGTA).
- How Many Directions Determine a Shape and other Sufficiency Results for Two Topological Transforms with Sayan Mukherjee and Katharine Turner. 38 pages. Accepted on April 27, 2022 by the Transactions of the AMS (TAMS).
- Decorated Merge Trees for Persistent Topology with Haibin Hang, Washington Mio, Tom Needham and Osman Berat Okutan. 39 pages. Accepted on February 9, 2022 by the Journal of Applied and Computational Topology (JACT).
- Classification of Constructible Cosheaves with Amit Patel. 37 pages. Published by Theory and Application of Categories (TAC) on June 24, 2020.
- Moduli Spaces of Morse Functions for Persistence with Michael J. Catanzaro, Brittany Terese Fasy, Jānis Lazovskis, Greg Malen, Hans Riess, Bei Wang, and Matthew Zabka. 33 pages. Published online by Journal of Applied and Computational Topology (JACT) on June 27, 2020.
- The Fiber of the Persistence Map for Functions on the Interval. 19 pages. Published by the Journal of Applied and Computational Topology (JACT) on January 22, 2019.
- Dualities between Cellular Sheaves and Cosheaves. 32 pages. Published by the Journal of Pure and Applied Algebra (JPAA) on June 27, 2017.
- Discrete Morse Theory for Computing Cellular Sheaf Cohomology with Robert Ghrist and Vidit Nanda. 23 pages. Published by Foundations of Computational Mathematics (FoCM) on June 20, 2015.
- Topological Data Analysis and Cosheaves. 39 pages. Published by the Japanese Journal of Industrial and Applied Math (JJIAM) on June 18, 2015.
- Euler Calculus with Applications to Signals and Sensing with Robert Ghrist and Michael Robinson. 71 pages. Published by Proceedings of Symposia in Applied Mathematics on February 1, 2012.
- Soliton Solutions of Integrable Systems and Hirota’s Method. 16 pages. Published by the Harvard College Mathematics Review (HCMR) Spring 2008.
- Mental Health in the Mathematics Community with Mikael Vejdemo-Johansson and Julie Corrigan. 6 pages. Published by the Notices of the American Mathematical Society in the August 2019 issue, Volume 66, Number 7, pp. 1079-1084.