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.
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 105 times and has been the subject of many study groups around the world. You can read it here:
- Decorated Merge Trees for Persistent Topology with Haibin Hang, Washington Mio, Tom Needham and Osman Berat Okutan. 76 pages. Last Update: March 31, 2021.
- A Relative Theory of Interleavings with Magnus Bakke Botnan and Elizabeth Munch. 50 pages. Last Update: April 29, 2020.
- How Many Directions Determine a Shape and other Sufficiency Results for Two Topological Transforms with Sayan Mukherjee and Katharine Turner. 30 pages. Last Update: October 10, 2019.
- Functors on Posets Left Kan Extend to Cosheaves: an Erratum. 10 pages. Last Update: July 22, 2019.
- 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.