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 35 times and has been the subject of many study groups around the world. You can read it here:
- When Left and Right Turns Inside Out: A Geometric and Categorical Introduction to an Inverse Problem in Persistence. Last Update: November 30, 2018.
- How Many Directions Determine a Shape and other Sufficiency Results for Two Topological Transforms with Sayan Mukherjee and Katharine Turner. Last Update: May 24, 2018.
- Classification of Constructible Cosheaves with Amit Patel. 20 pages. Last Update: May 3, 2017.
- The Fiber of the Persistence Map for Functions on the Interval. 19 pages. Published by JACT. Last Update: January 22, 2019.
- Dualities between Cellular Sheaves and Cosheaves. 32 pages. Published by JPAA: June 27, 2017.
- Discrete Morse Theory for Computing Cellular Sheaf Cohomology with Robert Ghrist and Vidit Nanda. 23 pages. Published by FoCM: June 20, 2015.
- Topological Data Analysis and Cosheaves. 39 pages. Published by JJIAM: 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: February 1, 2012.
- Soliton Solutions of Integrable Systems and Hirota’s Method. 16 pages. Published by the Harvard College Mathematics Review: Spring 2008.