Research and Writing
Papers
- An explicit leafwise branched covering for a transverse pseudo-Anosov flow (draft). (2026) pdf.
- Curve Stitching and Dancing Planets. (2024) pdf. and published in The Mathematical Intelligencer.
Abstract: We construct a specific branched submersion for a 3-manifold with a taut codimension one foliation transverse to a pseudo-Anosov flow. This map is a leafwise branched covering for the foliation: the restriction to each leaf is a branched covering. Moreover, its critical locus is a set of closed orbits of the flow, containing all the singular orbits. The construction depends only on the pseudo-Anosov flow and is done via an associated veering triangulation. We also develop framework for branched submersions, homotopies through branched immersions, and taut cone fields. In the last section, we work out an explicit example for the figure eight knot complement.
Abstract: Curve stitching is a classic educational activity where one constructs elegant curves from a family of straight lines. We perform curve stitching around a circle to make a modular stitch graph. Take m points equally spaced around a circle, choose an integer multiplier a, and draw a chord from point p to a p mod m What design will appear as the envelope of these chords? We connect these discrete objects to a continuous-time dynamical system and apply a topological perspective to understand the answer to this question.
Expository notes
- 3-Manifolds Notes (2025)
These are class notes from a course on 3-manifolds taught by Danny Calegari and Benson Farb in Fall 2024. There are still some comments and unfinished ideas. All mistakes are my own.
- Foliating Three-Dimensional Manifolds by Surfaces (2025)
This is my topics proposal for my PhD candidacy exam. It covers Riemann surfaces, mapping class groups, and foliations on three-manifolds.
- Unpacking Thurston's Theorem for post-critically finite maps (2024)
This is my final paper for Differential Topology at U Chicago.
- All knotted up (2024)
An exploration of the mathematics of designing crochet patterns for Seifert surfaces. This writeup was for the Big Internet Math-Off in 2024.
- (Con)figuring out spaces (2024)
An introduction to the configuration space of points using dancers on a stage as an example. This writeup was for the Big Internet Math-Off in 2024. I discuss these same ideas in a dance film.
- The Platonic Solids Strike Again! (2020)
This was my final paper for my undergraduate complex analysis class. It covers Mobius transformations, some basic group theory, and the classification of finite 3-dimensional rotation groups.
Undergraduate Research
- A characterization of two-dimensional Buchsbaum matching complexes. (2023) Joint with Bennet Goeckner, Legrand Jones, Rowan Rowlands. Published in the Electric Journal of Combinatorics. arxiv.
- Iterated Jump Graphs. (2022) Joint with Legrand Jones. Published in Rose-Hulman undergraduate journal. arxiv.
- All cellular strings on polytopes at once.(2022) Senior thesis. print.
