My research interests are in low-dimensional topology and symplectic/contact geometry. Most recently I have been thinking about trisections of 4-manifolds, knotted surfaces, and Lagrangian cobordisms between Legendrian knots. For a layperson’s “explainer” on some of the context in which my research sits, check out this article I wrote as part of the “Snapshots of Modern Mathematics from Oberwolfach” series, or this awesome Quanta article.

Publications & Preprints

1. “Bridge position of 3-manifolds embedded in the 5-sphere”
   • Joint with Román Aranda, Geunyoung Kim, Patrick Naylor, and Puttipong Pongtanapaisan. We study bridge decompositions of 3-manifolds embedded in the 5-sphere, encoding this decomposition diagrammatically.
2. “Danceability, A New Definition of Bridge Index”
   • Joint with Sol Addison, Puttipong Pongtanapaisan, Nancy Scherich, Lila Snodgrass, Everett Sullivan. We show that a math-dance-inspired link invariant called danceability is the same as the bridge index.
3. “Cornered skein lasagna theory”
   • Joint with Slava Krushkal and Yangxiao Luo. We extend skein lasagna theory to the setting of cornered 4-manifolds, motivated by trisections, which are decompositions of 4-manifolds into simple (but cornered) pieces.
4. “Classical invariants of spiral knots”
   • Joint with my students Ashish Das, Sydney Mayer, Luke Moyar, and Faisal Leo Quraishi, and co-group-leader Ryan Stees. This project, a product of the 2024 UVA Topology REU, investigates spiral knots, a braid-theoretic generalization of torus knots.
5. “Bridge indices of spatial graphs and diagram colorings”
   • Joint with Puttipong Pongtanapaisan and Hanh Vo. We extend the Wirtinger number to spatial graphs, and use this along with some lower bound techniques to compute the bridge number of a large family of spatial graphs.
6. “Pants distances of knotted surfaces in 4-manifolds”
   • Joint with Román Aranda, Devashi Gulati, Homayun Karimi, Geunyoung Kim, Nicholas Paul Meyer, and Puttipong Pongtanapaisan. We extend the Kirby-Thompson L-invariant to surfaces in 4-manifolds.
7. “Constructing Lagrangians from triple grid diagrams”
   • Joint with David Gay and Peter Lambert-Cole, in which we show how to encode Lagrangian surfaces in the complex projective plane with combinatorial diagrams, via trisections.
8. “A group-theoretic framework for low-dimensional topology”
   • Joint with Robion Kirby, Michael Klug, Vincent Longo, Benjamin Ruppik. In this project, which began at the Summer Trisectors Workshop in 2020, we detail how to recover a bridge trisected surface in a 4-manifold from its group trisection.
9. “Constructions of Lagrangian cobordisms”
   • Joint with my group from the WiSCon workshop in 2019: Noémie Legout, Caitlin Leverson, Maÿlis Limouzineau, Ziva Myer, Yu Pan, Samantha Pezzimenti, Lara Simone Suárez, and Lisa Traynor.
10. “Good Reduction of Preperiodic Points”
    • During my senior year at St. Louis University I completed an independent research project on dynamical systems under the guidance of Benjamin Hutz.
11. “A Generalization of a Method of Mordell to Ternary Quadratic Forms”
    • In the summer of 2015 I participated in WADE Into Research, a number theory REU. Joint with my groupmates Gabriel Durham and Tiffany Treece, and project leader Katherine Thompson.

Selected Recent Talks

Recent & Upcoming Travel

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In 2020 I co-organized GOATS, a virtual mini-conference series for graduate students in topology and geometry, held on April 25, June 6, and July 18. See details here.

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Here are some postcards I made for NGNCT 2021.

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Here is a postcard (front and back) I made for Braids in Low-Dimensional Topology (at ICERM) in 2022.