My research is in low-dimensional topology, with a focus on the relationship between Legendrian knot theory and category theory. Specifically, I study framed Legendrian tangles which have been assigned objects from a fixed strict monoidal category. Legendrian knot theory is a rich area of mathematics branching off from classical knot theory in the 1970s by Vladimir Arnold, Yakov Eliashberg, and others who were motivated by ideas from classical mechanics. Applications have been found in many areas including optics, biology, data science, and robotics. As with classical knot theory, the main objective of Legendrian knot theory is differentiating between different Legendrian knots. This is done by constructing invariants which are functions from the set of Legendrian knots to some other set usually numbers. In my work, I develop invariants for framed Legendrian tangles using objects from a specific class of strict monoidal categories. This continues the work started by Nicolai Reshetikhin and Vladimir Turaev in 1990 in which they demonstrated how to construct invariants for classical framed tangles (or ribbons) using techniques involving strict monoidal categories.
To see my full research statement, click on the link here - RESEARCH STATEMENT
A Presentation for Categories of Legendrian Tangles, with Dr. Benjamin Cooper, preprint in preparation, 2025
Framed Legendrian Link Invariants via Legendrian Ribbon Categories, Knots in Washington 52, December 2025
How to Mathematically Play with "Weird" Ribbons, Central College Mathematics Colloquium, November 2025
How to Mathematically Play with "Weird" Ribbons, Cornell College SIG Seminar, November 2025
How to Mathematically Play with "Weird" Ribbons, Wartburg College MCSP Department Seminar, November 2025
Modeling Bat Wings Through the Lens of Legendrian Knot Theory, Truman State University Math Colloquium, October 2025
How to Mathematically Play with "Weird" Ribbons, Beloit College Mathematics Colloquium, October 2025
Legendrian Ribbon Categories, University of Iowa Algebra Seminar, October 2025
A Categorical Perspective on Legendrian Knots, AMCS-MATH Day 2025, September 2025: Slides
Quantum Groups, University of Iowa Reading Topology Seminar, April 2025: Slides
Ribbon Hopf Algebras Viewed Topologically, University of Iowa Algebra Seminar, March 2025: Slides
Understanding Ribbon Categories with Ribbons, St. Ambrose University Math Club, December 2024
Groupoids Fibered in Schemes, University of Iowa Reading Topology Seminar, November 2024: Slides
Category Theory Via Legendrian Knots, AMCS-MATH Day 2024, September 2024: Slides
The Legendrian Monoidal Category, University of Iowa Algebra Seminar, April 2024
Freedman's Classification of Topological 4-Manifolds, University of Iowa Reading Topology Seminar, February 2024
Understanding Ribbon Categories with Ribbons, Graduate and Undergraduate Student Seminar, November 2023: Slides
Completeness of Riemannian Manifolds, University of Iowa Reading Topology Seminar, October 2023
Homotopy Relations on Maps, University of Iowa Student Category Seminar, October 2023
Cofibrantly Generated Model Categories, University of Iowa Student Category Seminar, September 2023
Knot Floer Homology and Applications, University of Iowa Reading Topology Seminar, March 2023
Rigid Monoidal Categories, University of Iowa Student Algebra Seminar, February 2023
Category Theory Via String Diagrams, University of Iowa Reading Topology Seminar, December 2022: Slides
Fast Khovanov Homology and Lee Homology, University of Iowa Reading Topology Seminar, March 2022: Slides