Research
My primary research interests are in low dimensional topology, and in particular quantum topology. The main tools I work with are diagrammatic algebras and diagrammatic representation theory. This usually means looking at actions of algebraic objects on vector spaces of diagrams. A general goal is to recover classical topological information using methods in quantum topology.
I am also interested in applications of constructions in quantum topology to quantum computing. Roughly speaking the idea is to encode a quantum computational scheme into a representation. Then questions about quantum computing can be interpreted and answered by studying the representation.
Research Papers:
6. Asymptotic faithfulness of quantum SP(4) mapping class group representations. Significant re-writes in progress.
5. Degenerations of skein algebras and quantum traces. Joint with Hiroaki Karuo and Thang Le. Submitted.
4. Quotients of braid groups by their congruence subgroups. Joint with Peter Patzt and Nancy Scherich. To appear in Proc. Amer. Math. Soc. Ser. B.
3. The Chebyshev-Frobenius homomorphism for stated skein modules of 3-manifolds. Joint with Thang Le. Mathematische Zeitschrift (2022) 301:1063–1105.
2. On topological quantum computing with mapping class group representations. Joint with Zhenghan Wang. Journal of Physics A: Mathematical and Theoretical, Volume 52, Number 1 (2018).
1. Comparing skein and quantum group representations and their application to asymptotic faithfulness. Joint with Zhenghan Wang. Pure and Applied Mathematics Quarterly, Volume 12, Number 4, 473-492, 2016.
Undergraduate Supervising:
2021 Georgia Tech REU- Filling permutations and minimally intersecting filling pairs
Joint with Elyssa Cirillo, Jiamin Li, Alice Ponte
2020 Georgia Tech REU
Joint with Andrea Barton, Jose Guzman, and Hannah Moon
2019 Georgia Tech REU
Joint with Jessica Appel, Katie Gravel, and Annie Holden
Arxiv: On quotients of congruence subgroups of braid groups. Submitted for Publication.
2017 UC Santa Barbara REU
Joint with Andres Mejia.
Arxiv:Admissibility and the C_2 spider.