Research
I am broadly interested in algebraic geometry. I’m currently working on some questions related to splitting loci and Hurwitz stacks of finite covers of P^1. Below are some publications from my undergraduate research.
Papers
Two Formulas for F-Polynomials, with Gregg Musiker and Tomoki Nakanishi. International Mathematics Research Notices (2023), rnad074.
We discuss a product formula for F-polynomials in cluster algebras, and provide two proofs. One proof is inductive and uses only the mutation rule for F-polynomials. The other is based on the Fock-Goncharov decomposition of mutations. We conclude by expanding this product formula as a sum and illustrate applications. This expansion provides an explicit combinatorial computation of F-polynomials in a given seed that depends only on the c-vectors and g-vectors along a finite sequence of mutations from the initial seed to the given seed.
Symmetric Group Action of the Birational R-matrix, with Sunita Chepuri. Journal of Combinatorics 14 (2023), no. 2, 213-256.
The birational R-matrix is a transformation that appears in the theory of geometric crystals, the study of total positivity in loop groups, and discrete dynamical systems. This R-matrix gives rise to an action of the symmetric group S_m on an m-tuple of vectors. While the birational R-matrix is precisely the formula corresponding to the action of the simple transposition si, explicit formulas for the action of other permutations are generally not known. One particular case was studied by Lam and Pylyavskyy as it relates to energy functions of crystals. In this paper, we will discuss formulas for several additional cases, including transpositions, and provide combinatorial interpretations for the functions that appear in our work.
On Virtually Cohen-Macaulay Simplicial Complexes, with Nathan Kenshur, Sean McNally, Zixuan Xu and Teresa Yu. Journal of Algebra 631(2023), 120-135.
We examine virtual resolutions of Stanley-Reisner ideals for a product of projective spaces. In particular, we provide sufficient conditions for a simplicial complex to be virtually Cohen-Macaulay (to have a virtual resolution with length equal to its codimension). We also show that all balanced simplicial complexes are virtually Cohen-Macaulay.
Other writings
Metaplectic Analogues of Gelfand-Graev Models, with Q. D. Vu, N. Kenshur, C. Meng, Z. Stier, C. Yost-Wolff. REU report (2019)
In this report, we study the representations of the Schur cover of the eleven (untwisted) finite groups of Lie type whose Schur multiplier is non-trivial.