Research

I'm interested in everything having to do with polynomials so I tend to gravitate towards problems in areas like algebraic geometry and commutative algebra but I love when applications for polynomials show up in other areas like combinatorics as well. As an undergraduate, I worked under the guidance of Adam Boocher (now at the University of San Diego) on an interesting combinatorial corner of commutative algebra that led to my senior thesis project on the Herzog-Kühl Equations. This theorem establishes a correspondence between the degree sequence of a finite graded free resolution of dimension zero and its Betti numbers and how we can use this correspondence along with a theorem by David Hilbert and John-Pierre Serre to find the dimension of a module over a polynomial ring. Under certain conditions, this can act as a very straightforward alternative to the traditionally difficult problem of finding the dimension of an algebraic variety.

Now, my research focuses on the use of combinatorial techniques in algebraic geometry. Specifically, I study blowups of toric varieties by studying the inherent combinatorial structure of toric varieties. In particular, I am currently studying the Cox rings of blowups of toric varieties that arise from triangular polytopes.

You can read my undergraduate thesis here.