My research interests are in commutative algebra, combinatorial commutative algebra, and algebraic geometry. Commutative algebra and algebraic geometry are closely linked through the study of solution sets V(S) to systems S of polynomial equations. The subtlety comes from different collections of polynomials defining the same solution set. It is helpful then, and non-trivial, to determine the largest set or ideal of polynomials defining V(S). The geometric structure of V(S) is reflected in the algebraic structure of its defining ideal — this is the phenomenon Hilbert’s Nullstellensatz describes.
My research profile reflects an outward trajectory from my initial interest in principal minor ideals, to projects which use a wide range of combinatorial techniques. Areas of combinatorial commutative algebra that appear in my work include matroids and matroid varieties, posets, graph theory, toric varieties, Stanley-Riesner rings, and sandpile groups.
Toric and tropical Bertini theorems in positive characteristic, joint with F. Gandini, M. Hering, D. Maclagan, F. Mohammadi, J. Rajchgot, and J. Yu. (arXiv:2111.13214)
Geometric equations for matroid varieties, joint with J. Sidman and W. Traves. J. Combin. Theory Ser. A 178 (2021), Paper No. 105360, 15 pp. MR4175891. (arXiv:1908.01233v3)
The sandpile group of a thick cycle graph, joint with D. Alar, J. Celaya, M. Henson, L. Garcia Puente. (arXiv:1710.06006)
Finiteness of associated primes of local cohomology modules over Stanley-Reisner rings, joint with R. Barrera and C. Madsen. (arXiv:1609.05366)
Principal minor ideals and rank restrictions on their vanishing sets, J. Algebra 469 (2017), 267–287. MR3563014. (arXiv:1503.05799)
Ideals generated by principal minors, Illinois J. Math. 59 (2015) no. 3, 675–689. MR3554228. (arXiv:1410.1910)