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.
I earned my PhD in 2014 from the University of Michigan under Mel Hochster. My undergraduate degree is from Kansas State University.
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)