Research Interests
My research lies at the intersection of commutative algebra, invariant theory, and representation theory. I enjoy investigating questions that can be studied with a computational or combinatorial approach. Here is a video pitch of my work and a talk at the CHAMPS seminar .
I have proved degree bounds of invariant skew polynomials, bounds on the Castelnuovo-Mumford regularity and formulas for equivariant Hilbert series of ideals of subspace arrangements, collaborated on writing a software package for Invariant Theory, and collaborated on proving a toric and tropical version of Bertini's theorem in characteristic p.
Degree Bounds for Invariant Skew Polynomials
Using previous work on the regularity of ideals of subspace arrangements over the exterior algebra, I established degree bounds for invariant skew polynomials. This approach to invariant theory started from a connection to ideals of subspace arrangements due to Derksen. His results show that finding a minimal sets of invariants is equivalent to finding ideal generators for the vanishing ideal of a special subspace arrangement. To translate this approach to the context of the exterior algebra, I generalized some classical notions from invariant theory to (possibly non-commutative) graded algebras and adapted Derksen's approach to prove Noether's Degree Bound in the exterior algebra. The preprint is on arXiv 2108.01767 and is currently under peer review.
Toric and Tropical Bertini Theorems in Positive Characteristics
Bertini's theorem in algebraic geometry states that the intersection of an irreducible variety with a generic hyperplane is another irreducible variety. This theorem has been generalized in various ways, but my group of collaborators was particularly interested in Fuchs, Mantova, and Zannier's Toric Bertini theorem, which establishes a version of Bertini's theorem for toric varieties in characteristics zero. Our group was able to prove a version of this result in characteristic p. Two of my collaborators, Maclagan and Yu used Fuchs, Mantova, and Zannier's result to prove a tropical Bertini theorem and higher connectivity of tropical varities. Our toric Bertini in characteristic p is then used to prove in characteristic p the same results. The preprint is on arXiv 2111.13214
(Equivariant) Resolutions of Ideals Associated to Subspace Arrangements
Conca and Herzog showed that the Castelnuovo-Mumford regularity of the product ideal of a subspace arrangement is equal to the number of subspaces in the arrangement. Derksen and Sidman showed that the Castelnuovo-Mumford regularity of the intersection ideal of the subspace arrangement is at most d. In my thesis I will show that analogous results hold when we replace the polynomial ring with the exterior algebra and work over a field of characteristic 0. I rely on the functoriality of free resolutions and construct a functor from the category of polynomial functors to itself. This functor transforms resolutions of ideals in the polynomial ring to resolutions of ideals in the exterior algebra. These results will appear in a forthcoming article (accepted by the Journal of Commutative Algebra).
Computational Tools and Macaulay2
I have adapted an algorithm of Derksen (Algorithm 4.3.1) for invariants under the diagonal actions of tori to diagonal actions of abelian groups (Chapter 3). Starting at the virtual M2 workshop at Cleveland State 2020, I lead a group project on implementing these (and other algorithms) in the computer algebra system Macaulay 2. Here are the features of our package which has been submitted for review and is available with the current distribution of M2. I am working with a group of undergraduate students to use the package InvariantRings to discover features of invariants of abelian groups. I also collaborate on writing a package for resolutions of length three and I have used this package to explore these resolutions with some collaborators, but this project is still at its preliminary stage.
Undergraduate Projects
Working with the Laboratory of Geometry at Michigan, I have mentored an undergraduate project aimed at computing the chromatic number of flip graphs.