I'm currently a visiting scholar at the University of California, Berkeley, where I'm working on problems in algebraic geometry and representation theory with a combinatorial flavor. The two things I'm spending most of my time thinking about right now are:
GL-equivariant Boij-Söderberg theory: Given a finitely-generated graded module over a polynomial ring, we can construct its "Betti Table" by taking a minimal free resolution and counting the number of generators in each degree at each step of the resolution. There's a beautiful story that's developed relatively recently about classifying all possible such Betti tables up to a rational multiple involving certain pairings with vector bundles on projective space. Together with Jake Levinson, a graduate student at Michigan, I'm working on a way to generalize this story to modules that are equivariant under an action of GL(V), where there should be a corresponding pairing involving vector bundles on the Grassmannian.
A preprint, written with Jake Levinson and Steven Sam, describing the first part of this story is now available on the arXiv.
- Maximal green sequences for positroid varieties: Together with Khrystyna Serhiyenko, a postdoc at Berkeley, I'm working on a method for constructing maximal green sequences for the conjectured cluster algebra structure on a positroid variety, generalizing some work of Marsh and Scott on the Grassmannian itself in this paper.
As a graduate student at the University of Michigan, I studied combinatorial algebraic geometry under David Speyer. At Michigan, my work was focused on the geometry and combinatorics of matroid varieties, the subvarieties of the Grassmannian defined by the vanishing and nonvanishing of Plücker coordinates.
There are two documents on the arXiv from the work that I did at Michigan:
- The first is a paper, which was published in the February 2015 issue of the Journal of Algebraic Combinatorics (JACO). It's about a method for estimating the codimension of a matroid variety using only the combinatorics of the matroid itself, that is, the list of Plücker coordinates which are allowed to vanish. While problems even simpler than this are known to be intractable, I show that for a particular class of matroid varieties, called positroid varieties, my procedure always produces the actual codimension.
- The second is my thesis. In addition to a version of the above paper, it also has a different result about positroid varieties. There is a very geometrically well-motivated method (not due to me) of finding the cohomology class of certain positroid varieties, called interval rank varieties, that involves repeatedly deforming them within the Grassmannian until one is left with a collection of Schubert varieties, which generate the cohomology ring. I show that this method extends to all positroid varieties on rank-3 Grassmannians, but that it cannot be extended any further.