Research Statement

Broadly speaking, my research studies relationships between combinatorics, algebra, topology, and geometry. These fields come together in the theory of matroids; a matroid is a combinatorial abstraction of linear dependence among vectors in a vector space. The ubiquity of linear algebra means that matroids are naturally connected to many other areas of mathematics. This rich interaction has lead me to a variety of projects, outlined below, and has made me eager to continue learning and researching mathematics.

My first projects were related to generalised notions of linear dependence; papers in this area investigate these dependences where we replace the base field by a hyperfield, i.e., a field with multi-valued addition. Baker and Bowler cite:BB19, building on work of Dress and Wenzel, proved that matroids over hyperfields provide a unifying framework for many existing theories of matroids with additional structure, including oriented matroids and valuated matroids. Oriented matroids abstract linear dependence over a totally ordered field and have deep connections to combinatorial topology, whereas valuated matroids abstract from fields with valuation; the latter are intimately related to tropical and algebraic geometry. Jun, Szczesny, and I cite:EJS20b extended Schmitt's matroid-minor Hopf algebra to matroids over hyperfields; our work defined the direct sum of matroids over hyperfields and investigated properties of their minors.

Matroids and oriented matroids have many equivalent axiomatisations, but there are few known for the generalisation over hyperfields. Constructing such axiomatics is a major open problem in this area. One positive result in this direction comes from my co-advisor Laura Anderson, who generalised vector axioms from oriented matroids to matroids over hyperfields in cite:And19. In cite:Epp22b I prove that every vector space over the hyperfield of signs—a three-element hyperfield generalising arithmetic over a linearly-ordered field—is infinite, so the notion of vector space dimension over simple hyperfields cannot be defined in the usual way. I further show that independence and span do not reasonably generalise to hyperfields without serious (as yet undiscovered) modifications.

In an ongoing project I investigate matroids over the triangle hyperfield, a hyperfield derived from the metric triangle inequality. I have proved that their vectors form a convex polyhedral cone, and have constructed examples for which the combinatorial type of this cone is not determined solely by the underlying matroid; in particular, the hyperfield coefficients can influence the combinatorial type of this cone. Furthermore, I modified an argument of Viro to show that the set of lengths in any sufficiently symmetric metric space form a hyperfield similar to the triangle hyperfield. Similar ideas can be used to build new classes of hyperfield with geometric meaning; I intend to continue studying this class of hyperfields and their matroids.

The next portion of my research concerns categories of matroids. Such categories have seen new interest in recent years, e.g., see cite:EH20. I am most interested in strong maps of matroids, which generalise the fact that preimages of subspaces under linear maps are again subspaces. Jun, Szczesny and I proved in cite:EJS20a that the category of matroids with strong maps has the structure of a proto-exact category, and thus has a \( K \)-theory. We then proved that the Grothendieck group of this category is isomorphic to \( \mathbb{Z}^2 \) and that Schmitt's matroid-minor Hopf algebra is dual to the Hall algebra of this category. Jun and I later defined and explored strong maps for infinite matroids in cite:EJ21, and proved that the resulting category is proto-exact as well. Moreover, we explored finiteness conditions in this category, characterising the finitary matroids as colimits of finite matroids; we further showed that the finitely presentable objects of this category are the finite matroids. We hope to extend these results to matroids over hyperfields, a project which requires a presently non-existent theory of strong maps for such matroids.

My most recent research explores Pythagorean hyperplane arrangements. A hyperplane is a codimension-one subspace of affine space, and a hyperplane arrangement is a finite collection of these hyperplanes. The arrangements I consider are defined using a gain graph, i.e., a graph where each edge orientation is labelled with a group element where reverse orientations having inverse labels; gain graphs generalise the signed graphs from cite:Zas82. Given a collection of reference points and a gain graph, we construct a hyperplane arrangement with one hyperplane for each edge of the gain graph. We typically use the additive group of real numbers as our gain group—that is, we consider real-weighted graphs where reversing the direction of an edge negates its weight. My co-advisor Thomas Zaslavsky initiated the study of Pythagorean arrangements in cite:Zas02; he focused on generic arrangements, i.e. those for which perturbation of the reference points does not change the arrangement's combinatorial type. His work obtained face-counting formulas for generic Pythagorean arrangements.

My thesis cite:Epp22a and subsequent results cite:Epp23 make two major advances in this area. First, given only the reference points and underlying graph, I construct the lattice of all combinatorial types of Pythagorean arrangements with this data, and I proved that this lattice is isomorphic to the lattice of flats of a derived hyperplane arrangement. In doing so my work gives a recipe for face-counting formulas for all Pythagorean arrangements, both generic and nongeneric, via an associated algebraic structure. Secondly, I proved that every generic arrangement has an explicit locus constructed from spheres and affine hyperplanes so that adding a new reference point preserves genericity if and only if the new point is not in this locus. The spheres in the collection correspond naturally with forests of the gain graph, and the combinatorial type of the new hyperplane arrangement (after adding the new point) is determined by its location with respect to the locus of spheres and hyperplanes. This ongoing work has a myriad of possible future directions with connections to matroid theory and metric and projective geometry. I plan to abstract techniques pioneered in cite:Zas75 to give explicit face-counting formulas via both methods described here.

As detailed above, my research centers mostly on combinatorics, graph theory, and matroids. That being said, I have broad mathematical interests, and have worked on a variety of projects related to combinatorics, algebra, and geometry in their relationships with these topics. I hope to continue contributing to and learning from the mathematics community by way of future collaborations and research in mathematics at large.

bibliography:sources.bib