Research

This thesis studies matroids and related objects from combinatorial, geometric, categorial, and algebraic perspectives. We present three projects, all of which study matroid-adjacent structures.

The first of these projects is on Pythagorean hyperplane arrangements, obtained from a real gain graph and a configuration of reference points in affine space. Our work determines the combinatorics of all such arrangements via a derived hyperplane arrangement in the real edge space of the graph. We also prove an extension result for configuration generic arrangements, i.e., those Pythagorean arrangements which have stable intersection pattern under perturbation of the reference points.

The second project is on the category of infinite matroids with strong maps. We prove that every full, minor-closed subcategory thereof is proto-exact. We further investigate finiteness conditions on this category, which results in a functor from infinite matroids to finitary matroids. Finally, we prove that the finitary matroids are precisely the colimits of finite matroids.

The third project is on matroids over hyperfields. Our work in this direction puts the theory of minors on solid ground with proofs of the usual properties of minors. We also enrich this theory with direct sums, including cryptomorphic descriptions thereof in terms of both circuits and Grassmann-Pl\"ucker functions. Finally, we provide obstructions to matroidal theories of vector arrangements over a hyperfield.

The final chapter discusses a variety of open questions and plans for future work.

This is joint work with Jaiung Jun. Theory Appl. Categ., Vol. 38, 2022, No. 11, pp 319-327. PDF.

This paper investigates infinite matroids from a categorical perspective. We define strong maps for infinite matroids and prove that the category of infinite matroids is proto-exact. We also characterize finitary matroids as co-limits of finite matroids. Finally, we show that the finitely presentable objects in this category are precisely the finite matroids.

This is joint work with Jaiung Jun and Matt Szczesny. Math. Z. 296 (2020), no. 1-2, 147–167. DOI 10.1007/s00209-019-02429-z.

This paper examines the category \( \mathbf{Mat}_\bullet \) of pointed matroids with pointed strong maps. We prove that this is a finitary proto-exact category with a purely matroid-theoretic argument. We next examine the \( K \)-theory obtained via the Waldhausen construction, and show that this is nontrivial via injections from the stable homotopy groups of spheres. Finally we show that Schmitt's matroid minor Hopf algebra is dual to the Hall algebra of \( \mathbf{Mat}_\bullet \).

This is joint work with Jaiung Jun and Matt Szczesny. J. Algebra 556 (2020), 806–835. DOI 10.1016/j.jalgebra.2020.02.042.

This paper investigates matroids over hyperfields with the goal of generalising Schmitt's matroid-minor Hopf algebra. We define direct sums of matroids over hyperfields for this purpose and give two cryptomorphic descriptions. Furthermore, we show that the usual properties one expects from minors of matroids hold also over hyperfields.

Submitted (2023).

This paper studies Pythagorean hyperplane arrangements, originally defined by Zaslavsky. A Pythagorean hyperplane arrangement is an affine hyperplane arrangement constructed from two pieces of data: (1) a configuration of reference points in affine space, and (2) an additive real gain graph, i.e., a graph with a real number \( g(e) \) associated to each oriented edge \( e \), satisfying \( g(\bar{e}) = - g(e) \). Zaslavsky studied the so-called generic Pythagorean arrangements, i.e., those whose combinatorial type is stable under perturbation of the reference points.

This paper studies a new notion of genericity, which I've dubbed gain genericity, wherein we require that the arrangement is stable under perturbation of the gains in our gain graph. As a natural consequence of my investigation, I am able to completely determine the combinatorics of any Pythagorean arrangement via two derived structures: the first is a matroid constructed from the data of the arrangement, and the second is an auxiliary hyperplane arrangement. The main theorem establishes the correspondence between these structures and their relationship with (failures of) gain genericity.

This is the first part of a series on such arrangements. It establishes the fundamentals of the theory on these arrangements, setting up the future results. Numerous examples and applications are also detailed.

Submitted (2022).

This paper grew out of my desire to create a theory of vector arrangements for matroids over hyperfields. I show that several generalisations of the usual notions of linear span and linear independence are unsatisfactory over hyperfields.

My main examples are over the Krasner hyperfield (i.e., the set \( \mathbb{K} = \{0, 1\} \) with the usual multiplication and hyperaddition \( 0 \oplus x = \{x\} \) and \( x \oplus x = \{0, x\} \)) and the hyperfield of signs (i.e., the set \( \mathbb{S} = \{0, 1, -1\} \) with the usual multiplication and hyperaddition \( 0 \oplus x = \{x\} = x \oplus x \) and \( x \oplus (-x) = \{0, x, -x\} \)). I exhibit a single set of vectors in \( \mathbb{K}^3 \) which foils each notion of span and independence, i.e., they fail to yield a matroid via these notions of independence and span. I then prove that every \( \mathbb{S} \)-module is either \( \mathbb{S} \) itself or infinite; thus \( \mathbb{S}^r \) is an \( \mathbb{S} \)-module if and only if \( r = 1 \).