Papers & Preprints
Papers & Preprints
Proto-Exact Categories of Matroids over Idylls and Tropical Toric Reflexive Sheaves (with Jaiung Jun & Alex Sistko). Preprint.
Torsor Structures on Spanning Trees (with Farbod Shokrieh). SIAM Journal on Discrete Mathematics, Volume 37, No. 3, Pages 2126-2147, 2023.
Monotone Catenary Degree in Numerical Monoids (with Daniel Gonzalez and Jenna Zomback). Preprint.
Snapshots from Non-Archimedean Geometry. 1-2-3 Seminar at the University of Washington. 2025.
Abstract: Complex analytic geometry is among the most successful fusions of analysis and algebra in mathematics, and much of this is reliant on the fact that the usual absolute value on $$\mathbb{C}$$ is highly unique. But what happens when we try to do analytic geometry with other absolute values or other fields? In the (more-typical) non-archimedean situation, the metric spaces defined on rational points are particularly badly-behaved, and we are in need of a "fix" to have a coherent notion of analytic geometry. We survey some of these pathological properties, describe how they can be remedied using inspiration from functional analysis, and see how this remedy affords us powerful analytic techniques familiar from the complex setting.
Convex Geometry and Compactified Jacobians. Western Algebraic Geometry Symposium (University of British Columbia). Spring 2025.
Abstract: For a smooth projective curve X, the Jacobian is an abelian variety which parametrizes line bundles on X. In the situation where X contains nodal singularities, the same construction produces a Jacobian which is no longer projective. The problem of compactifying these Jacobians has a long history, going back to the work of Mumford. Oda and Seshadri gave a class of well-behaved compactifications using polyhedral geometry and GIT. We revisit this construction from a combinatorial perspective, illustrating that their compactifications are equivalently described as coherent subdivisions of a zonotope associated to X. We enumerate the faces of these subdivisions and obtain a formula for the class of an Oda-Seshadri compactified Jacobian in the Grothendieck ring of varieties.
Matroids and their Combinatorial Geometry. 1-2-3 Seminar at the University of Washington. 2025.
Abstract: As any matroid theorist will enthusiastically tell you, matroids exhibit many strong and subtle relationships with other areas of mathematics. The past two decades have brought a resurgence of interest in matroid theory, with a crowning jewel being the positive solution of Mason's conjecture in 2018 (by two independent groups). To the surprise of many, one proof of this result uses deep tools from algebraic geometry. We provide an introduction to the theory of matroids through graph theory and polyhedral geometry before building toward some deep connections with algebraic geometry which were used in the proof. In particular, we will see that modern matroid theory benefits from relations with the geometry of toric varieties, intersection theory, and Hodge theory.
Hyperplanes, Posets, and Cohomology. 1-2-3 Seminar at the University of Washington. 2024.
Abstract: In this talk we examine some aspects of the modern theory of hyperplane arrangements, a theory which stars an interesting interplay between combinatorics and topology. Over the course of the talk, we will familiarize ourselves with two main characters associated with any arrangement: the intersection poset and the complex complement. The former is a combinatorial object associated to the arrangement and the latter is an interesting topological space given by taking the complement of the arrangement in a complex vector space. Over the reals, the complement of an arrangement consists of finitely many contractible polyhedra; over the complex numbers, the complement of an arrangement is connected in general and has nontrivial topology. We shall see that several interesting topological invariants of the complex complement are determined by combinatorial invariants associated to the arrangement.
Torsor Structures on Spanning Trees. Graduate Online Combinatorics Colloquium. 2021. (slides, video)
Abstract: The classical Kirchoff matrix-tree theorem states that the number of spanning trees of a finite connected graph is equal to the determinant of the graph's reduced Laplacian matrix. This result actually evinces the equality of the number of spanning trees of a graph and the cardinality of a finite group, referred to alternately as the sandpile group, Jacobian group, or Picard group of the graph. Further, one can endow the graph with a ribbon structure, an ordering of the edges around each vertex, in order to turn the set of spanning trees into a torsor for the Picard group. We present two such torsor structures in particular, survey some results comparing the two, and discuss a conjecture of Baker and Wang on these structures in nonplanar ribbon graphs.
Basic Ergodic Theory. Series of short expository works on ergodic theory.