Research and Scholarly Works

My research is broadly in the realm of computational algebraic geometry, with particular focus on tropical and non-Archimedean geometry.  Here are my preprints and publications, with some pictures and brief descriptions.  Authors who were undergrads when the research was undertaken are listed in bold.


(24) “Multiplicity-free gonality  on graphs”, with F. Dean and M. Everett.  (arXiv)

The gonality of a graph measures how few chips we need so that we can move the chips anywhere using chip-firing moves.  We introduce multiplicity-free gonality, which asks how few chips we need if they’re all placed on different vertices.

(23) “On the scramble number of graphs”, with M. Echavarria, M. Everett, R. Huang, L. Jacoby, and B. Weber. (arXiv)

The scramble number of a graph is a new invariant that helps us study chip-firing games.  We prove lots of results about scramble number, including that it’s NP-hard to compute!  (This work was part of the 2020 SMALL REU.)

(22) “Iterated and mixed discriminants”, with A. Dickenstein and S. di Rocco.  (arXiv)

Discriminants tell us when polynomials have solutions of higher multiplicity. In this paper we study when certain discriminants can be computing by taking discriminants of other discriminants.

(21) “Tropically planar graphs”, with D. Coles, N. Dutta, S. Jiang, and A. Scharf. (arXiv; supplemental material)

Tropically planar graphs appear in smooth tropical plane curves, and are dual to triangulations of polygons.  We compute how many graphs are tropically planar up to genus 7, and give upper and lower bounds on their number in general. (This work was part of the 2017 SMALL REU.)


(20) “Moduli dimensions of lattice polygons”, with M. Echavarria, M. EverettS. HuangL. Jacoby, A. Tewari, R. Vlad, and B. Weber, to appear in The Journal of Algebraic Combinatorics. (arXiv)

Given a lattice polygon, we can ask: how many degrees of freedom d are there in constructing a tropical (or algebraic!) curve with that Newton polygon?  We determine all possible values of d for lattice polygons.  (This work was part of the 2020 SMALL REU.)

(19) “The moduli space of tropical curves with fixed Newton polygon”, with D. Coles, N. Dutta, S. Jiang, and A. Scharf, to appear in Advances in Geometry. (arXiv)

Tropical plane curves contain graphs with lengths assigned to their edges.  We show how to find the number of degrees of freedom we have in choosing those lengths, either for a single triangulation or for a whole  polygon.  (This work was part of the 2017 SMALL REU.)

(18) “Gonality sequences of graphs”, with I. Aidun, F. Dean, T. Yu, and J. Yuan, SIAM Journal on Discrete Mathematics 35 (2021), no. 2, 814–839.  (arXiv)

For any graph, we can study an infinite family of chip-firing games on that graph called the gonality games. The gonality sequence of a graph is a list of numbers that measure how hard it is to win those games.  We determine what gonality sequences look like for small graphs, how gonality sequences can start, and how to compute them. (This work was part of the 2018 SMALL REU.)

(17) “Prism graphs in tropical plane curves”, with L. Jacoby and B. Weber, Involve, a Journal in Mathematics,14-3 (2021), 495–510. (arXiv)

The tropical plane curve pictured above has a prism graph with 11 bounded regions.   We prove that there exists no larger prism in a tropical plane curve.  (This work was part of the 2020 SMALL REU.)

(16) “On the gonality of Cartesian products of graphs”, with I. Aidun, Electronic Journal of Combinatorics 27 (2020), no. 4, Paper No. 4.52, 35 pp.  (arXiv)

If you know how to win a chip-firing game on two graphs G and H, then there’s a strategy to win on their product.  We study when this strategy is optimal, and when there are even better ones.  (This work was part of the 2018 SMALL REU.)

(15) “Higher-distance commuting varieties”, with M. Elyze, A. Guterman, and K. Sivic, to appear in Linear and Multilinear Algebra.  (arXiv)


Two matrices A and B commute if AB=BA.  If two matrices don’t commute, they still might commute with a common non-scalar matrix.  We show that this condition is determined by polynomial equations.  (This includes thesis work by Madeleine Elyze, Williams ’18.)

(14) “Convex lattice polygons with all lattice points visible”, with A. K. Tewari, to appear in Discrete Mathematics.  (arXiv)

The three polygons above have a circled grid point that can “see” all the other grid points in the polygon.  What can such a polygon look like? We provide a complete answer, and give some cool applications to tropical geometry.

(13)  “Tropical Geometry” (2020). In P. Harris, E. Inkso, & A. Wootton (eds.), A Project-Based Guide to Undergraduate Research in Mathematics: Starting and Sustaining Accessible Undergraduate Research (pp. 63-105).  Birkhäuser Basel. (arXiv)

This chapter contains a quick introduction to tropical geometry, and then lots of open problems accessible to undergrads.  Also check out the other chapters from the book!

(12) “Nullstellenfont,” with B. Logsdon and A. Michaelsen, Math Horizons, 27:4, 5-7 (2020).

An algebraic plane curve is defined by a polynomial in two variables.  Given any string of text, we can give you a polynomial whose corresponding curve looks like that string.  Try it out on this website!

(11) “Tropical hyperelliptic curves in the plane,” to appear in The Journal of Algebraic Combinatorics.  (arXiv)

Tropical hyperelliptic curves are graphs that are built out of two copies of a tree glued together.  We show which of these graphs can appear in smooth tropical plane curves.

(10) “Treewidth and gonality of glued grid graphs”, with I. Aidun, F. Dean, T. Yu, and J. Yuan, to appear in Discrete Applied Mathematics.  (arXiv)

Treewidth measures how close a graph is to being a tree.  We find the treewidth for grid graphs glued along their boundaries, and show how to win chip-firing games on them.  (This work was part of the 2018 SMALL REU.)

(9) “Graphs of gonality three”, with I. Aidun, F. Dean, T. Yu, and J. Yuan, Algebraic Combinatorics, Volume 2 (2019) no. 6 p. 1197-1217. (arXiv)

The chip-firing game on a graph lets you place a collection of chips on a graph, which you then move around to eliminate debt placed by an opponent.  In this paper, we study the graphs where you can win the game by starting with just three chips.  (This work was part of the 2018 SMALL REU.)

(8) “The smallest art gallery not guarded by every third vertex”, Geombinatorics 29 (2019), no. 1, 24-32.  (arXiv)

Any polygonal art gallery can be guarded by placing guards on a third of its vertices. But we can’t always place a guard on every third vertex; in this paper, we find the smallest possible example that shows this.

(7) “The tropical commuting variety”, with N. M. Tran, Linear Algebra Appl. 507 (2016), 300–321.  (arXiv)


Two matrices A and B commute if AB=BA.  We study what it means for two matrices to commute tropically, which means that addition is replaced with taking a minimum, and multiplication is replaced with addition.

(6) “Bitangents of tropical plane quartic curves”, with M. Baker, Y. Len, N. Pflueger, and Q. Ren, Mathematische Zeitschrift, 282 (2016), no. 3-4, 1017–1031. (arXivclassical_bitangentstropical_bitangents

Classically, a smooth plane quartic curve has 28 bitangent lines.  We prove that tropically, such a curve has 7 bitangent lines.

(5) “Moduli of tropical plane curves”, with S. Brodsky, M. Joswig, and B. Sturmfels, Research in the Mathematical Sciences, Vol. 2 (2015), 2:4.  (arXiv)


Split a polygon up into triangles, and draw the dual graph.  Which graphs can we see?  We answer this question for polygons with at most 5 lattice points in their interior.

(4) “Tropical images of intersection points”, Collectanea Mathematica, Vol. 66 (2015), Issue 2, 273-283.  (arXiv)tropical_intersection rational_function

Classical intersection points map to tropical intersection points.  I give a partial answer to the reverse questions:  which tropical intersection points can be lifted to classical intersection points?

(3) “Algorithms for Mumford curves”, with Q. Ren, Journal of Symbolic Computation, Vol. 68 (2015), 259-284.  (arXiv)


If we let certain groups act on a field, the quotient can be interpreted as an algebraic curve.  We develop algorithms for studying such curves over the field of p-adic numbers.

(2) “An elliptic curve family test of the L-functions Ratios Conjecture:, with D. Huynh and S. Miller, Journal of Number Theory, Vol. 131 (2011), 1117-1147.  (arXiv)

When studying the zeros of L-functions, it’s helpful to approximate their behavior.  We show that for a family of these functions, a certain approximation is very accurate. (This is based in part on my undergraduate thesis at Williams College with Steve Miller.)

(1) “The spiral index of knots”, with C. Adams, W. George, R. Hudson, L. Starkston, S. Taylor, and O. Turanova, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 149 (2010), Issue 2, 297-315.  (arXivPDF)


We study spiral projections of knots, which are pictures of knots without any inflection points. (This work came out of Colin Adams’ Knot Theory group in the 2008 SMALL REU.)

Other scholarly works

My dissertation is titled “Tropical and non-Archimedean curves”, written at UC Berkeley with Bernd Sturmfels as my advisor.  You can find a PDF of it here.  It includes material from publications (3) through (7) listed above.