{"id":5,"date":"2016-07-16T08:01:49","date_gmt":"2016-07-16T12:01:49","guid":{"rendered":"http:\/\/sites.williams.edu\/10rem\/?page_id=5"},"modified":"2026-06-23T09:58:56","modified_gmt":"2026-06-23T13:58:56","slug":"research","status":"publish","type":"page","link":"https:\/\/sites.williams.edu\/10rem\/research\/","title":{"rendered":"Research and Scholarly Works"},"content":{"rendered":"

My research is broadly in the realm of tropical geometry, with particular emphasis on tropical curves and chip-firing games on graphs. \u00a0Here are my preprints and publications, with some pictures and brief descriptions.\u00a0 Authors who were undergrads when the research was undertaken are listed in bold.<\/p>\n

Preprints<\/strong><\/p>\n

(36) “The\u00a0d<\/em>-gonal locus in the moduli space of tropical plane curves”, with Desmond Leitz<\/strong>, S\u00f8ren Newman-Taylor<\/strong>, and Vincent X. Wang<\/strong>. (arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

We group tropical curves of a fixed genus together into moduli space two ways:\u00a0 one by fixing their gonality (the number of chips we need to chip-fire anywhere on the curve), and one by fixing the width of their Newton polygon.\u00a0 We believe these two spaces are identical, and prove that as long as the genus is big enough compared to the width of the polygons, the two spaces have the same dimension. (This work was part of the 2025 SMALL REU.)<\/em><\/p>\n

(35) “The gonality of circulant graphs”, with Lisa Cenek<\/strong>, Lizzie Ferguson<\/strong>, Eyobel Gebre<\/strong>, Cassandra Marcussen<\/strong>, Jason Meintjes<\/strong>, Liz Ostermeyer<\/strong>, and Shefali Ramakrishna<\/strong>. (arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

A circulant graph has vertices arranged in a circle with a consistent pattern of connections between nearby vertices.\u00a0 In this paper we play chip-firing games on these graphs, and find an upper bound on how many chips need to move them anywhere in the circle.\u00a0 Remarkably, our upper bound depends only on the pattern of connections, and not on how many vertices there are! (This work was part of the 2021 SMALL REU.)<\/em><\/p>\n

 <\/p>\n

(34) “The gonality of chess graphs” with Nila Cibu<\/strong>, Kexin Ding<\/strong>, Steven DiSilvio<\/strong>, Sasha Kononova<\/strong>, Chan Lee<\/strong>, and Krish Singal<\/strong>. (arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

A chess graph encodes the ways that a chess piece can move around on a rectangular chess board.\u00a0 We study chip-firing games on these graphs, and find upper and lower bounds on the gonality for king’s, bishop’s, and knight’s graphs.\u00a0 (This work was part of the 2023 SMALL REU.)<\/em><\/p>\n

(33) “On the size and complexity of scrambles” with Seamus Connor<\/strong>, Steven DiSilvio<\/strong>, Sasha Kononova<\/strong>, and Krish Singal<\/strong>.\u00a0 (arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

A scramble of a graph is a collection of connected subgraphs (called “eggs”), and we measure the order of scramble by how hard it is to hit every egg, and how hard it is to separate some pair of eggs.\u00a0 We show that in order to build a scramble on maximum order, sometimes we need exponentially many eggs!\u00a0 (This work was part of the 2023 SMALL REU.)<\/em><\/p>\n

(32) “Computing higher graph gonality is hard”, with Lucas Tolley<\/strong>. (arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

The r-th gonality of a graph is the minimum number of chips we need so that we can place r of them wherever we would like, using chip-firing moves.\u00a0 We prove that it’s hard (more precisely, NP-hard) to compute r-th gonality for a graph, and that the same is true for many related problems.<\/em><\/p>\n

(31) “Bounds on higher graph gonality”, with Lisa Cenek<\/strong>, Lizzie Ferguson<\/strong>, Eyobel Gebre<\/strong>, Cassandra Marcussen<\/strong>, Jason Meintjes<\/strong>, Liz Ostermeyer<\/strong>, and Shefali Ramakrishna<\/strong>. (arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

The r-th gonality of a graph is the minimum number of chips we need so that we can place r of them wherever we would like, using chip-firing moves.\u00a0 We prove some new upper and lower bounds on r-th gonality. (This work was part of the 2021 SMALL REU.)<\/em><\/p>\n

Publications<\/strong><\/p>\n

(30) “Commuting graphs of p<\/em>-adic matrices”, to appear in Linear Algebra and its Applications. (arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

Build a graph whose vertices are (non-scalar) matrices with entries from the p-adic number, with two matrices joined by an edge if they commute under matrix multiplication.\u00a0 We prove precisely when these graphs are connected, and show that when connected the possible diameters are 4, 5, and 6, with all achieved in certain cases!<\/em><\/p>\n

(29) “Scramble number and tree-cut decompositions” with Lisa Cenek<\/strong>, Lizzie Ferguson<\/strong>, Eyobel Gebre<\/strong>, Cassandra Marcussen<\/strong>, Jason Meintjes<\/strong>, Liz Ostermeyer<\/strong>, Shefali Ramakrishna,\u00a0<\/strong>and\u00a0Ben Weber<\/strong>, to appear in The Art of Discrete and Applied Mathematics.\u00a0(arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

A tree-cut decomposition takes a graph and puts it inside of a tree.\u00a0 We introduce a new invariant, called the screewidth of a graph, which measures how “full” the tree needs to be.\u00a0 We relate this new invariant to scramble number and to graph gonality. (This work was part of the 2021 SMALL REU.)<\/em><\/p>\n

(28) “Chip-firing on the Platonic solids: a primer for studying graph gonality”, with Marchelle Beougher<\/strong>, Kexin Ding<\/strong>, Max Everett<\/strong>, Robin Huang<\/strong>, Chan Lee<\/strong>, and Ben Weber<\/strong>, to appear in Involve. (arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

This is a friendly and accessible introduction to studying chip-firing games and graph gonality.\u00a0 We use graphs coming from the five Platonic solids to demonstrate theorems, techniques, and open problems.<\/em><\/p>\n

(27) “The gonality of queen’s graphs”, with Noah Speeter, to appear in Discrete Mathematics.\u00a0 (arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

A queen’s graph encodes the ways that a queen can move around on a rectangular chess board.\u00a0 We study chip-firing games on these graphs, and show that finding divisors of positive rank and minimum degree is intimately related to finding maximum collections of non-attacking queens.<\/em><\/p>\n

(26) “Uniform scrambles on graphs”, with Lisa Cenek<\/strong>, Lizzie Ferguson<\/strong>, Eyobel Gebre<\/strong>, Cassandra Marcussen<\/strong>, Jason Meintjes<\/strong>, Liz Ostermeyer<\/strong>, and Shefali Ramakrishna<\/strong>.\u00a0 Australasian Journal of Combinatorics 87 (2023), 129–147.\u00a0(arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

A scramble is a collection of connected subgraphs of a graph.\u00a0 We study scrambles where every subgraph has the same number of vertices, and use that to study chip-firing games.\u00a0 (This work was part of the 2021 SMALL REU.)<\/em><\/p>\n

(25) “Multiplicity-free gonality\u00a0 on graphs”, with Franny Dean<\/strong> and Max Everett<\/strong>. The Electronic Journal of Graph Theory and Applications 11\u00a0<\/strong>(2023), no. 2, 357–380.\u00a0 (arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

The gonality of a graph measures how few chips we need so that we can move the chips anywhere using chip-firing moves.\u00a0 We introduce multiplicity-free gonality, which asks how few chips we need if they’re all placed on different vertices.<\/em><\/p>\n

(24) “Graphs of scramble number two” with Robin Eagleton<\/strong>. Discrete Mathematics,
\nVolume 346, Issue 10, 2023. (arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

The scramble number of a graph is a new invariant that helps us study chip-firing games.\u00a0 In this paper, we classify all graphs of scramble number two.<\/em><\/p>\n

(23) “Iterated and mixed discriminants”, with Alicia Dickenstein and Sandra di Rocco. J. Comb. Algebra 7 (2023), no. 1, 45–81. \u00a0(arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

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.<\/em><\/p>\n

(22) “Tropically planar graphs”, with Desmond Coles<\/strong>, Neelav Dutta<\/strong>, Sifan Jiang<\/strong>, and Andrew Scharf<\/strong>, Collect. Math. 74 (2023), no. 1, 27–60\u00a0(arXiv<\/a>; supplemental material<\/a>)<\/p>\n

\"\"<\/p>\n

Tropically planar graphs appear in smooth tropical plane curves, and are dual to triangulations of polygons.\u00a0 We compute how many graphs are tropically planar up to genus 7, and give upper and lower bounds on their number in general.\u00a0(This work was part of the 2017 SMALL REU.)<\/em><\/p>\n

(21) “On the scramble number of graphs”, with Marino Echavarria<\/strong>, Max Everett<\/strong>, Robin Huang<\/strong>, Liza Jacoby<\/strong>, and Ben Weber<\/strong>, Discrete Appl. Math.<\/em>\u00a0310 (2022), 43–59.\u00a0(arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

The scramble number of a graph is a new invariant that helps us study chip-firing games.\u00a0 We prove lots of results about scramble number, including that it’s NP-hard to compute!\u00a0 (This work was part of the 2020 SMALL REU.)<\/em><\/p>\n

(20) “Moduli dimensions of lattice polygons”, with Marino Echavarria<\/strong>, Max Everett<\/strong>, Robin Huang<\/strong>,\u00a0Liza Jacoby<\/strong>, Ayush K. Tewari, Raluca Vlad<\/strong>, and Ben Weber<\/strong>, The Journal of Algebraic Combinatorics 55\u00a0<\/strong>(2022), no. 2, 559-589. (arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

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?\u00a0 We determine all possible values of d for lattice polygons.\u00a0 (This work was part of the 2020 SMALL REU.)<\/em><\/p>\n

(19) “The moduli space of tropical curves with fixed Newton polygon”, with Desmond Coles<\/strong>, Neelav Dutta<\/strong>, Sifan Jiang<\/strong>, and Andrew Scharf<\/strong>, Advances in Geometry 22 (2022), no. 1, 49-68. (arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

Tropical plane curves contain graphs with lengths assigned to their edges.\u00a0 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\u00a0 polygon.\u00a0 (This work was part of the 2017 SMALL REU.)<\/em><\/p>\n

(18) “Gonality sequences of graphs”, with Ivan Aidun<\/strong>, Franny Dean<\/strong>, Teresa Yu<\/strong>, and Julie Yuan<\/strong>, SIAM Journal on Discrete Mathematics 35\u00a0<\/strong>(2021), no. 2, 814–839.\u00a0 (arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

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.\u00a0 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.)<\/em><\/p>\n

(17) “Prism graphs in tropical plane curves”, with Liza Jacoby <\/strong>and\u00a0Ben Weber<\/strong>, Involve, a Journal in Mathematics,14-3 (2021), 495–510. (arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

The tropical plane curve pictured above has a prism graph with 11 bounded regions.\u00a0 \u00a0We prove that there exists no larger prism in a tropical plane curve.\u00a0 (This work was part of the 2020 SMALL REU.)<\/em><\/p>\n

(16) “On the gonality of Cartesian products of graphs”, with Ivan Aidun<\/strong>, Electronic Journal of Combinatorics 27 (2020), no. 4, Paper No. 4.52, 35 pp. \u00a0(arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

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.\u00a0 We study when this strategy is optimal, and when there are even better ones.\u00a0 (This work was part of the 2018 SMALL REU.)<\/em><\/p>\n

(15) “Higher-distance commuting varieties”, with Madeleine Elyze,<\/strong> Alexander Guterman, and Klemen Sivic, Linear and Multilinear Algebra 70\u00a0<\/strong>(2022), no. 17, 3248–3270.\u00a0 (arXiv<\/a>)<\/p>\n

 <\/p>\n

\"\"<\/p>\n

Two matrices A and B commute if AB=BA.\u00a0 If two matrices don’t commute, they still might commute with a common non-scalar matrix.\u00a0 We show that this condition is determined by polynomial equations.\u00a0 (This includes thesis work by Madeleine Elyze, Williams ’18.)<\/em><\/p>\n

(14) “Convex lattice polygons with all lattice points visible”, with Ayush K. Tewari, Discrete Mathematics<\/em>\u00a0344\u00a0<\/strong>(2021), no. 1, Paper No. 112161, 19 pp.\u00a0 (arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

The three polygons above have a circled grid point that can “see” all the other grid points in the polygon.\u00a0 What can such a polygon look like? We provide a complete answer, and give some cool applications to tropical geometry.<\/em><\/p>\n

(13)\u00a0 “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 <\/em>(pp. 63-105).\u00a0 Birkh\u00e4user Basel. (arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

This chapter contains a quick introduction to tropical geometry, and then lots of open problems accessible to undergrads.\u00a0 Also check out the other chapters from the book<\/a>!<\/em><\/p>\n

(12) “Nullstellenfont,” with\u00a0Ben Logsdon <\/strong>and\u00a0Anya Michaelsen<\/strong>, Math Horizons,<\/span>\u00a027:4,<\/span>\u00a05-7 (2020).<\/span><\/p>\n

\"\"<\/p>\n

An algebraic plane curve is defined by a polynomial in two variables.\u00a0 Given any string of text, we can give you a polynomial whose corresponding curve looks like that string.\u00a0 Try it out on this website!<\/a><\/em><\/p>\n

(11) “Tropical hyperelliptic curves in the plane,” The Journal of Algebraic Combinatorics<\/i>\u00a053\u00a0<\/strong>(2021), no. 2, 369-388.<\/em>\u00a0 (arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

Tropical hyperelliptic curves are graphs that are built out of two copies of a tree glued together.\u00a0 We show which of these graphs can appear in smooth tropical plane curves.<\/em><\/p>\n

(10) “Treewidth and gonality of glued grid graphs”, with Ivan Aidun<\/strong>, Franny Dean<\/strong>, Teresa Yu<\/strong>, and Julie Yuan<\/strong>, Discrete Applied Mathematics 279 (2020), 1-11.\u00a0\u00a0(arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

Treewidth measures how close a graph is to being a tree.\u00a0 We find the treewidth for grid graphs glued along their boundaries, and show how to win chip-firing games on them.\u00a0 (This work was part of the 2018 SMALL REU.)<\/em><\/p>\n

(9)\u00a0“Graphs of gonality three”, with\u00a0Ivan Aidun<\/strong>, Franny Dean<\/strong>, Teresa Yu<\/strong>, and Julie Yuan<\/strong>, Algebraic Combinatorics, Volume 2 (2019) no. 6 p. 1197-1217. (arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

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.\u00a0 In this paper, we study the graphs where you can win the game by starting with just three chips.\u00a0\u00a0(This work was part of the 2018 SMALL REU.)<\/em><\/p>\n

(8) “The smallest art gallery not guarded by every third vertex”, Geombinatorics 29 (2019), no. 1, 24-32.\u00a0 (arXiv<\/a>)<\/p>\n

\"\"<\/p>\n

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.<\/em><\/p>\n

(7) “The tropical commuting variety”, with Ngoc M. Tran, Linear Algebra Appl. 507 (2016), 300–321. \u00a0(arXiv<\/a>)<\/p>\n

\"two_dim_fan\"<\/p>\n

Two matrices A and B commute if AB=BA. \u00a0We study what it means for two matrices to commute tropically, which means\u00a0that addition is replaced with taking a minimum, and multiplication is replaced with addition.<\/em><\/p>\n

(6) “Bitangents of tropical plane quartic curves”, with Matt Baker, Yoav Len, Nathan Pflueger, and Qingchun Ren, Mathematische Zeitschrift, 282 (2016), no. 3-4, 1017–1031. (arXiv<\/a>)\u00a0\"classical_bitangents\"\"tropical_bitangents\"<\/p>\n

Classically, a smooth plane quartic curve has 28 bitangent lines. \u00a0We prove that tropically, such a curve\u00a0has 7 bitangent lines.<\/em><\/p>\n

(5) “Moduli of tropical plane curves”, with Sarah Brodsky, Michael Joswig, and Bernd Sturmfels, Research in the Mathematical Sciences, Vol. 2 (2015), 2:4. \u00a0(arXiv<\/a>)<\/p>\n

\"mewtwo_graph\"<\/p>\n

Split a polygon up into triangles, and draw the dual graph. \u00a0Which graphs can we see? \u00a0We answer this\u00a0question for polygons with at most 5 lattice\u00a0points in their interior.<\/em><\/p>\n

(4) “Tropical images of intersection points”, Collectanea Mathematica, Vol. 66 (2015), Issue 2, 273-283. \u00a0(arXiv<\/a>)\"tropical_intersection\" \"rational_function\"<\/p>\n

Classical intersection points map\u00a0to tropical intersection points. \u00a0I give a partial answer to the reverse questions: \u00a0which tropical intersection points can be lifted to\u00a0classical intersection points?<\/em><\/p>\n

(3) “Algorithms for Mumford curves”, with Qingchun Ren, Journal of Symbolic Computation, Vol. 68 (2015), 259-284. \u00a0(arXiv<\/a>)<\/p>\n

\"mumford_curve\"<\/p>\n

If we let certain groups act on a field, the quotient can be interpreted as an algebraic curve. \u00a0We develop algorithms for studying such curves over the field of p-adic numbers.<\/em><\/p>\n

(2) “An elliptic curve test of the L-functions Ratios Conjecture”, Duc Khiem Huynh and Steven J. Miller, Journal of Number Theory, Vol. 131 (2011), 1117-1147. \u00a0(arXiv<\/a>)
\n\"ratios_prediction\"<\/p>\n

When studying the zeros of L-functions, it’s helpful to approximate their behavior. \u00a0We 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.)<\/em><\/p>\n

(1) “The spiral index of knots”, Colin Adams, William George, Rachel Hudson, Laura Starkston, Samuel Taylor, and Olga Turanova, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 149 (2010), Issue 2, 297-315. \u00a0(arXiv<\/a>,\u00a0PDF<\/a>)<\/p>\n

\"nested_beauty\"<\/p>\n

We study\u00a0spiral 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.)
\n<\/em><\/p>\n

Other scholarly works<\/strong><\/p>\n

My dissertation is titled “Tropical and non-Archimedean curves”, written at UC Berkeley with Bernd Sturmfels as my advisor. \u00a0You can find a PDF of it\u00a0here<\/a>. \u00a0It includes material from publications (3) through (7) listed above.<\/p>\n","protected":false},"excerpt":{"rendered":"

My research is broadly in the realm of tropical geometry, with particular emphasis on tropical curves and chip-firing games on graphs. \u00a0Here are my preprints and publications, with some pictures and brief descriptions.\u00a0 Authors who were undergrads when the research … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1294,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-5","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/10rem\/wp-json\/wp\/v2\/pages\/5","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.williams.edu\/10rem\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sites.williams.edu\/10rem\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/10rem\/wp-json\/wp\/v2\/users\/1294"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/10rem\/wp-json\/wp\/v2\/comments?post=5"}],"version-history":[{"count":98,"href":"https:\/\/sites.williams.edu\/10rem\/wp-json\/wp\/v2\/pages\/5\/revisions"}],"predecessor-version":[{"id":440,"href":"https:\/\/sites.williams.edu\/10rem\/wp-json\/wp\/v2\/pages\/5\/revisions\/440"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/10rem\/wp-json\/wp\/v2\/media?parent=5"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}