## Student Research

I have mentored undergraduate research students since 2013 at Yale, MIT, and Cambridge.

If you are an undergraduate and would like to work with me, take a look at the descriptions, slides, papers, and posters below. They give a good sense of what areas I like tend to suggest problems. The main players tend to be toric geometry, tropical and algebraic moduli spaces, hyperplane arrangements, and Brill-Noether theory.

Realizability of tropical curves via Hurwitz theory, with Mia Lam and Chi Kin Ng (2023). This is another project that relates to the "realizability" problem that Koyama studied a couple of years earlier. The proof of Speyer's famous "well-spacedness" theorem for genus one tropical curves characterizes when such tropical curves come from algebraic geometry. Speyer's theorem (and various other proofs) are quite technical in nature, relying on non-archimedean uniformization, deformations of Gorenstein singularities, and other techniques. These techniques also seem very special to genus one. Mia and Derek found a proof using a much more elementary point of view: tropical admissible covers and relative stable maps. Their results also gave the first nontrivial realizability criteria in genus two. Slides.

Chow rings of toric graph associahedra, with Irina Djankovic (2022). By combining constructions in toric geometry and polyhedral combinatorics, a graph on n+1 vertices gives rise to a topological space of complex dimension n – the toric graph associahedron. In special cases, the spaces include toric varieties of the permutohedron and stellahedron. The latter spaces have been made famous recently by the Fields Medal work of June Huh. Irina began an exploration of the intermediate cases. She investigated the combinatorial structure of these cohomology rings. As the graphs get simpler, the rings are replaced with subrings. Qualitatively her result showed that if one considers cohomology classes up to symmetry, the complete graph can be replaced with a much smaller graph without making the ring too much smaller. Slides.

Combinatorics of superabundant tropical curves in low genus, with Sae Koyama (2021). The tropical realizability problem asks which embedded tropical curves, which are fundamentally objects of combinatorial geometry, come from algebraic geometry via a process called tropicalization. The question is wide open in general, but partial answers have led to progress in enumerative geometry, the study of moduli, and Brill-Noether theory. Sae investigated the combinatorial phenomena that lead to negative answers to this question, completely classifying such phenomena in low genus, and thereby producing the first fundamentally new types of non-realizable tropical curves since Speyer's PhD thesis in 2005. Slides.

Quasisymmetric functions and piecewise polynomials on cone stacks, with Parth Shimpi and Dan Townsend (2021). The ring of quasi-symmetric functions is a beautiful ring from the world of algebraic combinatorics, generalizing the more well-studied ring of symmetric functions. In his PhD thesis, Oesinghaus showed that this arises as the Chow ring of a certain natural object in algebraic geometry: the stack of expansions of a smooth pair. There is a network of equivalences that equates this stack to a piece of combinatorial datum known as a cone stack. In their project, Dan and Parth showed how to directly extract QSym from the combinatorial datum, using its associated ring of piecewise polynomial functions. Slides.

Characteristic numbers for curves on a quadric surface, with Joe Benton and James Rawson (2020). The characteristic numbers of an algebraic surface count curves on that surface that satisfy geometric constraints, such as passing through points and being tangent to curves. These have been the subject of intense study since the 19th century, particularly for the projective plane. The corresponding geometry for the quadric surface P1xP1 was the subject of what Joe and James did during the summer. They used a transparent geometric argument using the moduli space of stable maps, following in the footsteps of Vakil and Kock-Graber-Pandharipande, to compute some of the basic characteristic numbers of P1xP1. The "full" characteristic numbers problem for both P2 and P1xP1 remains wide open! I'll post a report here some day!

Euler characteristics of tropical descendant cycles, with Diogo Fonseca (2019). On the moduli space of pointed rational curves, there are natural cohomology classes, called descendants, that play an important role in enumerative geometry. Tropical geometry associates a simplicial complex to each one of these cohomology classes. Diogo studied the topology of these descendants, and calculated recursive formulae for their Euler characteristics in a number of cases. We also calculated the homotopy type of the the complex associated to the standard cotangent divisor class, known as the psi class.

Incidence geometry and universality in the tropical plane, with Milo Brandt, Michelle Jones, and Catherine Lee (2017). Journal of Combinatorial Theory, Series A. The Sylvester-Gallai theorem is a beautiful theorem in classical geometry, which states that for any collection of non-collinear points in the real plane, there is a line passing through exactly 2 of the points. Interestingly, the result is false over the complex numbers - the inflection points of a smooth plane cubic yield a counterexample. This makes it unclear whether the statement should be true for points in the tropical plane. Milo, Michelle, and Catherine proved that the statement is true tropically, and also establish a colored variant, classically known as the Motzkin-Rabin theorem. Finally, they proved a polyhedral analogue of Mnev's universality theorem, roughly stating that the realization space of tropical line arrangements have unconstrained local geometry.

Topology of tropical moduli of weighted stable curves, with Alois Cerbu, Luke Peilen, and Andrew Salmon, co-supervised with Steffen Marcus (2017). Advances in Geometry. The tropical moduli spaces of curves are topological spaces that parameterize metric structures on graphs, in analogous fashion to how the moduli space curves parametrizes hyperbolic metrics on Riemann surfaces. Alois, Luke, and Andrew investigated the topology of a network of moduli spaces determined by a stability condition known as a weight. Under mild hypotheses, we showed that the moduli spaces of weighted stable curves in genus zero and one are homotopic to wedge sums of spheres of varying dimension. Without these hypotheses, the spaces run wild - they can be disconnected, have torsion fundamental group, and have nonzero homology in a wide range of degrees. This contrasts with and builds on work of Chan, Chan-Galatius-Payne, and Vogtmann. Slides.

A note on Brill-Noether existence for graphs of low genus, with Stanislav Atanasov (2016). Michigan Mathematical Journal. In an influential 2008 paper, Baker proposed a number of conjectures relating the divisor theory of algebraic curves with an analogous combinatorial theory on finite graphs. The last remaining open conjecture, known as the existence conjecture, is a purely combinatorial statement predicting that divisors on graphs with certain numerical parameters always exist. We prove the Brill-Noether existence conjecture in low genus, and a number of interesting infinite families. Slides.

Tropicalizing line arrangements over finite fields, with Derek Boyer, Andre Moura, and Scott Weady (2016). Line arrangements in the plane come up frequently in tropical geometry, but have thus far been studied via the tropicalization of their complement. Derek, Andre, and Scott studied a more naive but appealing approach, by studying the collection of all tropicalizations of the arrangement, obtained from all projective coordinate changes. We consider the question of when a given set of tropical lines can be obtained by tropicalizing a translate of the arrangement of all lines in the projective plane over a finite field . The main result is an effective combinatorial characterization answering this question. Slides.

Toric graph associahedra and compactifications of M0,n, with Rodrigo da Rosa, co-supervised with Dave Jensen (2014). Journal of Algebraic Combinatorics. To any graph G one can associate a toric variety X(G), obtained as a blowup of projective space along coordinate subspaces corresponding to connected subgraphs of G. The combinatorics of this blowup is controlled by a polytope, known as a graph associahedron, a class of polytopes that includes the permutohedron, associahedron, and stellahedron. We show that the space X(G) is isomorphic to a weighted stable modular compactification of M0,n precisely when G is an iterated cone over a discrete set. This generalizes a well-known result of Losev-Manin and Kapranov, identifying many new compactifications of the moduli space of pointed rational curves, giving point of access to their geometry. Poster.

Realization of groups with pairing as Jacobians of finite graphs, with Lou Gaudet, Nick Wawrykow, and Teddy Weisman, co-supervised with Dave Jensen (2014). Annals of Combinatorics. The Jacobian construction associates to a finite graph G, a finite abelian group Jac(G). This group comes up frequently in algebraic and arithmetic geometry, particularly in the study of Neron models of Jacobians, and has played a prominent role in the last decade in combinatorial Brill-Noether theory. A fundamental question on the discrete side of the subject is to understand which finite abelian groups, with an additional structure known as a pairing, occur as Jacobians. Lou, Nick, and Teddy studied these question, and produced explicit constructions which, conditional on the generalized Riemann hypothesis (!), produce all groups with pairing of odd order as Jacobians. They also proved results showing that if one restricts themselves to simple graphs or biconnected graphs, many finite abelian groups never occur at all. Poster.

Brill-Noether theory of maximally symmetric graphs, with Timothy Leake (2013). European Journal of Combinatorics. The existence of a Brill-Noether general graph was conjectured by Baker in 2008 and established in a seminal paper of Cools, Draisma, Payne, and Robeva. In a 2011 paper, Caporaso conjectured that in each genus, the graph with largest possible automorphism group is Brill-Noether general. Timothy and I took this as motivation to systemically analyze the full Brill-Noether theory of such graphs. We found that, with some exceptions in low genus, these graphs are never Brill-Noether general. The results are sharp, and we undertook the study separately for simple graphs and multigraph with or without loops.

Undergraduate students that have worked with me, in chronological order

Timothy Leake (Summer '13, combinatorial Brill-Noether theory), Rodrigo Ferreira da Rosa (Summer '14, M0,n and graph associahedra), Andrew Deveau, Jenna Kainic, and Dan Mitropolsky (Summer '14, Gonality of random graphs), Louis Gaudet, Nicholas Wawrykow, and Teddy Weisman (Summer '14, Jacobians of finite graphs), Derek Boyer, Andre Moura, and Scott Weady (Summer '16, Tropical line arrangements), Stanislav Atanasov (Summer '16, Combinatorial Brill-Noether theory), Johnny Gao (Fall '16-Spring '17, Betti numbers of toric graph associahedra), Milo Brandt, Michelle Jones, and Catherine Lee (Summer '17, Incidence geometry of tropical lines), and Alois Cerbu, Luke Peilen, and Andrew Salmon (Summer '17, topology of tropical moduli spaces), Abhijit Mudigonda (Spring '18, partial reciprocal planes and matroids), Hung-Hsun Yu (Spring '18, tropical bitangents and quartics), Diogo Fonseca (Summer '19, Topology of tropical cotangent classes), Joe Benton and James Rawson (Summer '20, Characteristic numbers of the quadric surface), Parth Shimpi and Dan Townsend (Summer '21, Chow rings of stacks of expansions), Sae Koyama (Summer '21, Superabundant tropical curves), Irina Đanković (Summer '22, Matroids and toric graph associahedra), Mia Lam and Chi Kin Ng (Summer '23, Tropical realizability via Hurwitz theory), Dylan Toh (Summer '24, Higher double ramification cycles).