## 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, the slides, papers, and posters below should give you a fairly good inclination of what areas I like to look for problems. In broad strokes, the main players tend to be toric geometry, tropical and algebraic moduli spaces, hyperplane arrangements, and combinatorial aspects of Brill-Noether theory, but there is a lot of mathematics out there!

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.**Chow rings of toric graph associahedra, with Irina Djankovic (2022).****Slides****.****Combinatorics of superabundant tropical curves in low genus, with Sae Koyama (2021).****Slides.****Quasisymmetric functions and piecewise polynomials on cone stacks, with Parth Shimpi and Dan Townsend (2021).****Slides****.**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!**Characteristic numbers for curves on a quadric surface, with Joe Benton and James Rawson (2020).****I'll post a report here some day!**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.**Euler characteristics of tropical descendant cycles, with Diogo Fonseca (2019), paper overdue.**,**Incidence geometry and universality in the tropical plane****with Milo Brandt, Michelle Jones, and Catherine Lee (2017). Journal of Combinatorial Theory, Series A.****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.****Slides***.***A note on Brill-Noether existence for graphs of low genus,**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.**with Stanislav Atanasov (2016). Michigan Mathematical Journal.****Slides.****Tropicalizing line arrangements over finite fields***,***with Derek Boyer, Andre Moura, and Scott Weady (2016).****Slides***.***Toric graph associahedra and compactifications of M**_{0,n}_{,}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 M**with Rodrigo da Rosa, co-supervised with Dave Jensen (2014). Journal of Algebraic Combinatorics.**_{0,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.**.**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, M_{0,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), Tom Edwards (Summer '21, Chromatic compactifications of M_{0,n}; supervised with Navid Nabijou), Sae Koyama (Summer '21, Superabundant tropical curves), Irina Đanković (Summer '22, Matroids and toric graph associahedra).