## Student Research

I have mentored undergraduate research students since 2013. During my time at Yale, I worked closely with the **SUMRY program**, a 10 week research program at Yale University. At MIT I took students through the **UROP program**. If you are an undergraduate and would like to work with me, the slides 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, and combinatorial aspects of Brill-Noether theory, but there is a lot of mathematics out there!

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 in-preparation.**,**Incidence geometry and universality in the tropical plane**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**with Milo Brandt, Michelle Jones, and Catherine Lee (2017). Journal of Combinatorial Theory, Series A.****Topology of tropical moduli of weighted stable curves***,*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.**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,****with Stanislav Atanasov (2016). Michigan Mathematical Journal.****Slides.**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.**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,}**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***,*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.**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 (Summer '20).