## Research & Publications

I am broadly interested in algebraic geometry, with a focus on the use of logarithmic and tropical methods to establish results about the geometry of curves, moduli spaces, and enumerative geometry. A complete list of papers is available **here****, **as well as on **arXiv**** **and** GoogleScholar**.

**Selected Publications and Preprints**

**Logarithmic Gromov-Witten theory with expansions (2019).***expansions*of a target along the strata of a normal crossings divisor. The result can be viewed as an algebraic manifestation of a multifold symplectic sum formula.*(Under review)***Moduli of stable maps in genus one & logarithmic geometry, Volumes I & II (2017)****.**These two papers, written jointly with Keli Santos-Parker and Jonathan Wise, investigate the relationship between tropical geometry and the structure of elliptic curve singularities. This reveals a connection between earlier work of Smyth related to the Hassett-Keel program, and work of Vakil-Zinger on the geometry of the Kontsevich space. The main results are the construction of moduli spaces desingularizing the Kontsevich space in genus one and its logarithmic analogues, a strong factorization theorem for maps between moduli spaces of pointed elliptic curves, and a solution to the tropical inverse problem in genus one. The theory is developed further in**joint work**with Luca Battistella and Navid Nabijou*(Volume I in Geometry & Topology, Volume II in Algebra & Number Theory)***Brill-Noether theory for curves of a fixed gonality (2017)****.**The Brill-Noether theorem governs the complexity of embeddings of smooth algebraic curves in projective space, when the complex structure on the curve is general. This paper, written with Dave Jensen, is a generalization of this theorem for curves which are general within a prescribed special locus. More precisely, for curves that are general of a fixed gonality. The result was conjectured in earlier work of Pflueger, and provides an interpolation between the Brill-Noether theorem for general curves and Clifford's theorem for hyperelliptic curves.*(Under review)***Skeletons of stable maps I: rational curves in toric varieties (2016)****.**This paper constructs the moduli space of logarithmic stable maps from genus 0 curves to toric varieties**joint work**with Cavalieri and Markwig when the target is a projective line. The result provides a very direct avenue of access to the geometry of these spaces. The fundamental correspondence theorems in tropical geometry are immediate consequences.*(Journal of the London Mathematical Society)***Tropicalizing the space of admissible covers (2014)****.**The moduli space of admissible covers is a compactification of the Hurwitz scheme. In this paper with Renzo Cavalieri and Hannah Markwig, we identify a precise relationship between the non-archimedean analytic skeleton of this space and a tropical moduli space, using a framework outlined by Abramovich-Caporaso-Payne. The equality of tropical and algebraic Hurwitz numbers is established as a consequence.*(Mathematische Annalen)*