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 and preprints can be found here and my papers are available on arXiv, GoogleScholar.
Selected Publications and Preprints
- Logarithmic Gromov-Witten theory with expansions (2019). This project concerns the Gromov-Witten theory of degenerations, and especially those of normal crossings type. The main result establishes a formula for the Gromov-Witten invariants of a smooth fiber in a degeneration in terms of the invariants of strata in a degenerate fiber. It generalizes a well-known formula due to Jun Li, building on foundational work in logarithmic Gromov-Witten theory by Abramovich-Chen-Gross-Siebert. (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, provide a reworking of the foundations of the theory of stable maps in genus one. We use tropical methods to realize a connection between earlier work of Smyth and Vakil-Zinger, leading to the systematic introduction of elliptic singularities into the stable maps theory. We apply this to construct moduli theoretic desingularizations of various spaces of genus one curves. (Volume I in Geometry & Topology, Volume II under review)
- 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 is a broad generalization of a classical theorem of Clifford on 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 in an elementary geometric fashion using non-archimedean and tropical geometry. The result is a stable maps version of work of Tevelev for the space of pointed genus 0 curves and provides easy access to the geometry of these moduli spaces, and yields tropical correspondence theorems as simple 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. (Mathematische Annalen)