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 Donaldson-Thomas theory (2020). This paper, written with Davesh Maulik, constructs an enumerative geometry of ideal sheaves on a smooth threefold equipped with tangency conditions along a normal crossings divisor. This completes a sheaf theory parallel to logarithmic Gromov-Witten theory. The geometric picture is meant to mirror the one laid out in the Gromov-Witten paper below, though the details are rather different. The fundamental geometry concerns the locus in the Hilbert scheme comprising subschemes that are transverse to the divisor. The main insights in building the theory come from tropical geometry. The origins of these sheaf counting theories are very inspiring.
  • 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 celebrated formula due to Jun Li. The geometric input is the construction of new moduli spaces of stable maps to 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.
  • Moduli of stable maps in genus one & logarithmic geometry, Volumes I & II (2017) & Curve counting in genus one: relative geometry & elliptic singularities (2019). These three papers, written with subsets of Luca Battistella, Navid Nabijou, Keli Santos-Parker, and Jonathan Wise, investigate the relationship between tropical geometry, Gromov-Witten theory, and the structure of elliptic curve singularities. The first paper establishes the basic theory of radially aligned curves and contractions, and relates it to Smyth's work on pointed elliptic curves. The second paper applies this to solve the tropical inverse problem in genus one and desingularizes genus one logarithmic stable map spaces for toric targets. The final paper turns theory into practice by establishing a calculation scheme for relative geometries, and explains the interaction between elliptic singularities and degeneration formulas.
  • Brill-Noether theory for curves of a fixed gonality (2017). The Brill-Noether theorem is fundamental in the theory of algebraic curves. It governs the complexity of embeddings of smooth curves in projective space, when the curve is general in moduli. This paper, written with Dave Jensen, is a generalization of this theorem that replaces the moduli space of curves with the Hurwitz space. We give a closed formula for the dimension of Brill-Noether varieties for general curves of a fixed gonality. The result was conjectured in earlier work of Nathan Pflueger, and interpolates between the Brill-Noether theorem for general curves and Clifford's theorem for hyperelliptic curves. Hannah Larson has pushed these results in a beautiful direction.
  • 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 paper generalizes Tevelev's work on tropical compactifications for the space of pointed rational curves, as well as joint work with Cavalieri and Markwig when the target is a projective line. The result provides an avenue of access to the geometry of these spaces.
  • 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. The latter is a general blueprint that explains why enumerative information survives the tropicalization process.

Collaborators on ongoing work: Luca Battistella, Francesca Carocci, Yoav Len, Davesh Maulik, Navid Nabijou, Martin Ulirsch.