Research & Publications
I am interested in algebraic geometry, and the use of tropical methods to study algebraic varieties, their moduli spaces, and enumerative geometry. A complete list of papers is available here, on arXiv, and GoogleScholar. Here's a selection.
Logarithmic Donaldson-Thomas theory (2020). Davesh and I construct an enumerative geometry of ideal sheaves on a smooth threefold, with tangency conditions along a normal crossings divisor. This completes a sheaf theory parallel to logarithmic Gromov-Witten theory. 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 normal crossings degenerations. The main result is a formula for the Gromov-Witten invariants of a smooth fibre in a degeneration in terms of the invariants of strata in a degenerate fibre.
Moduli of maps in genus one & logarithmic geometry, Volumes I & II (2017) & Curve counting in genus one: relative geometry & elliptic singularities (2019). These papers, written with subsets of Luca, Navid, Keli, and Jonathan, investigate the relationship between tropical geometry, Gromov-Witten theory, and elliptic curve singularities. The first establishes the basic theory of radially aligned curves, and connects work of Vakil-Zinger and Smyth. The second solves the tropical inverse problem in genus one and desingularizes genus one logarithmic stable map spaces. The final paper is a calculation scheme for relative geometries.
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 the moduli space of curves. Dave and I generalize this theorem by working over the Hurwitz space and give formulas for the dimensions of Brill-Noether varieties for general curves of a fixed gonality.
Skeletons of stable maps I: rational curves in toric varieties (2016). This paper constructs the space of logarithmic stable maps from genus 0 curves to toric varieties in an elementary geometric fashion. The paper generalizes Tevelev's work on tropical compactifications, and earlier joint work with Renzo and Hannah. 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 work with Renzo and Hannah, we identify a simple 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 a consequence.
My coauthors: Stanislav Atanasov, Luca Battistella, Dori Bejleri, Milo Brandt, Renzo Cavalieri, Alois Cerbu, Rodrigo Ferreira da Rosa, Tyler Foster, Louis Gaudet, Simon Hampe, David Jensen, Paul Johnson, Michelle Jones, Dagan Karp, Timothy Leake, Catherine Lee, Yoav Len, Steffen Marcus, Hannah Markwig, Davesh Maulik, Samouil Molcho, Navid Nabijou, Luke Peilen, Paul Riggins, Keli Santos-Parker, Andrew Salmon, Mattia Talpo, Martin Ulirsch, Jeremy Usatine, Ravi Vakil, Nick Wawrykow, Teddy Weisman, Jonathan Wise, Ursula Whitcher.