Research & Publications
Gromov-Witten theory via roots and logarithms (2022). The geometry of orbifolds and logarithmic structures provide two ways to study rational curves in a manifold with fixed tangency along a boundary divisor. The orbifold theory has elegant structural properties, but the logarithmic theory has better theoretical properties. The paper provides a bridge by proving that the orbifold theory of blowups of the target manifold converges to the logarithmic theory.
A case study of intersections on blowups of the moduli of curves (2021). Sam and I explore intersection theory on the system of blowups of a simple normal crossings pair induced by tropical geometry. The protagonist is the toric contact cycle: curves in a toric variety framed at infinity. We relate this cycle to virtual strict transforms of the double ramification cycle. En route, we outline ideas that should form the basis of a logarithmic intersection theory. We build on ideas in this paper with Navid.
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). These papers, written with 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 papers of Vakil-Zinger and Smyth. The second solves the tropical inverse problem in genus one and desingularizes genus one logarithmic stable map spaces. Luca, Navid, and I pushed the story further. Later, Francesca and Luca did some amazing work in genus two.
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.
My coauthors: Stanislav Atanasov, Luca Battistella, Dori Bejleri, Milo Brandt, Renzo Cavalieri, Alois Cerbu, Dan Corey, 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, Ajith Urundolil Kumaran, Jeremy Usatine, Ravi Vakil, Nick Wawrykow, Teddy Weisman, Jonathan Wise, Ursula Whitcher.