Research

Drafts/preprints are available upon request. Click here for my CV.

Universal relations on moduli spaces of twisted maps  with Sam Johnston, in preparation.

We provide a splitting of the virtual class of twisted maps to the universal snc pair. This in particular allows us to prove Conjecture W of BNR22 that the constant term of the polynomial of orbifold Gromov--Witten invariants is the Naive Log invariant. We then use this to deduce Gromov--Witten classes are tautological in a variety of higher genus contexts. The first step in the virtual splitting is constructing a global embedding of the moduli space in to a universal Jacobian over a space of log twisted maps. This allows us to relate the stratifications of the paper below to spaces of logarithmic maps.

Moduli spaces of twisted maps to smooth pairs (Preprint, Jan. 2025 https://arxiv.org/abs/2501.15171)

We study moduli spaces of twisted maps to a smooth pair in arbitrary genus, and give geometric explanations for previously known comparisons between orbifold and logarithmic Gromov--Witten invariants. Namely, we study the space of twisted maps to the universal target and classify its irreducible components in terms of combinatorial/tropical information. We also introduce natural morphisms between these moduli spaces for different rooting parameters and compute their degree on various strata. Combining this with additional hypotheses on the discrete data, we show these degrees are monomial of degree between 0 and max(0,2g−1) in the rooting parameter. We discuss the virtual theory of the moduli spaces, and relate our polynomiality results to work of Tseng and You on the higher genus orbifold Gromov--Witten invariants of smooth pairs, recovering their results in genus 1. We discuss what is needed to deduce arbitrary genus comparison results using the previous sections. We conclude with some geometric examples, starting by re-framing the original genus 1 example of Maulik in this new formalism.

Moduli spaces of twisted maps to smooth pairs (PhD Thesis, October 2024)

The tropical geometry of orbifolds (Preprint, Oct. 2023) (last updated 11th Dec 2023)

Introducing and studying the tropicalisation of orbifolds and logarithmic orbifolds. I study the interactions between logarithmic and orbifold maps by defining and tropicalising orbi-log stable maps. I also give a significant generalisation of a result of Cavalieri, Chan, Ulirsch and Wise between cone stacks and Artin fans to the orbifold setting. These ideas have allowed me to prove a tropical lifting theorem for twisted stable maps. This is the first example of tropical/combinatorial techniques in orbifold theory.

Investigating transversals as generating sets for finite groups with M. Chiodo, O. Donlan, P. Piwek (2019)