# Research

Drafts/preprints are available upon request.

The geometry of moduli spaces of twisted maps to smooth pairs (In preparation)

A study of moduli spaces of twisted maps to the ``universal pair", Orb(A_r). We endow Orb(A_r) with a stratification from tropical-style data and classify the irreducible components of this space in terms of this stratification. We also study the natural comparison morphisms between these spaces as we vary the rooting parameter, and show generically on an irreducible component the degrees of such morphisms vary monomially of degree between 0 and max(2g - 1,0). This gives a geometric heuristic of Tseng and You's polynomiality result of orbifold Gromov--Witten invariants to smooth pairs. We upgrade this to a new proof of Tseng and You's result in genus 1. A higher genus decomposition of the virtual class will be studied in future work with Sam Johnston.

Analytification of log schemes with S. Karwa (In preparation)

We construct an analytification functor which, to a log scheme $(X,\mathcal{M}_X)$ over a valued field, associates a locally ringed space $(X,\mathcal{M}_X)^{\mathrm{an}}$. This formalises the notion of that the Berkovich analytification of $X$ is equal to the tropicalisation of $(X,\Ocal_X)$. The functor passes through the category of valuative log schemes, and therefore we obtain an explicit description of the valuativisation of a log scheme in the trivially valued case. As a result, any functor which is invariant under log modifications induces an associated functor on the corresponding analytic spaces. We offer a detailed description in the case of LogChow. Furthermore, to each log scheme $(X,\mathcal{M}_X)$ we associate a polyhedral complex $\text{Sk}(X,\mathcal{M}_X)$, called the skeleton which embeds in $(X,\mathcal{M}_X)^{\mathrm{an}}$ along with a deformation retraction to this complex. When we work over $\mathbb{C}$ with the Euclidean norm, we recover real torus fibrations; in the case of a non-Archimedean valued field, we recover affinoid torus fibrations. We use these skeleta to study log modifications of the moduli space of curves.

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)