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, after capping with a lambda class, however we provide a counterexample for rank 2 in genus 1. 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 (Submitted, 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.
Investigating transversals as generating sets for finite groups with M. Chiodo, O. Donlan, P. Piwek (2019)
A Wall-Crossing formula for logarithmic quasimaps with Q. Shafi
We prove a birational invariance result for logarithmic quasimaps for a GIT quotient $X$ with snc divisor $D$. Using ideas of BNR22 we then show that in genus 0 there exists a log modification of $(X|D)$ for which the associated orbifold quasimap virtual class for the multi-root stack of X along D agrees with the logarithmic quasimap virtual class, for any stability condition. Combining these two results with Zhou's orbifold quasi-map wall crossing formula allows us to obtain a new wall-crossing formula, expressing the logarithmic Gromov-Witten virtual class in terms of the logarithmic quasimap virtual class plus explicit correction terms.
TroPrym and stacky covers of graphs with T. Poiret
We construct a notion of stacky double covers of graphs and their associated tropical Prym varieties. This leads to a notion of a universal Prym variety over the moduli space of double covers of graphs; the introduction of stacky graphs fixes an index 2 issue that stops other attempts at forming a universal family.
Canonical compactifications of cluster varieties in log geometry with L. Herr, S. Karwa, P. Zaika
We construct the notions of $\mathbb{G}_{log}$ and $\mathbb{G}_{trop}$-cluster varieties. The former is a canonical compactification of a cluster variety U for which every snc compactification X of U arises as a blowup of, and the latter object can be seen as it's tropicalisation. $\mathbb{G}_{trop}$-cluster varieties can naturally be viewed as integral affine manifolds with singularities.
Analytifications of log schemes and skeleta with S. Karwa
We introduce a notion of analytic Artin fan and also $\mathbb{F}_1$-Artin fan using Berkovich's $\mathbb{F}_1$-space construction, when working over a non-Archimedean valued field $K$ with ring of integers $R$. For the rank of the valuation at most $1$, we prove that there is an equivalence of categories between usual Artin fans over $R$ localised at log modifications, the category of analytic Artin fans over $K$ and the category of $\mathbb{F}_1$-Artin fans over $K$. The assignment of a log scheme to the corresponding analytic or $\mathbb{F}_1$-Artin fan can then be thought of as constructing an appropriate skeleton.
Moduli spaces of twisted maps to smooth pairs (PhD Thesis, October 2024)
The tropical geometry of orbifolds (Unpublished notes, 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.