If you want a clean list with dates, see my CV. Papers below are arranged partly thematically, and may have a brief summary, or notes pointing to more recent results that may be relevant.
Enumerative geometry and moduli of curves
Enumerative geometry aims to solve counting problems like 'how many conics lie on a generic quintic 3-fold?' (609,250, a surprisingly useful number). Quite a number of the papers below concern the 'double ramification cycle'; this may be viewed as a cycle on the moduli space of curves measuring how often a linear combination of markings is a principal divisor, or as a relative Gromov-Witten invariant of BG_m.
(with Y. Bae, R. Pandharipande, J. Schmitt, and R. Schwarz) `Pixton's formula and Abel-Jacobi theory on the Picard stack', Acta Math. - [ arXiv ] This is the 'universal double ramification cycle' paper, which is the other half the proof of Conjecture A of [JPPZ1].
(with G. Orecchia) `Logarithmic moduli of roots of line bundles on curves', Expositiones Mathematicae, special edition in memory of Bas Edixhoven. - [journal | arXiv]
We show that the torsion in the Logarithmic Picard stack of Molcho-Wise gives a clean way to compactify spaces of roots of line bundles.
(with R. Schwarz) `Logarithmic intersections of double ramification cycles', Algebraic Geometry - [journal | arXiv] We prove that double-double ramification cycles are tautological, via proving that the log double ramification cycle is log tautological. Molcho and Ranganathan proved similar things at a similar time, though the proofs seem to me very different; they prove a general result comparing strict and total pullbacks of cycles under log blowups, we use the shape of the terms appearing in Pixton's formula.
(with J. Schmitt) `Infinitesimal structure of the pluricanonical double ramification locus', Compositio Mathematica, Vol. 157, No. 10 (2021), pp. 2280 2337. - [ arXiv ] We do a lot of deformation theory to understand the multiplicities of the components of the double ramification locus. This is basically one half of the proof of Conjecture A of [JPPZ1].
(with O. Biesel ) `Fine compactified moduli of enriched structures on stable curves', Memoirs of the AMS - [arXiv]. This solved a number of conjectures of Maino. My student Pim Spelier has now written a logarithmic version of this paper, and it's quite a bit shorter and more general.
(with A. Pixton and J. Schmitt ) `Multiplicativity of the double ramification cycle', Documenta - [ journal | arXiv ] Here we introduced the log double ramification cycle, and showed that it is in some ways better-behaved than the classical one.
(with J. Kass and N. Pagani ) `Extending the Double Ramification Cycle using Jacobians', European Journal of Mathematics (2018). doi - [ journal | arXiv ] We relate two different approaches to DR cycles, via log geometry and via Brill-Noether loci on compactificed jacobians. Contains a comparison lemma that has been useful in a number of subsequent works.
`Extending the double ramification cycle by resolving the Abel-Jacobi map', JIM Jussieu - [journal | arXiv] This was where I first applied the ideas from `A Neron model of the universal jacobian' to DR problems, constructing the k-pluricanonical DR for all k (see also Marcus-Wise).
Arakelov Geometry
Arakelov geometry gives conceptual machinery to understand height functions in number theory, in terms of intersection theory on schemes over ZZ enhanced with metric data at infinity. The 'height jump' appearing in some of the papers below is a measure of the failure of the formation of the height to commute with non-flat base change, and is intimately related to structures of Néron models (for which see lower on this page).
(with A. M, Botero, J. I. Burgos Gil, R. de Jong) `Rings of Siegel-Jacobi forms of bounded relative index are not finitely generated', to appear in Duke - [arXiv] Siegel-Jacobi forms are analogues of modular forms on the universal abelian variety over the moduli stack of abelian varieties. We prove a conjecture of Kramer and disprove a theorem of Runge.
(with A. M, Botero, J. I. Burgos Gil, R. de Jong) `Chern-Weil and Hilbert-Samuel formulae for singular hermitian line bundles', Documenta Mathematica. - [journal | arXiv] This sets up lots of intersection theory of b-divisors (divisors on blowups, closely related to the log Chow ring), used in the 'Rings of Siegel Jacobi forms' paper just above.
(with Steffen Muller and Raymond van Bommel) `Explicit arithmetic intersection theory and computation of Neron-Tate heights', Mathematics of Computation - [ journal | arXiv ] A very general algorithm for computing Néron-Tate heights of points on jacobians, with a MAGMA implementation. Mostly written at a workshop in Baskerville Hall in the Wye Valley.
(with Jose Burgos Gil and Robin de Jong ) `Positivity of the height jump divisor', IMRN (2017) - [ journal | arXiv ] Proving Hain's conjecture on the positivity of the height jump (see also work of Brosnan-Pearlstein).
(with Jose Burgos Gil and Robin de Jong ) `Singularity of the biextension metric for families of abelian varieties', Forum of Mathematics, Sigma - [ journal | arXiv ] Here we proved an analogue of Hain's positivity conjecture mentioned just above, for families of abelian varieties. The result is less general than the one above, but we obtain more refined information on the shape of the functions involved.
(with Owen Biesel and Robin de Jong ) `Neron models and the height jump divisor', Trans. AMS 369 (2017), 8685--8723 - [ journal | arXiv] We prove an even more special case of Hain's positive conjecture, this time for families of jacobians of curves. The proof is combinatorial, eventually reducing to some statement about electrical networks. We give a new proof of Silverman and Tate's results on specialisation of heights in families of abelian varieties.
`Torsion points and height jumping in higher-dimensional families of abelian varieties', IJNT - [journal | arXiv] This paper discusses some consequences for rational torsion points of various theorems on Néron models. We conjecture a generalisation of Silverman and Tate's results on heights in families, which we show would imply the Torsion Conjecture for abelian varieties. We show that families admitting separated Neron models (and satisfying a technical positivity condition) satisfy this conjecture. Unfortunately, most families do not admit separated Neron models.
(with Robin de Jong ) `Asymptotics of the Neron height pairing', Math. Res. Lett. (2015) - [ journal | arxiv] We prove a very special case of Hain's positvity conjecture (see above for more general version).
`An Arakelov-Theoretic approach to naive heights on hyperelliptic Jacobians', New York J. Math. (2014) - [ journal | arXiv] Naive heights are a useful computational tool for finding the rational points on curves and their jacobians. The key properties one needs are that the naive height is close to the Neron-Tate height, and that one can compute all the rational points of bounded naive height. Classically Nice heights are defined in terms of projective embeddings of the jacobian. Here we propose a new approach via Arakelov intersection theory, requiring no projective embeddings.
`Computing Neron-Tate heights of points on hyperelliptic Jacobians', J. Number Theory (2012) - [ journal | arXiv ] The Neron-Tate height pairing is a quadratic form on the group of rational points on an abelian variety. Here we give an algorithm (with MAGMA implementation) to compute it on jacobians of hyperelliptic curves. Steffen Muller did similar work at the same time; the implementation currently used in MAGMA is his.
Néron models
Néron models are 'optimal' models of degenerating abelian varieties. Classically they were generally considered only for 1-parameter degenerations, but the papers below mostly explore what happens over higher-dimensional bases.
(with G. Orecchia, S. Molcho and T. Poiret) `Models of Jacobians of curves', Crelle - [arXiv]
This largely supersedes my earlier paper `Neron models of jacobians over base schemes of dimension greater than 1' in the same journal. We no longer assume Néron models are separated, but rather give a tropical criterion for separatedness, and in fact classify all separated group models.
`A Neron model of the universal jacobian', Annales Henri Lebesgue - [ journal | arXiv ]. This paper asks when the jacobian of the universal stable curve admits a separated Neron Model (rarely), and constructs a universal base-change after which a separated Neron model does exist. Contains the germs of lots ideas that have been useful to me in enumerative geometry.
`Quasi-compactness of Neron models, and an application to torsion points', Manuscripta Math. (2016) - [ journal | arXiv ] When a family of abelian varieties has a separated Néron model, one has good control of torsion points. But this does not happen very often.
`Neron models of jacobians over base schemes of dimension greater than 1', Journal fur die reine und angewandte Mathematik - [ journal | arXiv | video lecture ] In large part initiating study of Néron models over higher dimensional bases. By now this has mostly been superseded, in particular by `Models of Jacobians of curves' (with G. Orecchia, S. Molcho and T. Poiret). Marcus-Wise have greatly simplified and improved the results of the most technical section (Section 4: Classification of vertical Cartier divisors on certain complete local rings). Unfortunately in this paper we assume Néron models are separated; dropping this assumption gives a more flexible and general theory, which we explore in 'Models of Jacobians of curves'. Unlike in the case over a Dedekind scheme, in general the uniqueness part of the NMP does not imply separatedness.
Rational points
This is number theory in the most classical sense: trying to understand 'directly' the rational solutions to polynomial equations.
(with N. Rome) `Fields of definition of curves of a given degree', Journal de Théorie des Nombres de Bordeaux, Vol. 32, No. 1 (2020), pp. 291-310 - [journal | arXiv ] Kontsevich and Manin computed the number of genus-0 curves of degree d through 3d-1 points. We investigate when these curves are defined over the rational numbers.
(with Rene Pannekoek ) `The Brauer-Manin obstruction on Kummer varieties and ranks of twists of abelian varieties', Bulletin of the London Mathematical Society (2015) - [ journal | arXiv ] We show that, if the Brauer-Manin obstruction is the only one to weak approximation on all Kummer varieties, then ranks of twists of any positive-dimensional abelian variety are unbounded.
Modelling and statistics
`The norm of the saturation of a binomial ideal, and applications to Markov bases', Algebraic Statistics, Vol. 11 (2020), No. 2, 169–187 - [journal | arXiv] Computing saturations of binomial ideal is a small industry, with interesting applications to statistical problems. We introduce a coarser approach, where we do not try to compute the saturation directly, but only bound how far a given ideal is from being saturated. This is often enough to solve the original statistical problem. We give some computation examples where our algorithms outperform the previous state-of-the-art.
(with M. Dellar, S. P. Boerlijst) `Improving estimations of life history parameters of small animals in mesocosm experiments: A case study on mosquitoes in Methods', Methods in Ecology and Evolution - [journal] The goal of this research is to understand life-cycles of mosquitos, with a view to understanding how changes in climate and land-use will affect future disease risk. Collecting data is quite hard, and the models have many parameters. We introduced the idea of formally differentiating the model with respect to its parameters, which allowed us to perform gradient descent using significantly less data than needed by other methods.
(with M. Dellar, K. Topp, G. Pardo, A. del Prado, N. Fitton, G. Banos, and E. Wall), `Empirical and dynamic approaches for modelling the yield and N content of European grasslands', Environmental modelling and Software - [journal] The goal here was to model the yield of grasslands under various climate conditions. We compared a pre-existing process-based dynamic model with a simple regression model. Generally, regression was better for identifying large-scale trends, the process-based model gave more precise local information.
Non-scientific
(with J. Briët and R.J. Kang) `Steps towards openness and fairness in scientific publishing', https://ir.cwi.nl/pub/31565, Nieuw Archief voor Wiskunde, 5(23), 53 - 55. ISSN 0028-9825. This article is aimed at mathematicians working in the Netherlands, giving an overview of open access issues, and practical information for how to navigate them.