Publications
Papers are listed in reverse chronological order. Links to preprints and published versions are provided when available.
Preprints
Integration in finite terms and exponentially algebraic functions
With Jonathan Kirby.
We develop differential-algebraic and model-theoretic methods for studying exponential algebraicity and generalize classical results of Abel and Liouville on integration in finite terms. Applications include exponential transcendence and independence results for several classical functions, together with corresponding decidability results.
Accepted for publication
On the density of strongly minimal algebraic vector fields
To appear in the Annales scientifiques de l’École normale supérieure.
We prove that generic algebraic vector fields in arbitrary dimension define strongly minimal and geometrically trivial differential equations. More generally, analogous results are established for generic vector fields with sufficiently large pole order on complements of smooth hyperplane sections.
Published papers
Abelian reduction in differential-algebraic and bimeromorphic geometry
With Rahim Moosa.
Annales de l’Institut Fourier 75 (2025), no. 4, 1811–1853.
We develop a Galois-theoretic notion of abelian reduction in finite-rank stability theory and apply it to differential-algebraic and bimeromorphic geometry. Among the consequences are generalizations of several results of Fujiki and applications to the birational geometry of algebraic vector fields.
On the equations of Poizat and Liénard
With James Freitag, David Marker, and Joel Nagloo.
International Mathematics Research Notices 2023 (2023), no. 19, 16478–16539.
We study Poizat equations, viewed as a class of Liénard equations, from the perspective of geometric stability theory. We characterize strong minimality, classify the algebraic relations among solutions in the strongly minimal cases, and study Liouvillian and Pfaffian solutions.
The degree of nonminimality is at most 2
With James Freitag and Rahim Moosa.
Journal of Mathematical Logic 23 (2023), no. 3, Article 2250031, 6 pp.
We prove that the degree of nonminimality of a finite-rank type in a differentially closed field is at most two. In particular, nonminimality can always be witnessed over a pair of realizations of the type.
Relative internality and definable fibrations
With Léo Jimenez and Anand Pillay.
Advances in Mathematics 415 (2023), Article 108870, 38 pp.
We study relative internality and uniform relative internality in stable theories. For finite-rank fibrations with a base orthogonal to the constants, we relate uniform internality to the existence of a product decomposition and obtain applications to differential Galois theory and complex geometry.
When any three solutions are independent
With James Freitag and Rahim Moosa.
Inventiones Mathematicae 230 (2022), no. 3, 1249–1265.
We show that if an algebraic differential equation admits a nontrivial algebraic relation among any number of distinct nonalgebraic solutions and their derivatives, then such a relation already occurs among three solutions. For autonomous equations over the constants, two solutions suffice.
Generic planar algebraic vector fields are strongly minimal and disintegrated
Algebra & Number Theory 15 (2021), no. 10, 2449–2483.
We prove that a generic polynomial vector field on the affine plane of degree at least three defines a strongly minimal and geometrically trivial differential equation.
Rational factors, invariant foliations and algebraic disintegration of compact mixing Anosov flows of dimension 3
Confluentes Mathematici 12 (2020), no. 2, 49–78.
We develop a geometric approach to semiminimality based on rational factors and invariant algebraic foliations. We apply this framework to algebraic differential equations inducing compact mixing Anosov flows of dimension three and prove that their generic types are disintegrated.
Corps différentiels et flots géodésiques I: Orthogonalité aux constantes pour les équations différentielles autonomes
Bulletin de la Société Mathématique de France 148 (2020), no. 3, 529–595. In French.
We establish a dynamical criterion for orthogonality to the constants for real algebraic differential equations. The criterion applies when the associated real flow admits a Zariski-dense compact invariant set on which it is topologically weakly mixing.
Differential fields and geodesic flows II: Geodesic flows of pseudo-Riemannian algebraic varieties
Israel Journal of Mathematics 230 (2019), no. 2, 527–561.
We introduce pseudo-Riemannian algebraic varieties and study their geodesic equations model-theoretically. For algebraic presentations of compact negatively curved Riemannian manifolds, we prove that the corresponding geodesic differential equation is absolutely irreducible and orthogonal to the constants.