Selected talks and lectures
This page contains a selection of slides and recorded lectures related to my research.
Exponential algebra and integration in finite terms
Integration in finite terms and exponentially algebraic functions
Joint work with Jonathan Kirby.
We study integration in finite terms for exponentially algebraic functions. Our results extend classical theorems of Abel and Liouville and connect questions from differential algebra, exponential algebra, and functional transcendence.
Preprint · Slides — DART XII, Kassel, April 2024 · Slides — CIRM, 30 September 2024 · Video — Kolchin Seminar
Algebraic vector fields and strong minimality
On the density of strongly minimal algebraic vector fields
Strong minimality is a central model-theoretic notion in differential algebra. In this work, I prove that generic algebraic vector fields in arbitrary dimension define strongly minimal and geometrically trivial differential equations. Together with work of Devilbiss and Freitag, this resolves a question going back to Poizat.
Preprint · Slides — DART XI, London, June 2023 · Video — Kolchin Seminar
Algebraic independence of solutions
When any three solutions are independent
Joint work with James Freitag and Rahim Moosa.
Suppose that an algebraic differential equation admits a nontrivial algebraic relation among some finite collection of distinct nonalgebraic solutions and their derivatives. We prove that such a relation already occurs among three solutions. For autonomous equations over the constants, two solutions suffice.
Paper · Related lecture by Rahim Moosa — Fields Institute, 2021
Definable Galois theory and complex geometry
Galois groups of non-standard curves in CCM
This lecture presents a version of definable Galois theory in the model theory of compact complex manifolds. It explains, in dimension one, how Galois-theoretic methods can be used in the bimeromorphic classification of compact complex manifolds.
The underlying work is joint with Rahim Moosa.
Paper · Slides — Notre Dame Model Theory Seminar
Foliations and web geometry
Holomorphic foliations and webs provide geometric tools for carrying out parts of the model-theoretic analysis of algebraic differential equations. Understanding this interaction more systematically is a long-term theme of my research.
A model-theoretic invitation to web geometry
This lecture discusses the semiminimal analysis of two-dimensional types in the theories of compact complex manifolds and differentially closed fields.
Slides — Notre Dame Model Theory Seminar, 2020
Ax–Schanuel through holomorphic foliations
This lecture presents a proof of the Ax–Schanuel theorem using the language of holomorphic foliations.
Video — Brazil–France School on Foliation Theory, CIRM, 2024
Geodesic differential equations
A model-theoretic analysis of geodesic equations in negative curvature
This lecture studies geodesic differential equations associated with real algebraic Riemannian manifolds from the perspectives of differential algebra and model theory, with particular emphasis on negatively curved manifolds.