Rémi Jaoui

Rémi Jaoui

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Talks and conferences

On the problem of integration in finite terms for exponentially algebraic functions

I have been working during the last two years with Jonathan Kirby on the problem of integration in finite terms for exponentially algebraic functions. The paper is still in preparation but here are the slides of two talks below summarize the main ideas of this work.

  • The talk was part of the DART XII conference in Kassel, Germany (April 2024)
  • The talk was part of the Model Theory and Applications to Groups and Combinatorics conference in the CIRM (October 2024)

On the density of strongly minimal algebraic vector fields

The notion of strong minimality is without doubt one of the main contribution of model theory to differential algebra. In this preprint (2023), I show that “most” algebraic vector fields do satisfy this property. Together with a paper of Devilbiss and Freitag, this answers a long standing open question formulated by Poizat around 1981.

  • The talk was part of the DART XI conference in London, England (July 2023).

A full video of a lecture given at the Kolchin seminar of differential algebra on this article can be found here

When any three solutions are independent

Given an algebraic differential equation, what is the minimal number n of solutions that I have look at if I want to conclude that any sequence of solutions of this equation is algebraically independent?

This is the kind of questions that led James Freitag and Rahim Moosa to introduce the nonminimality degree of a type here. In this paper, we show that in the previous question, n = 3 is enough and that actually one can even take n = 2 if the equation is autonomous.

A detailed account of this result can be found in the talk given by Rahim Moosa as one of the special lectures in honor of Anand Pillay’s 70th birthday (Fields Institute, 2021).

Some definable Galois theory of compact complex manifolds

Pillay’s formulation of definable Galois theory allows to develop a definable Galois in the theory of compact complex manifolds analogous to the usual differential Galois theory. In this paper, with Rahim Moosa, we use this idea to prove Galois-theoretic generalizations of several results of Fujiki concerning the bimeromorphic classification of compact complex manifolds.

Here is a talk given at the Notre Dame logic seminar which explains the one-dimensional case of this variant of differential Galois theory.

Foliations and webs in model theory

In several of my papers, an important idea is that one can use complex holomorphic foliations to perform effectively some of the model-theoretic calculus. The investigation of a wider connection between the two subjects is a long-term goal of my research. Here are two examples

  • The talk (Notre Dame, 2020) concerns the semi-minimal analysis of two-dimensional types in CCM and DCF_0.

*This lecture (CIRM, Brazil-France School on Foliation Theory, 2024) presents a proof of the Ax-Schanuel Theorem in the language of holomorphic foliations.

Model theory of geodesic differential equations

This is a subject I did not work on for a long time. The central problem is to study from the perpective of differential algebra and model theory, the geodesic differential equations describing the geodesics of real-algebraic Riemannian manifolds.

The state of the art on this question is presented in this talk (BIRS, 2020)