Welcome to my webpage!

My name is Rémi Jaoui and I am a researcher working in the Institut Camille Jordan of the Université Claude Bernard Lyon 1 on the interaction between model theory, a branch of mathematical logic, and differential algebra.

Autonomous algebraic differential equations

A central object of my research are systems of autonomous algebraic differential equations which are differential equations of the form \[ P(y,y’, \ldots, y^{(n)}) = 0 \] where \(P\) is a polynomial with complex coefficients . Such a differential equation can be represented geometrically as a complex algebraic variety X (defined by \( P = 0 \) and \( \frac {\partial P} {\partial x_n} \neq 0 \)) endowed with a vector field \( v\).

Similarly, one can represent differential equations appearing in classical mechanics in this form, as long as they only involve algebraic functions. This is the case of the geodesic flows, the movement of a rotating solid and many variants of the n-body problems. Such equations are often represented as pairs \( (M,v_H) \), where \( (M,\omega) \) is a symplectic variety and \( v_H \) is the Hamiltonian function \( H \in \mathcal O (M) \)

Some motivating questions on autonomous differential equations \( (X,v) \) are:

• Can the differential equation \((X,v)\) be solved using only “classical functions”? If it is possible, can it be achieved algorithmically?
• Given a finite collection of differential equations \( (X_1,v_1), \ldots , (X_r,v_r) \), can \((X,v)\) be solved using only “classical functions” and solutions of \( (X_1,v_1), \ldots , (X_r,v_r) \) ? If it is possible, can it be achieved algorithmically?
• Do the solutions of \( (X,v) \) form large independent sets of meromorphic functions or do they share many algebraic relations?

Geometric stability theory

A characteristic feature of these questions is that they can be formulated in the language of geometric stability theory at the level of the definable set associated to a differential equation \( (X,v) \) in the theory \( DCF_0\). So my point of view to study these questions is to use tools from the classical theory of differential equations and amplify them them using tools of geometric stability theory.

In the language of model theory, motivating questions are:

• Locate classical differential equations on the map of differentially closed fields provided by Zilber’s trichotomy.
• Develop effective tools to study the generic solutions of algebraic differential equation using geometric linearization data along a known particular solution.
• Study the variations of the semi-minimal analysis of a differential equations in families.