Remi Jaoui, Department of Pure Mathematics, University of Waterloo
"On the model theory of geodesic differential equations I"
Geodesy is an old and well-established subject in mathematics. Given a smooth manifold $S$ embedded in the Euclidean space $\mathbb R^n$ and two points on $S$, the main question is to determine the shortest path (geodesic) drawn on $S$ joining these two points.
It is well-known that all the geodesics of $S$ satisfy the same differential equation (namely, the equation describing the movement of a particle constrained to move without friction along the manifold $S$). Of course, the behavior of this differential equation and of its solutions heavily depends on the geometric properties of $S$.
In this first talk, I will state a (still incomplete) description of the model-theoretic properties of this algebraic differential equation when $S$ is a smooth and compact real-algebraic surface with negative curvature. Then, I will explain the core of the model-theoretic strategy leading to the aforementioned description.
MC 5403