Remi Jaoui, Department of Pure Mathematics, University of Waterloo
"On the model theory of geodesic differential equations II"
Geodesic differential equations form a classical family of differential equations, describing the free movement of a particle in a curved space. The purpose of this talk, divided in two parts, is to describe the possible model-theoretic behavior of such differential equations when the space is compact with negative curvature.
Using model-theoretic stability tool-box and Hrushovski-Sokolovic's results on differentially closed fields, I explained in my first talk how to reduce such a description to the study of rational factors and rational integrals for these differential equations.
A key feature of this study is to understand what may happen for the indeterminacy locus, where the rational morphism is not defined. If it is natural to expect that rational factors and rational integrals constrain heavily the possible dynamics, outside of the indeterminacy locus, one also needs to understand what the indeterminacy locus may look like.
In this second talk, I will prove some properties of the indeterminacy locus for both rational integrals and (variants of) rational factors. Then, I will explain how to deduce the aforementioned properties for geodesic differential equations in negative curvature from their hyperbolic dynamical properties, the so-called Anosov structure.
MC 5403