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A model-theoretic analysis of geodesic equations in negative curvature

Banff International Research Station for Mathematical Innovation and Discovery

To any algebraic differential equation, one can associate a first-order structure which encodes some of the properties of algebraic integrability and of algebraic independence of its solutions. To describe the structure associated to a given algebraic (non linear) differential equation (E), typical questions are: Is it possible to express the general solutions of (E) from successive resolutions of linear differential equations? Is it possible to express the general solutions of (E) from successive resolutions of algebraic differential equations of lower order than (E)? Given distinct initial conditions for (E), under which conditions are the solutions associated to these initial conditions algebraically independent? In my talk, I will discuss in this setting one of the simplest examples of non completely integrable Hamiltonian systems: the geodesic motion on an algebraically presented compact Riemannian surface with negative curvature. I will explain a qualitative model-theoretic description of the associated structure (and its content in the differential algebraic language used above) based on the global hyperbolic dynamical properties identified by Anosov in the 70â s (today called Anosov flows) for the geodesic motion in negative curvature.

Mathematics; Mathematical logic and foundations; Algebraic geometry

video/mp4

Author affiliation: University of Notre Dame

2020-12-02

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/76627

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/