In this talk we give background on the dynamics of the geodesic flow for locally symmetric spaces of variable negative curvature (such as complex hyperbolic manifolds). We describe the structures associated to the geodesic flow which survive under perturbation of the metric. We then relate the structure of the Lyapunov spectrum of the geodesic flow of these perturbed metrics to the existence of certain horizontal quasiconformal structures on unstable horospheres. We will use these structures in the second talk to locally characterize closed negatively curved locally symmetric spaces up to isometry by their Lyapunov spectra. Speaker(s): Clark Butler (U Chicago)