We discuss some recent progress in the smooth ergodic theory of geodesic flows. We will start with an intuitive overview of the classic results developed by luminaries such as Anosov, Bowen and Ruelle in the well understood setting of surfaces with variable negative curvature. Efforts to understand the much more difficult case of non-positive curvature were initiated by Pesin in the 1970's. However, despite substantial successes, the picture has remained far from complete. There has been a great deal of recent progress in this area, which has required, and motivated, the development of new machinery in the abstract theory. I will give an overview of some recent developments, including:

1) General machinery developed by Vaughn Climenhaga and myself, which gives "non-uniform" dynamical criteria for uniqueness of equilibrium measures;

2) Joint work with Keith Burns, Vaughn Climenhaga and Todd Fisher, where we apply this machinery to geodesic flow on non-positive curvature manifolds;

3) If time permits, I will also mention related joint work with Jean-Francois Lafont and Dave Constantine, where we develop the theory of equilibrium measures for geodesic flow on locally CAT(-1) spaces; these are geodesic metric spaces which generalize negative curvature Riemannian manifolds by having the "thin triangle" property.

This is in some sense both a prequel and a sequel to a talk I gave in the RTG seminar in December 2016. Speaker(s): Dan Thompson (The Ohio State University)