Locally Homogeneous Flows and Anosov Representations - 4/5

Explore the intersection of dynamical systems and geometric group theory in this advanced mathematics lecture that examines locally homogeneous flows and Anosov representations. Delve into how Anosov representations, which form an open set of homomorphisms from discrete hyperbolic groups into semi-simple Lie groups, can be understood through dynamical language and geometric structures. Learn about recent collaborative research with B. Delarue and A. Sanders that utilizes non-compact quotients of homogeneous spaces equipped with flows commuting with Lie group actions, producing locally homogeneous flows with uniformly hyperbolic dynamics satisfying Smale's axiom A. Begin with projective Anosov representations into SL(d,R), understanding their construction and dynamical properties, before progressing to the general case of arbitrary semi-simple Lie groups and flag manifolds. Discover how linear algebra techniques from the SL(d,R) case translate to differential geometric concepts in more general settings, with particular emphasis on examples describable as non-Riemannian geodesic flows. Gain insights into how this approach enables the application of modern analytic techniques from smooth dynamics to Anosov representations, opening new avenues for research in this field.