Locally Homogeneous Flows and Anosov Representations - 3/5

Explore the intersection of dynamical systems and geometric group theory in this advanced mathematics lecture that examines locally homogeneous flows and Anosov representations. Learn how Anosov representations, which form an open set of homomorphisms from discrete hyperbolic groups into semi-simple Lie groups, can be understood through a dynamical framework originally introduced by Labourie. Discover recent collaborative work with B. Delarue and A. Sanders that constructs 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. See how linear algebraic techniques from the SL(d,R) case extend to differential geometric concepts in more general settings, with particular emphasis on examples describable as non-Riemannian geodesic flows. Gain insight into how this approach enables the application of modern analytic techniques from smooth dynamics to Anosov representations, opening new avenues for research in this active area of mathematics.