Locally Homogeneous Flows and Anosov Representations - 2/5

Explore the second lecture in a comprehensive mathematical series examining the intersection of dynamical systems and geometric group theory through the lens of Anosov representations. Delve into the fundamental concepts of Anosov representations as open sets of homomorphisms from discrete hyperbolic groups into semi-simple Lie groups, originally introduced by Labourie using dynamical language. Learn how these representations require sections of associate flat bundles to provide hyperbolic sets for specific flows, and discover the various equivalent characterizations that have emerged to avoid direct involvement with flow dynamics. Examine recent collaborative work with B. Delarue and A. Sanders that utilizes non-compact quotients of open subsets of homogeneous spaces equipped with flows commuting with Lie group actions, producing locally homogeneous flows with uniformly hyperbolic dynamics satisfying Smale's axiom A. Focus initially on projective Anosov representations into SL(d,R), understanding both the construction of locally homogeneous flows and their 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 situations, with particular attention to examples describable as non-Riemannian geodesic flows. Gain insights into how this approach enables the application of modern analytic techniques from smooth dynamics that were previously inapplicable to Anosov representations, while building foundational knowledge in Lie theory, dynamical systems, differential geometry, and geometric group theory.