Locally Homogeneous Flows and Anosov Representations - 1/5

Explore the foundational concepts of Anosov representations and their connection to locally homogeneous flows in this first lecture of a five-part series. Begin with an introduction to Anosov representations as open sets of homomorphisms from discrete hyperbolic groups into semi-simple Lie groups, originally formulated by Labourie using dynamical language. Discover how these representations require sections of associate flat bundles to provide hyperbolic sets for flows, and examine the evolution toward equivalent characterizations that avoid flow dynamics while producing geometric structures through compact quotients of flag manifold subsets. Learn about recent collaborative work with B. Delarue and A. Sanders that employs non-compact quotients of homogeneous spaces equipped with G-commuting flows, creating locally homogeneous flows with uniformly hyperbolic dynamics satisfying Smale's axiom A. Understand how this innovative approach enables the application of modern smooth dynamics analytical techniques previously inaccessible to Anosov representations. Focus initially on projective Anosov representations into SL(d,R), examining the construction and dynamical properties of locally homogeneous flows, before preparing for subsequent lectures covering general semi-simple Lie groups and arbitrary flag manifolds. Gain insight into how linear algebra concepts from the SL(d,R) case translate to differential geometric notions in broader contexts, with examples emphasizing non-Riemannian geodesic flows, while building essential background in Lie theory, dynamical systems, differential geometry, and geometric group theory.