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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 recent work utilizing non-compact quotients of homogeneous spaces equipped with flows that commute with group actions. Learn about the construction of locally homogeneous flows with uniformly hyperbolic dynamics that satisfy Smale's axiom A, enabling the application of modern analytic techniques from smooth dynamics to Anosov representations. Begin with projective Anosov representations into SL(d,R), understanding their definition and the construction of associated locally homogeneous flows with their dynamical properties. Progress to the general case of arbitrary semi-simple Lie groups and flag manifolds, discovering how linear algebra concepts from the SL(d,R) case translate to differential geometric notions in broader contexts. Examine concrete examples with emphasis on non-Riemannian geodesic flows, while gaining exposure to essential concepts from Lie theory, dynamical systems, differential geometry, and geometric group theory presented with minimal background requirements.
