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Explore the geometric properties of Anosov flows and their applications to rigidity problems in this advanced mathematics lecture. Begin with an introduction to Anosov flows as dynamical systems where vectors experience uniform expansion or contraction, using the geodesic flow on negatively curved manifolds as a fundamental example. Learn about key concepts including invariant manifolds, ergodicity, and regularity questions that arise in the study of these systems. Discover compact group extensions of Anosov flows and understand the role of the associated "Brin group," which functions as a Galois group for these extensions. Apply these theoretical tools to examine a significant result demonstrating that compact negatively curved real-analytic Riemannian manifolds with infinitely many totally geodesic hypersurfaces must have constant sectional curvature, a theorem developed collaboratively with David Fisher and Ben Lowe.
