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Explore the geometry of Anosov flows and their rigidity properties in this advanced mathematical lecture delivered by Simion Filip from the University of Chicago at IHES. Begin with an introduction to the fundamental concepts of Anosov flows, using the geodesic flow on manifolds of negative sectional curvature as the archetypal example, where every vector experiences uniform expansion or contraction. Examine the geometric structure of these dynamical systems, including invariant manifolds, ergodicity properties, and various regularity questions that arise in their study. Delve into flows that are compact group extensions of Anosov flows and discover the associated "Brin group," which functions as a type of Galois group for the extension. Learn how these sophisticated techniques apply to prove a significant rigidity result obtained jointly with David Fisher and Ben Lowe: if a compact negatively curved real-analytic Riemannian manifold contains infinitely many totally geodesic hypersurfaces, then it must have constant sectional curvature. Gain insight into the deep connections between dynamical systems theory, differential geometry, and rigidity phenomena in modern mathematics.
