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The classification of totally geodesic (i.e., rigid) submanifolds is a fundamental question in Riemannian geometry. Following work of Eskin, McMullen, Mukamel, and Wright, these submanifolds have played a fundamental role in the discovery of new phenomena in Teichmüller dynamics. I will discuss recent progress towards a complete classification of these submanifolds. This is joint work with Alex Wright. No previous familiarity with the subject will be assumed.