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Penn Mathematics Colloquium

Wednesday, December 7, 2022 - 3:45pm

Matthew Stover

Temple University


University of Pennsylvania


Tea will be served in the math lounge, 4E16 DRL

Among closed manifolds admitting a metric of constant sectional curvature the most mysterious are arguably hyperbolic manifolds. Further, they divide into "arithmetic" and "nonarithmetic", but it is not at all evident from the definition, which is rooted in algebra and number theory, whether (non)arithmeticity has anything to do with differential geometry. Recently, Bader, Fisher, Miller and I gave a completely geometric characterization of arithmeticity in terms of properly immersed totally geodesic submanifolds. Assuming only basics from algebra and differential topology, I will first describe the history of our understanding of hyperbolic manifolds and its intimate ties to totally geodesic submanifolds, then how we import tools from dynamics originating in groundbreaking work of Margulis from the 1970s to prove our geometric characterization. Our main tool is a superrigidity theorem for representations of fundamental groups of (real and complex) hyperbolic manifolds, and if there is time I will explain its applications to other topics like geometric properties of Dehn filling, complex geometry, and Hodge theory.