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Harmonic 1-forms on hyperbolic 3-manifolds: connections and computations


Speaker: Nathan Dunfield, U. Illinois Urbana Champaign
Location: Zoom
Start time: Tue, Jan 26 2021, 1:10PM

Each cohomology group of a Riemannian manifold has a natural inner product, namely the $L^2$-inner product on harmonic representatives. If we want to understanding torsion growth in hyperbolic 3-manifolds, as in the conjectures of Bergeron-Venkatesh, Lê, and Lück, it is crucial to understand this “harmonic norm” on H^1 as it forms an important component of the Ray-Singer analytic torsion. I will first sketch these connections in the context of experimental evidence for torsion growth. Then, I will relate the harmonic norm to the purely topological Thurston norm, following Bergeron-Şengün-Ventkatesh and my own work with Brock. I will end by describing work in progress with Anil Hirani to numerically compute harmonic 1-forms on a few thousand hyperbolic 3-manifolds and show some preliminary results. No prior knowledge about these topics will be assumed beyond basics about hyperbolic manifolds, lattices in Lie groups, and differential forms (specifically, the Hodge Theorem).