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Home » RAS - Research Seminar (Part 2): Harmonic 1-forms on hyperbolic 3-manifolds: connections and computations

Seminar

RAS - Research Seminar (Part 2): Harmonic 1-forms on hyperbolic 3-manifolds: connections and computations November 23, 2020 (10:00 AM PST - 11:00 AM PST)
Parent Program:
Location: MSRI: Online/Virtual
Speaker(s) Nathan Dunfield (University of Illinois at Urbana-Champaign)
Description

To participate in this seminar, please register here: https://www.msri.org/seminars/25205

This is one of the research seminars for the RAS program, that distinguishes itself from the postdocs and program associates seminars in that speakers are chosen among Research Members, Research Professors with occasional outside speakers.

Video

Harmonic 1-Forms On Hyperbolic 3-Manifolds: Connections And Computations

Abstract/Media

To participate in this seminar, please register here: https://www.msri.org/seminars/25205

This is one of the research seminars for the RAS program, that distinguishes itself from the postdocs and program associates seminars in that speakers are chosen among Research Members, Research Professors with occasional outside speakers.

Abstract: 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).

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Harmonic 1-Forms On Hyperbolic 3-Manifolds: Connections And Computations

H.264 Video 25358_28916_8653_Harmonic_1-Forms_on_Hyperbolic_3-Manifolds_Connections_and_Computations.mp4