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The Seiberg-Witten equations and the length spectrum of hyperbolic three-manifolds

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Tuesday, October 02, 2018
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3:15 pm - 4:15 pm
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Michael Lipnowski (McGill University)
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Geometry/topology Seminar

We exhibit examples of hyperbolic three-manifolds for which the Seiberg-Witten equations do not admit any irreducible solution. Our approach relies hyperbolic geometry in an essential way; it combines an explicit upper bound for the first eigenvalue on coexact 1-forms \lambda_1^* on rational homology spheres which admit irreducible solutions together with a version of the Selberg trace formula relating the spectrum of the Laplacian on coexact 1-forms with the volume and complex length spectrum of a hyperbolic three-manifold. Using these relationships, we also provide precise certified numerical bounds on \lambda_1^* for several hyperbolic rational homology spheres.