Mathematics

The spectral gap of the Hodge Laplacian on coexact 1-forms is a fundamental quantity associated to a Riemannian manifold which has attracted quite a lot of attention in recent years. I will begin by discussing the role it plays in topological problems about three-manifolds (such as the existence on taut foliations on rational homology spheres) via its relation with Floer theoretic invariants. Motivated by this, I will then focus on the case of hyperbolic manifolds, and describe techniques to determine explicitly such spectral gaps in concrete examples (including certain infinite families). This is joint work with M. Lipnowski.