Abstract：I will discuss some new bounds on the spectra of Laplacian operators on hyperbolic 3-manifolds. One example of such a bound is that the spectral gap of the Laplace-Beltrami operator on a closed orientable hyperbolic 3-manifold must be less than 47.32, or less than 31.57 if the first Betti number is positive. The bounds are derived using two approaches, both of which employ linear prog...
Let X be a compact arithmetic hyperbolic 3-manifold. Let f be a Hecke-Maass form on X, which is a joint eigenfunction of the Laplacian and Hecke operators. In this talk, I will talk about the L^2 restriction norm problems of f with high frequency. I will present a power-saving bound over the local bound by Burq, Gérard, and Tzvetkov for the L^2 norm of f restricted to a totally geodesic surface...
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