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 programming techniques: 1) the Selberg trace formula, and 2) identities derived from the associativity of spectral decompositions. The second approach is inspired by an area of physics called the conformal bootstrap. This talk is based on work done in collaboration with Dalimil Mazáč and Sridip Pal.