Bootstrapping hyperbolic manifolds

2022-02-16 18:09 - 19:09

Abstract: Hyperbolic manifolds are a class of Riemannian manifolds that are important in mathematics and physics, playing a prominent role in topology, number theory, and string theory. Associated with a given hyperbolic metric is a sequence of numbers corresponding to the discrete eigenvalues of the Laplace-Beltrami operator. While these eigenvalues usually cannot be calculated exactly, they can be found numerically and must also satisfy various bounds. In this talk, I will discuss a new approach for finding numerical bounds on the eigenvalues of closed hyperbolic manifolds using general consistency conditions and semidefinite programming, inspired by the approach of the conformal bootstrap from physics. Although these bootstrap bounds follow from seemingly trivial consistency conditions, they are surprisingly strong and are sometimes almost saturated by actual manifolds; for example, one such bound implies that the first nonzero eigenvalue of a closed hyperbolic surface must be less than 3.83890, and this is very close to being saturated by a particular genus-2 surface called the Bolza surface. I will show how to derive this and other bounds and will discuss some possible future directions for this approach.