**Title: **Automorphic Spectra and the Conformal Bootstrap

**Abstract: **I will explain that the spectral geometry of hyperbolic manifolds provides a remarkably faithful model of the modern conformal bootstrap. In particular, to each hyperbolic manifold, one can associate a Hilbert space of local operators, which is a unitary representation of a conformal group. The scaling dimensions of the operators are related to the eigenvalues of the Laplacian on the manifold. The operators satisfy an operator product expansion. Finally, one can define their correlation functions and derive bootstrap equations constraining the spectrum. As an application, I will use conformal bootstrap techniques to derive upper bounds on the lowest positive eigenvalue of the Laplacian on closed hyperbolic surfaces and 2-orbifolds. In a number of notable cases, the bounds are nearly saturated by known surfaces and orbifolds. For instance, the bound on all genus-2 surfaces is λ1≤3.8388976481, while the Bolza surface has λ1≈3.838887258. The talk will be based on https://arxiv.org/abs/2111.12716, which is joint work with P. Kravchuk and S. Pal.