Mathematics Colloquia and Seminars

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We describe a new method for constraining Laplacian spectra of hyperbolic surfaces and 2-orbifolds and Dirac spectra of hyperbolic spin surfaces and 2-orbifolds. The key ingredient is an infinite collection of spectral identities satisfied by integrated triple products of automorphic forms. These spectral identities follow the consistency of associativity of multiplication of functions on $\Gamma\PSL_2(\mathbb R)$ and $\tilde \Gamma\SL_2(\mathbb R)$ (for the spin surface) with the spectral decomposition, where $\Gamma$ is cocompact Fuchsian group and $\tilde\Gamma$ is a lift of $\Gamma$ inside $SL_ 2(\mathbb R)$. These spectral identities, with the use of semidefinite programming, produce rigorous upper bounds on the Laplacian spectral gap as well as on the Dirac spectral gap conditioned on the former. In several examples, the bound is nearly sharp. For instance, our bound on all genus-2 surfaces is $\lambda_1 ⩽ 3.8388976481$, while the Bolza surface has $\lambda_1 ≈ 3.838887258.$ A numerical algorithm based on the Selberg trace formula shows the [0; 3, 3, 5] orbifold and the Bolza surface nearly saturate the bounds on Dirac spectral gap, conditioned on Laplacian spectral gap at genus 0 and 2 respectively. We can further produce rigorous bounds on Laplacian spectral gaps for any genus g surface as well as refined bounds on any genus g hyperelliptic surface using the harmonic spinors. The bounds also allow us to determine the set of Laplacian spectral gaps attained by all hyperbolic 2-orbifolds and the set of Laplacian spectral gaps attained by all hyperbolic spin 2-orbifolds. We show that given a hyperbolic 2-orbifold, the Laplacian spectral gap is upper bounded by 44.8883537 and given a hyperbolic 2-orbifold, equipped with spin structure, the Laplacian spectral gap is upper bounded by 12.13798; these bounds are nearly saturated by [0; 2, 3, 7], having $λ_1 ≈ 44.88835$ and [0; 3, 3, 5] having $λ_1 ≈ 12.13623$ respectively. If time permits, I will talk about the extension of our method for deriving bounds on the low-energy spectra of Laplacians on powers of the cotangent bundle on compact hyperbolic 3-manifolds and orbifolds. The ideas were closely inspired by modern conformal bootstrap. The talk will be based on the joint work arXiv:2111.12716[hep-th] with Dalimil Mazac, Petr Kravchuk and ongoing joint work with Elliott Gesteau, David Simmons-Duffin, Yixin Xu as well as arXiv:2308.11174 [math.sp] with James Bonifacio, Dalimil Mazac.