Tag - Leech lattice

Dalimil Mazac: Sphere Packings, Spectral Gaps and the Conformal Bootstrap

8th November, 2023

I will discuss infinite-dimensional linear programs producing bounds on the spectral gap in various settings. This includes new bounds on the spectral gap of hyperbolic manifolds as well as the Cohn-Elkies bound on the density of sphere packings. The bounds allow us to essentially determine the complete set of spectral gaps achieved by hyperbolic 2-orbifolds. The linear programs involved have been the subject of intense study by mathematical physicists in the context of the conformal bootstrap.

I will review the method of analytic extremal functionals, introduced by the speaker to prove sharp bounds in the conformal bootstrap. When used within the Cohn-Elkies linear program, this method reproduces the groundbreaking solution of Viazovska et al of the sphere packing problem in dimensions 8 and 24, as well as the interpolation basis used in the proof of universal optimality of the E8 and Leech lattice. The connections covered in this talk offer a broader framework for studying optimality in infinite-dimensional linear programs.

Ching Hung Lam: A lattice theoretical interpretation of generalized deep holes

6th June, 2022

We will give a lattice-theoretical interpretation of generalized deep holes of the Leech lattice VOA VΛ. We show that a generalized deep hole defines a 'true' automorphism invariant deep hole of the Leech lattice. We will also discuss a correspondence between the set of isomorphism classes of holomorphic VOA V of central charge 24 having non-abelian V1 and the set of equivalence classes of pairs (𝜏,β̃ ) satisfying certain conditions, where 𝜏∈Co0 and β̃ is a 𝜏-invariant deep hole of squared length 2. It provides a new combinatorial approach towards the classification of holomorphic VOAs of central charge 24. Finally, we will discuss an observation of G. Höhn, which relates the weight one Lie algebra of holomorphic VOAs of central charge 24 to certain codewords associated with the glue codes of Niemeier lattices.