On the computation of three and four-point bounds in discrete geometry and analytic number theory
N.M. Leijenhorst (TU Delft - Discrete Mathematics and Optimization)
Dion Gijswijt – Promotor (TU Delft - Discrete Mathematics and Optimization)
D. de Laat – Copromotor (TU Delft - Discrete Mathematics and Optimization)
More Info
expand_more
Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons.
Abstract
In this thesis, we consider extremal problems in discrete geometry, and in particular, the Lasserre hierarchies for such problems. We give a unifying framework that encompasses the known hierarchies, and lay the foundation to use the second level of such hierarchies for problems on the sphere in practice. We perform explicit harmonic analysis and use polynomial optimization techniques to reduce the problems to a finite-dimensional semidefinite program. We introduce a specialized semidefinite programming solver that uses the structure of the problems, allowing us to use polynomials of significantly higher degree than previously possible, and a much faster rounding procedure to obtain exact optimal solutions to the semidefinite programs.
We use this to prove that the D4 root system is the unique optimal solution to the kissing number problem in dimension 4, and is an optimal spherical code. We also prove there are exactly two optimal spherical codes with 12 points in dimension 4. Furthermore, we show that the spectral embeddings of all known triangle-free strongly regular graphs are optimal spherical codes, as well as certain Kerdock spherical codes. We give numerical evidence that the second level of the Lasserre hierarchy for minimizing harmonic energy is sharp for several infinite families of configurations.
We also investigate the strength of the hierarchy for the polarization problem. Finally, we consider triple and quadruple correlation bounds in analytic number theory, which gives new bounds on the fraction of double and triple zeros of the Riemann ζ-function and related functions.