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.

We give universal bounds on the fraction of nontrivial zeros having given multiplicity for L-functions attached to a cuspidal automorphic representation of GLm/Q. For this, we apply the higher-level correlation asymptotic of Hejhal, Rudnick, and Sarnak in conjunction with semidefinite programming bounds.

We study a primal-dual interior point method specialized to clustered low-rank semidefinite programs requiring high precision numerics, which arise from certain multivariate polynomial (matrix) programs through sums-of-squares characterizations and sampling. We consider the interplay of sampling and symmetry reduction as well as a greedy method to obtain numerically good bases and sample points. We apply this to the computation of three-point bounds for the kissing number problem, for which we show a significant speedup. This allows for the computation of improved kissing number bounds in dimensions 11 through 23. The approach performs well for problems with bad numerical conditioning, which we show through new computations for the binary sphere packing problem.