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.@en

We propose a hierarchy of k-point bounds extending the Delsarte–Goethals–Seidel linear programming 2-point bound and the Bachoc–Vallentin semidefinite programming 3-point bound for spherical codes. An optimized implementation of this hierarchy allows us to compute 4, 5, and 6-point bounds for the maximum number of equiangular lines in Euclidean space with a fixed common angle.

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Assuming the generalized Riemann hypothesis, we give asymptotic bounds on the size of intervals that contain primes from a given arithmetic progression using the approach developed by Carneiro, Milinovich and Soundararajan [Comment. Math. Helv. 94, no. 3 (2019)]. For this we extend the Guinand-Weil explicit formula over all Dirichlet characters modulo q ≥ 3, and we reduce the associated extremal problems to convex optimization problems that can be solved numerically via semidefinite programming.

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In this paper we give an algorithm to round the floating point output of a semidefinite programming solver to a solution over the rationals or a quadratic extension of the rationals. This algorithm does not require the solution to be strictly feasible and works for large problems. We apply this to get sharp bounds for packing problems, and we use these sharp bounds to prove that certain optimal packing configurations are unique up to rotations. In particular, we show that the configuration coming from the E₈ root lattice is the unique optimal code with minimal angular distance π/3 on the hemisphere in R⁸, and we_prove that the three-point bound for the (3, 8, ϑ)-spherical code, where ϑ is such that cos ϑ = (2√2 - 1)/7, is sharp by rounding to Q[√2]. We also use our machinery to compute sharp upper bounds on the number of spheres that can be packed into a larger sphere.

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In this paper we study the distribution of the non-trivial zeros of the Riemann zeta-function (and other L-functions) using Montgomery's pair correlation approach. We use semidefinite programming to improve upon numerous asymptotic bounds in the theory of the zeta-function, including the proportion of distinct zeros, counts of small gaps between zeros, and sums involving multiplicities of zeros.@en

We carry out a numerical study of the spinless modular bootstrap for conformal field theories with current algebra U(1)^c× U(1)^c, or equivalently the linear programming bound for sphere packing in 2c dimensions. We give a more detailed picture of the behavior for finite c than was previously available, and we extrapolate as c → ∞. Our extrapolation indicates an exponential improvement for sphere packing density bounds in high dimen- sions. Furthermore, we study when these bounds can be tight. Besides the known cases c = 1/2, 4, and 12 and the conjectured case c = 1, our calculations numerically rule out sharp bounds for all other c < 90, by combining the modular bootstrap with linear programming bounds for spherical codes.

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We use moment techniques to construct a converging hierarchy of optimization problems to lower bound the ground state energy of interacting particle systems. We approximate (from below) the infinite-dimensional optimization problems in this hierarchy by block diagonal semidefinite programs. For this we develop the necessary harmonic analysis for spaces consisting of subsets of another space, and we develop symmetric sum-of-squares techniques. We numerically compute the second step of our hierarchy for Riesz s-energy problems with five particles on the two-dimensional unit sphere, where the s = 1 case is the Thomson problem. This yields new numerically sharp bounds (up to high precision) and suggests that the second step of our hierarchy may be sharp throughout a phase transition and may be universally sharp for five particles on the unit sphere. This is the first time a four-point bound has been computed for a problem in discrete geometry.

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In this thesis we develop techniques for solving problems in extremal geometry. We give an infinite dimensional generalization of moment techniques from polynomial optimization. We use this to construct semidefinite programming hierarchies for approximating optimal packing densities and ground state energies of particle systems. For this we define topological packing graphs as an abstraction for the graphs arising from geometric packing problems, and we prove results concerning convergence and strong duality. We use harmonic analysis to perform symmetry reduction and reduce to a finite dimensional variable space in the optimization problems. For this we explicitly work out the harmonic analysis for kernels on spaces consisting of subsets of another space. We show how sums of squares characterizations from real algebraic geometry can be used to reduce the infinitely many constraints to finitely many semidefinite constraints, where we focus in particular on numerical conditioning and symmetry reduction. We perform explicit computations for concrete problems: We give new bounds for binary spherical cap packings, binary sphere packings, and classical sphere packing problems. This can be used, for instance, to give a simple optimality proof of a binary spherical cap packing. We compute the second step of our hierarchy where the numerical results suggest the bound is sharp for the 5-particle case of the Thomson and related problems. This is the first time a 4-point bound has been computed for a continuous problem.@en