Three-point bounds for energy minimization

Three-point semidefinite programming bounds are one of the most powerful known tools for bounding the size of spherical codes. In this paper, we use them to prove lower bounds for the potential energy of particles interacting via a pair potential function. We show that our bounds are sharp for seven points in RP^2. Specifically, we prove that the seven lines connecting opposite vertices of a cube and of its dual octahedron are universally optimal. (In other words, among all configurations of seven lines through the origin, this one minimizes energy for all potential functions that are completely monotonic functions of squared chordal distance.) This configuration is the only known universal optimum that is not distance regular, and the last remaining universal optimum in RP^2. We also give a new derivation of semidefinite programming bounds and present several surprising conjectures about them.Comment: 30 page

Experimental study of energy-minimizing point configurations on spheres

In this paper we report on massive computer experiments aimed at finding spherical point configurations that minimize potential energy. We present experimental evidence for two new universal optima (consisting of 40 points in 10 dimensions and 64 points in 14 dimensions), as well as evidence that there are no others with at most 64 points. We also describe several other new polytopes, and we present new geometrical descriptions of some of the known universal optima.Comment: 41 pages, 12 figures, to appear in Experimental Mathematic

The Lasserre hierarchy for equiangular lines with a fixed angle

We compute the second and third levels of the Lasserre hierarchy for the spherical finite distance problem. A connection is used between invariants in representations of the orthogonal group and representations of the general linear group, which allows computations in high dimensions. We give new linear bounds on the maximum number of equiangular lines in dimension $n$ with common angle $\arccos \alpha$. These are obtained through asymptotic analysis in $n$ of the semidefinite programming bound given by the second level.Comment: 25 pages, 2 figures. Submitted versio

Exact semidefinite programming bounds for packing problems

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. 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 $\mathsf{E}_8$ root lattice is the unique optimal code with minimal angular distance $\pi/3$ on the hemisphere in $\mathbb R^8$, and we prove that the three-point bound for the $(3, 8, \vartheta)$-spherical code, where $\vartheta$ is such that $\cos \vartheta = (2\sqrt{2}-1)/7$, is sharp by rounding to $\mathbb Q[\sqrt{2}]$. We also use our machinery to compute sharp upper bounds on the number of spheres that can be packed into a larger sphere.Comment: 24 page

Spherical two-distance sets and related topics in harmonic analysis

This dissertation is devoted to the study of applications of harmonic analysis. The maximum size of spherical few-distance sets had been studied by Delsarte at al. in the 1970s. In particular, the maximum size of spherical two-distance sets in $\mathbb{R}^n$ had been known for $n \leq 39$ except $n=23$ by linear programming methods in 2008. Our contribution is to extend the known results of the maximum size of spherical two-distance sets in $\mathbb{R}^n$ when $n=23$, $40 \leq n \leq 93$ and $n \neq 46, 78$. The maximum size of equiangular lines in $\mathbb{R}^n$ had been known for all $n \leq 23$ except $n=14, 16, 17, 18, 19$ and $20$ since 1973. We use the semidefinite programming method to find the maximum size for equiangular line sets in $\mathbb{R}^n$ when $24 \leq n \leq 41$ and $n=43$. We suggest a method of constructing spherical two-distance sets that also form tight frames. We derive new structural properties of the Gram matrix of a two-distance set that also forms a tight frame for $\mathbb{R}^n$. One of the main results in this part is a new correspondence between two-distance tight frames and certain strongly regular graphs. This allows us to use spectral properties of strongly regular graphs to construct two-distance tight frames. Several new examples are obtained using this characterization. Bannai, Okuda, and Tagami proved that a tight spherical designs of harmonic index 4 exists if and only if there exists an equiangular line set with the angle $\arccos (1/(2k-1))$ in the Euclidean space of dimension $3(2k-1)^2-4$ for each integer $k \geq 2$. We show nonexistence of tight spherical designs of harmonic index $4$ on $S^{n-1}$ with $n\geq 3$ by a modification of the semidefinite programming method. We also derive new relative bounds for equiangular line sets. These new relative bounds are usually tighter than previous relative bounds by Lemmens and Seidel