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This expository paper describes Viazovskas breakthrough solution of the sphere packing problem in eight dimensions, as well as its extension to twenty-four dimensions by Cohn, Kumar, Miller, Radchenko, and Viazovska.
We obtain new restrictions on the linear programming bound for sphere packing, by optimizing over spaces of modular forms to produce feasible points in the dual linear program. In contrast to the situation in dimensions 8 and 24, where the linear pro
We prove that for any isometric action of a group on a unit sphere of dimension larger than one, the quotient space has diameter zero or larger than a universal dimension-independent positive constant.
We solve the oscillation stability problem for the Urysohn sphere, an analog of the distortion problem for the Hilbert space in the context of the Urysohn universal metric space. This is achieved by solving a purely combinatorial problem involving a
Consider a random set of points on the unit sphere in $mathbb{R}^d$, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the case $d=3$,
In 1969, Fejes Toth conjectured that in Euclidean 3-space any packing of equal balls such that each ball is touched by twelve others consists of hexagonal layers. This article verifies this conjecture.