Spherical codes, kissing numbers and linear programming bounds -- Abhinav Kumar, March 28, 2005

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A spherical code is a finite subset of the unit Euclidean sphere
S^{n-1} \subset \mathbb{R}^n. It is said to have angle \geq \phi if any 2
distinct points are separated by a spherical distance \phi. The question
arises: what is the maximum cardinality of a spherical code of angle \geq
\phi? In particular, if \phi = \pi/3, this is equivalent to the kissing
number problem: how many nonoverlapping spheres of the same size can you
fit around a central sphere? The technique of linear programming bounds is
a very useful method for obtaining upper bounds for these problems, and we
will illustrate it with a few remarkable examples. If time permits, we
will also discuss a generalization of packing to potential energy (joint
work with Henry Cohn), and linear programming bounds in that situation.