How should one distribute a certain number of points over a sphere, so that they are well separated from each other? One natural method is energy minimization: put an electric charge on each point and let them repel each other. This problem arises naturally in physics, but it extends to far more general spaces and potential functions; it can be viewed as a broad generalization of packing problems. Some of the most remarkable exceptional structures in mathematics (such as the E_8 root system, the Leech lattice minimal vectors, the Schlaefli configuration of 27 points in R^6 related to the 27 lines on a cubic surface, the icosahedron, and the regular 600-cell) are solutions of energy minimization problems of this sort. These examples have a rare property called universal optimality: they simultaneously minimize a broad class of potential functions.
This talk will survey what's known and conjectured about universally optimal configurations.