Yuval Peres (Microsoft Research) Gravitational allocation to uniform points on the Sphere

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Given $n$ uniform points on the surface of a sphere, how can we partition the sphere fairly among them in an equivariant way? "Fairly" means that each region has the same area. "Equivariant" means that if we rotate the sphere, the solution rotates along with the points. It turns out that if the given points apply a two-dimensional gravity force to the rest of the sphere, then the basins of attraction for the resulting gradient flow yield such a partition—with exactly equal areas, no matter how the points are distributed. Moreover, this partition minimizes, up to a bounded factor, the average distance between points in the same cell. I will also present an application to almost optimal matching of $n$ blue uniform points to $n$ red uniform points on the sphere. This is joint work with Nina Holden and Alex Zhai.