Título: Gravitational allocation of uniform points on the sphere

Palestrante: Yuval Peres (Kent State University)
Data: 12/04/2021
Horário: 15:00h
Local: Transmissão online

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Resumo: Given uniform points on the surface of a two-dimensional sphere, how can we partition the sphere fairly among them? "Fairly" means that each region has the same area. 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. This is related to work of Nazarov-Sodin-Volberg on Gaussian analytic functions. (See the cover of the AMS Notices at http://www.ams.org/publications/journals/notices/201705/rnoti-cvr1.pdf.) Our main result is that 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 on the sphere, connecting to a classical result of Ajtai, Komlos and Tusnady (Combinatorica 1984).

Joint work with Nina Holden and Alex Zhai.