Poppy-Seed Bagel Theorem

The poppy-seed bagel theorem, in potential theory, concerns large numbers of mutually repelling particles confined to a bounded surface or body, where the repulsion between two particles falls off as a power of the distance between them governed by a parameter s. The theorem states that once s is at least as large as the dimension of the set the particles are confined to, the particles' equilibrium configuration becomes nearly uniformly distributed across the set as their number grows, the same conclusion that would place poppy seeds uniformly across a two-dimensional bagel surface if the seeds obeyed at least an inverse-square repulsion. Applications of the underlying mathematics include Coulomb's law in electrostatics, Riesz potential theory, and the generalized Thomson problem of arranging repelling points on a sphere; in 2022 the mathematician Maryna Viazovska and coauthors used modular forms and linear programming to determine the exact constants involved in dimensions 8 and 24, the dimensions of the E8 and Leech lattices.