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Title:
Sphere packing, Fourier interpolation, and the Universal Optimality Theorem
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Abstract:
I will discuss recent work on the optimal arrangement of points in euclidean space. In addition to the solution to the sphere packing problem in dimensions 8 and 24 from 2016, the "Universal Optimality" conjecture has now been proved in these dimensions as well. This shows that E8 and the Leech lattice minimize energy for any completely monotonic function of distance-squared, a fact which was previously not known for any configuration of points in any dimension > 1. Beyond giving a new proof of these sphere packing results, Universal Optimality also gives information about long-range interactions. Another application is to find the global minimum of the log-determinant of the laplacian among flat tori in those dimensions. The techniques involve arranging both a function and its Fourier transform to vanish at certain points, which leads to a new interpolation formula that recovers a radial Schwartz function from the values of it, its Fourier transform, and their derivatives, at special arithmetic points. Finally, fitting with the theme of the "Automorphic Structure" workshop, the interpolation formula reduces to an identity involving modular forms. (joint with Henry Cohn, Abhinav Kumar, Danylo Radchenko, and Maryna Viazovska)
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