Sphere packing, Fourier interpolation, and the Universal Optimality Theorem

Stephen Miller, Rutgers University
Fine Hall 314

I will discuss recent work on the optimal arrangement of points in euclidean space. In addition to the proof of the sphere packing problem in dimensions 8 and 24 from 2016, the "Universal Optimality" conjecture has now been established in these dimensions as well. This shows that E8 and the Leech lattice minimize energy for any completely monotonic function of distance-squared.  Previously there were no proofs in any dimension > 1 that a particular configuration minimizes energy for some example of such a potential. Beyond giving a new proof of the 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, confirming a conjecture of Sarnak and Strombergsson. 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 at special arithmetic points of it, its Fourier transform, and their derivatives.

(Joint work with Henry Cohn, Abhinav Kumar, Danylo Radchenko, and Maryna Viazovska)