Representation Theory and Number Theory Seminar, 2019-2020

Fall 2019: Monday, 3:00-4:00 PM, LCB 215

August 29:
Title: Brief Organizational Meeting

August 26:
Speaker: Stephen Miller, Rutgers
Title: An update on the sphere packing problem
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 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 any nonzero example of such a potential. Beyond giving a new proof of the sphere packing results, Universal Optimality also gives information about long-range interactions. The techniques reduce the problem to a question in single-variable calculus, which is ultimately solved using modular forms and methods introduced by Viazovska. This will be a colloquium-style talk. (Joint work with Henry Cohn, Abhinav Kumar, Danylo Radchenko, and Maryna Viazovska)