# Representation Theory and Number Theory Seminar, 2019-2020

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

** August 29:**

**Speaker: **

**Title: ** Brief Organizational Meeting

**Abstract: **

** 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)