Por João Pedro Ramos (ETH Zürich).

The celebrated sphere packing problem, a cornerstone of metric geometry, asks what is the densest configuration of points in the Euclidean space such that all points lie at distance at least 2 from each other. A solution to this problem was only known in dimensions $n=1,2,3$, until in 2016 Maryna Viazovska was responsible for a breakthrough result which solved the problem for the additional dimensions $8$ and $24$. She proved, moreover, that the only optimal lattices are the $E_8$ and $\Lambda_{24}$ lattices, in dimensions $n=8$ and $24$, respectively. One may then ask whether any lattice close to attaining optimality in the sphere packing problem is suitably close to those structures, and whether anything can be said for more general (non-lattice) packings.

The purpose of this talk is then to discuss the sphere packing problem, to understand the recent breakthrough by Viazvoska, and to answer the questions in the previous paragraph in the affirmative. We will prove, moreover, a sharp stability result for lattice packings in dimensions $8$ and $24$, and, time-permitting, we shall discuss possible improvements of our results and open questions. This is based on joint work with K. Böröczky and D. Radchenko.

The Lisbon Webinar in Analysis in Differential Equations is a joint iniciative of CAMGSD, CMAFcIO and GFM, three research centers of the University of Lisbon. It is aimed at filling the absence of face-to-face seminars and wishes to be a meeting point of mathematicians working in the field.

Transmissão via Zoom (pw: lisbonwade).